*********************************************************************** * * SplittingFunctions.f: * * Collections of the QCD splitting fuctions up to NNLO. * ************************************************************************ * __ * ..The parametrized 3-loop MS non-singlet splitting functions P^(2) * for the evolution of unpolarized partons densities, mu_r = mu_f. * The expansion parameter is alpha_s/(4 pi). * * ..The distributions (in the mathematical sense) are given as in eq. * (B.26) of Floratos, Kounnas, Lacaze: Nucl. Phys. B192 (1981) 417. * The name-endings A, B, and C of the functions below correspond to * the kernel superscripts [2], [3], and [1] in that equation. * * ..The relative accuracy of these parametrizations, as well as of * the convolution results, is better than one part in thousand. * * ..References: S. Moch, J. Vermaseren and A. Vogt, * hep-ph/0209100 = Nucl. Phys. B646 (2002) 181, * hep-ph/0403192 = Nucl. Phys. B688 (2004) 101 * * ===================================================================== * * * ..This is the regular piece of P2_NS+. The rational coefficients are * exact, the rest has been fitted for x between 10^-6 and 1 - 10^-6. * The N_f^2 part is exact and was first determined in N-space by * J.A. Gracey in Phys. Lett. B322 (1994) 141. * FUNCTION P2NSPA (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) D81 = 1./81.D0 * P2NSPA = 1641.1 - 3135.* Y + 243.6 * Y**2 - 522.1 * Y**3 , + 128.*D81 * DL**4 + 2400.*D81 * DL**3 , + 294.9 * DL**2 + 1258.* DL , + 714.1 * DL1 + DL*DL1 * (563.9 + 256.8 * DL) , + NF * ( -197.0 + 381.1 * Y + 72.94 * Y**2 + 44.79 * Y**3 , - 192.*D81 * DL**3 - 2608.*D81 * DL**2 - 152.6 * DL , - 5120.*D81 * DL1 - 56.66 * DL*DL1 - 1.497 * Y*DL**3 ) , + NF**2 * ( 32.* Y*DL/(1.-Y) * (3.* DL + 10.) + 64. , + (48.* DL**2 + 352.* DL + 384.) * (1.-Y) ) * D81 * RETURN END * * --------------------------------------------------------------------- * * * ..This is the regular piece of P2_NS-. The rational coefficients are * exact, the rest has been fitted for x between 10^-6 and 1 - 10^-6. * The N_f^2 part is exact (and identical to that of P2_NS+). * FUNCTION P2NSMA (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) D81 = 1./81.D0 * P2NSMA = 1860.2 - 3505.* Y + 297.0 * Y**2 - 433.2 * Y**3 , + 116.*D81 * DL**4 + 2880.*D81 * DL**3 , + 399.2 * DL**2 + 1465.2 * DL , + 714.1 * DL1 + DL*DL1 * (684.0 + 251.2 * DL) , + NF * ( -216.62 + 406.5 * Y + 77.89 * Y**2 + 34.76 * Y**3 , - 256.*D81 * DL**3 - 3216.*D81 * DL**2 - 172.69 * DL , - 5120.*D81 * DL1 - 65.43 * DL*DL1 - 1.136 * Y*DL**3 ) , + NF**2 * ( 32.* Y*DL/(1.-Y) * (3.* DL + 10.) + 64. , + (48.* DL**2 + 352.* DL + 384.) * (1.-Y) ) * D81 * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular piece of both P2_NS+ and P2_NS-. It is exact * up to the truncation of the irrational coefficients. * FUNCTION P2NSB (Y, NF) IMPLICIT REAL*8 (A-Z) INTEGER NF * P2NSB = ( 1174.898 - NF * 183.187 - NF**2 * 64./81.D0 ) / (1.-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece of P2_NS+. The coefficients of delta(1-x) * have been partly shifted relative to the exact (truncated) values. * FUNCTION P2NSPC (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL1 = LOG (1.-Y) * P2NSPC = 1174.898 * DL1 + 1295.624 - 0.24 , - NF * ( 183.187 * DL1 + 173.938 - 0.011 ) , + NF**2 * ( - 64./81.D0 * DL1 + 1.13067 ) * RETURN END * * * --------------------------------------------------------------------- * * * ..This is the 'local' piece of P2_NS-. The coefficients of delta(1-x) * have been partly shifted relative to the exact (truncated) values. * FUNCTION P2NSMC (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL1 = LOG (1.-Y) * P2NSMC = 1174.898 * DL1 + 1295.624 - 0.154 , - NF * ( 183.187 * DL1 + 173.938 - 0.005 ) , + NF**2 * ( - 64./81.D0 * DL1 + 1.13067 ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is P2_NSS, the difference of P2_NSV and P2_NS-. * FUNCTION P2NSSA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * D27 = 1./27.D0 DL = LOG (Y) Y1 = 1.- Y DL1 = LOG (Y1) * P2NSSA = Y1* ( 151.49 + 44.51 * Y - 43.12 * Y**2 + 4.820 * Y**3 ) 1 + 40.*D27 * DL**4 - 80.*D27 * DL**3 + 6.892 * DL**2 2 + 178.04 * DL + DL*DL1 * ( - 173.1 + 46.18 * DL ) 4 + Y1*DL1 * ( - 163.9 / Y - 7.208 * Y ) * P2NSSA = NF * P2NSSA * RETURN END * * =================================================================av== * * ..The 1- and 2-loop MS(bar) non-singlet splitting functions P_NS^(2) * for the evolution of unpolarized partons densities, mu_r = mu_f. * The expansion parameter is alpha_s/(4 pi). * * ..The distributions (in the mathematical sense) are given as in eq. * (B.26) of Floratos, Kounnas, Lacaze: Nucl. Phys. B192 (1981) 417. * The name-endings A, B, and C of the functions below correspond to * the kernel superscripts [2], [3], and [1] in that equation. * * ..The code uses the package of Gehrmann and Remiddi for the harmonic * polylogarithms published in hep-ph/0107173 = CPC 141 (2001) 296. * * ===================================================================== * * * ..This is the regular 2-loop piece for P_NS^+. * FUNCTION X1NSPA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X2NSB and X2NSC * COMMON / P1SOFT / A2 * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) DM = 1D0/(1D0-X) * gqq1 = 2d0 * CF * NF * ( ( - 1.1111111111111112d0 - ( 2d0 * lnx ) 1 / 3d0 ) * pqq - ( 4d0 * ( 1d0 - x ) ) / 3d0 ) 2 + 4d0 * CA * CF * ( ( 3.7222222222222223d0 + ( 11d0 * lnx ) 3 / 6d0 + lnx**2 / 2d0 - pi**2 / 6d0 ) * pqq 4 + ( 20d0 * ( 1d0 - x ) ) / 3d0 + lnx * ( 1d0 + x) ) 5 + 4d0 * CF**2 * ( ( ( - 3d0 * lnx ) / 2d0 - 2d0 * ln1mx 6 * lnx ) * pqq - 5d0 * ( 1d0 - x ) - ( lnx**2 * ( 1d0 7 + x ) ) / 2d0 - lnx * ( 1.5d0 + ( 7d0 * x ) / 2d0 ) ) 8 + 4d0 * CF * ( CF - CA / 2d0 ) * ( 2d0 * pqqmx * S2x 9 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2 = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * GQQ1L = DM * A2 * * ...The regular piece of the coefficient function * X1NSPA = GQQ1 - GQQ1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the regular 2-loop piece for P_NS^-. * FUNCTION X1NSMA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X2NSB and X2NSC * COMMON / P1SOFT / A2 * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) DM = 1D0/(1D0-X) * gqq1 = 2d0 * CF * NF * ( ( - 1.1111111111111112d0 - ( 2d0 * lnx ) 1 / 3d0 ) * pqq - ( 4d0 * ( 1d0 - x ) ) / 3d0 ) 2 + 4d0 * CA * CF * ( ( 3.7222222222222223d0 + ( 11d0 * lnx ) 3 / 6d0 + lnx**2 / 2d0 - pi**2 / 6d0 ) * pqq 4 + ( 20d0 * ( 1d0 - x ) ) / 3d0 + lnx * ( 1d0 + x) ) 5 + 4d0 * CF**2 * ( ( ( - 3d0 * lnx ) / 2d0 - 2d0 * ln1mx 6 * lnx ) * pqq - 5d0 * ( 1d0 - x ) - ( lnx**2 * ( 1d0 7 + x ) ) / 2d0 - lnx * ( 1.5d0 + ( 7d0 * x ) / 2d0 ) ) 8 - 4d0 * CF * ( CF - CA / 2d0 ) * ( 2d0 * pqqmx * S2x 9 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2 = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * GQQ1L = DM * A2 * * ...The regular piece of the coefficient function * X1NSMA = GQQ1 - GQQ1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X1NSB (Y) IMPLICIT REAL*8 (A - Z) * COMMON / P1SOFT / A2 * X1NSB = A2/(1D0-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X1NSC (Y, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...The coefficient of delta(1-x) * P1DELT = & - 1D0/3D0*cf*nf & + 3D0/2D0*cf**2 & + 17D0/6D0*ca*cf & + 24D0*zeta3*cf**2 & - 12D0*zeta3*ca*cf & - 8D0/3D0*zeta2*cf*nf & - 12D0*zeta2*cf**2 & + 44D0/3D0*zeta2*ca*cf * * ...The soft (`+'-distribution) part of the splitting function * A2 = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * X1NSC = LOG (1D0-Y) * A2 + P1DELT * RETURN END * * ===================================================================== * * * ..This is the regular 1-loop piece. * FUNCTION X0NSA (X) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0NSA = - 2D0*CF * (1.+ X) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X0NSB (Y) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0NSB = 4D0*CF/(1D0-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X0NSC (Y) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0NSC = 4D0*CF * LOG (1D0-Y) + 3D0*CF * RETURN END * * =================================================================av== * * ..The parametrized 3-loop MS singlet splitting functions P^(2) for * the evolution of unpol. singlet parton densities at mu_r = mu_f. * The expansion parameter is alpha_s/(4 pi). * * ..The distributions (in the mathematical sense) are given as in eq. * (B.27) of Floratos, Kounnas, Lacaze: Nucl. Phys. B192 (1981) 417. * The name-endings A, B, and C of the functions below correspond to * the kernel superscripts [2], [3], and [1] in that equation. * * ..The relative accuracy of these parametrisations, as well as of * the convolution results, is better than one part in thousand. * ..The coefficients of 1/(1-x)_+, (ln x)/x and 1/x are exact (up * to a truncation of irrational coefficients). Furthermore all * coefficients written as fractions (e.g., 160./27.D0) are exact. * The other terms at x < 1 have fitted to the exact results for x * between 10^-6 and 1 - 10^-6. The coefficient of delta(1-x) of * P_gg^(2) have been slightly adjusted using the second moments. * * ..References: S. Moch, J. Vermaseren and A. Vogt, * hep-ph/0403192 = Nucl. Phys. B688 (2004) 101 * A. Vogt, S. Moch and J. Vermaseren, * hep-ph/0404111 = Nucl. Phys. B691 (2004) 129 * * ===================================================================== * * * ..The (regular) pure-singlet splitting functions P_ps^(2). * P_qq^(2) is obtained by adding the non-singlet quantity P_NS^(2)+. * A parametrization of the latter is provided in the file xpns2p.f. FUNCTION P2PSA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2PS1 = - 3584./(27.D0*Y) * DL - 506.0/ Y + 160./27.D0 * DL**4 , - 400./9.D0 * DL**3 + 131.4 * DL**2 - 661.6 * DL , - 5.926 * DL1**3 - 9.751 * DL1**2 - 72.11 * DL1 , + 177.4 + 392.9 * Y - 101.4 * Y**2 - 57.04 * DL*DL1 P2PS2 = 256./(81.*Y) + 32./27.D0 * DL**3 + 17.89 * DL**2 , + 61.75 * DL + 1.778 * DL1**2 + 5.944 * DL1 + 100.1 , - 125.2 * Y + 49.26 * Y**2 - 12.59 * Y**3 , - 1.889 * DL*DL1 * P2PSA = (1.-Y) * NF * ( P2PS1 + NF * P2PS2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The gluon->quark splitting functions P_qg^(2). * FUNCTION P2QGA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2QG1 = - 896./(3.D0*Y) * DL - 1268.3 / Y + 536./27.D0 * DL**4 , - 44./3.D0 * DL**3 + 881.5 * DL**2 + 424.9 * DL , + 100./27.D0 * DL1**4 - 70./9.D0 * DL1**3 , - 120.5 * DL1**2 + 104.42 * DL1 , + 2522. - 3316.* Y + 2126.* Y**2 , + DL*DL1 * (1823. - 25.22 * DL) - 252.5 * Y*DL**3 P2QG2 = 1112./(243.D0*Y) - 16./9.D0 * DL**4 , - 376./27.D0 * DL**3 - 90.8 * DL**2 - 254.0 * DL , + 20./27.D0 * DL1**3 + 200./27.D0 * DL1**2 - 5.496 * DL1 , - 252.0 + 158.0 * Y + 145.4 * Y**2 - 139.28 * Y**3 , - DL*DL1 * ( 53.09 + 80.616 * DL) - 98.07 * Y*DL**2 , + 11.70 * Y*DL**3 * P2QGA = NF * ( P2QG1 + NF * P2QG2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The quark->gluon splitting functions P_gq^(2). P2GQ2 is exact. * FUNCTION P2GQA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2GQ0 = 1189.3 * DL/Y + 6163.1 / Y - 4288./81.D0 * DL**4 , + 1568./9.D0 * DL**3 - 1794. * DL**2 + 4033. * DL , + 400./81.D0 * DL1**4 + 2200./27.D0 * DL1**3 , + 606.3 * DL1**2 + 2193.* DL1 , - 4307. + 489.3 * Y + 1452.* Y**2 + 146.0 * Y**3 , - 447.3 * DL**2*DL1 - 972.9 * Y*DL**2 P2GQ1 = 71.082 * DL/Y - 46.41 / Y + 128./27.D0 * DL**4 , + 704/81.D0 * DL**3 + 20.39 * DL**2 + 174.8 * DL , - 400./81.D0 * DL1**3 - 68.069 * DL1**2 - 296.7 * DL1 , - 183.8 + 33.35 * Y - 277.9 * Y**2 + 108.6 * Y*DL**2 , - 49.68 * DL*DL1 P2GQ2 = ( 64. * ( - 1./Y + 1. + 2.* Y) , + 320.* DL1 * ( 1./Y - 1. + 0.8 * Y) , + 96.* DL1**2 * ( 1./Y - 1. + 0.5 * Y) ) / 27.D0 * P2GQA = ( P2GQ0 + NF * (P2GQ1 + NF * P2GQ2) ) * RETURN END * * --------------------------------------------------------------------- * * * ..The regular piece of the gluon-gluon splitting function P_gg^(2). * FUNCTION P2GGA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2GGA0 = 2675.8 * DL/Y + 14214./ Y - 144. * DL**4 + 72. * DL**3 1 - 7471. * DL**2 + 274.4 * DL + 3589. * DL1 - 20852. 2 + 3968.* Y - 3363. * Y**2 + 4848. * Y**3 3 + DL*DL1 * ( 7305. + 8757. * DL ) P2GGA1 = 157.27 * DL/Y + 182.96 / Y + 512./27.D0 * DL**4 1 + 832./9.D0 * DL**3 + 491.3 * DL**2 + 1541. * DL 2 - 320.0 * DL1 - 350.2 + 755.7 * Y - 713.8 * Y**2 3 + 559.3 * Y**3 + DL*DL1 * ( 26.15 - 808.7 * DL ) P2GGA2 = - 680./(243.D0 * Y) - 32./27.D0 * DL**3 + 9.680 * DL**2 1 - 3.422 * DL - 13.878 + 153.4 * Y - 187.7 * Y**2 2 + 52.75 * Y**3 - DL*DL1 * (115.6 - 85.25* Y + 63.23* DL) * P2GGA = P2GGA0 + NF * ( P2GGA1 + NF * P2GGA2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The singular piece of the gluon-gluon splitting function P_gg^(2). * FUNCTION P2GGB (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * P2GGB = ( 2643.521 - NF * 412.172 - NF**2 * 16./9.D0 ) / ( 1.-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..The 'local' piece of the gluon-gluon splitting function P_gg^(2). * FUNCTION P2GGC (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL1 = LOG (1.-Y) * P2GGC = 2643.521 * DL1 + 4425.448 + 0.446 , - NF * ( 412.172 * DL1 + 528.720 + 0.003 ) , + NF**2 * ( - 16./9.D0 * DL1 + 6.4630) * RETURN END * * =================================================================av== * * ..The 1- and 2-loop MS(bar) singlet splitting functions P_ij^(2) * for the evolution of unpolarized partons densities, mu_r = mu_f. * The expansion parameter is alpha_s/(4 pi). * * ..The distributions (in the mathematical sense) are given as in eq. * (B.26) of Floratos, Kounnas, Lacaze: Nucl. Phys. B192 (1981) 417. * The name-endings A, B, and C of the functions below correspond to * the kernel superscripts [2], [3], and [1] in that equation. * * ..The code uses the package of Gehrmann and Remiddi for the harmonic * polylogarithms published in hep-ph/0107173 = CPC 141 (2001) 296. * * ===================================================================== * * * ..The 2-loop pure-singlet splitting functions P_ps^(1) * FUNCTION X1PSA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * INTEGER NF * * ...some abbreviations * DX = 1D0/X LNX = DLOG(X) HR200 = LNX * LNX / 2D0 * * ...The splitting function in terms of the harmonic polylogs * X1PSA = & + nf*cf * ( - 8D0 + 24D0*x - 224D0/9D0*x**2 + 80D0/9D0* & dx + 4D0*LNX + 20D0*LNX*x + 32D0/3D0*LNX*x**2 - & 8D0*HR200 - 8D0*HR200*x ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop gluon->quark splitting functions P_qg^(1) * FUNCTION X1QGA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqg = x**2 + ( 1d0 - x )**2 pqgmx = x**2 + ( 1d0 + x )**2 S2x = S2(x) * X1QGA = 2d0 * CF * NF * ( 4d0 + 4d0 * ln1mx + ( 10d0 - 4d0 1 * ( ln1mx - lnx ) + 2d0 * ( - ln1mx + lnx )**2 2 - 2d0 * pi**2 / 3d0 ) * pqg - lnx * ( 1d0 - 4d0 * x ) 3 - lnx**2 * ( 1d0 - 2d0 * x ) - 9d0 * x ) 4 + 2d0 * CA * NF * ( 20.22222222222222d0 - 4d0 * ln1mx 5 + ( - 24.22222222222222d0 + 4d0 * ln1mx - 2d0 * ln1mx**2 6 + ( 44d0 * lnx ) /3d0 - lnx**2 + pi**2 / 3d0 ) * pqg 7 + 2d0 * pqgmx * S2x + 40d0 / ( 9d0 * x ) 8 + ( 14d0 * x ) / 9d0 - lnx**2 * ( 2d0 + 8d0 * x ) 9 + lnx * ( - 12.666666666666666d0 + ( 136d0 * x ) / 3d0 ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop quark->gluon splitting functions P_gq^(1) * FUNCTION X1GQA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pgq = ( 1d0 + ( 1d0 - x )**2 ) / x pgqmx = - ( 1d0 + ( 1d0 + x )**2 ) / x S2x = S2(x) * X1GQA = 2d0 * CF * NF * ( - ( ( 2.2222222222222223d0 1 + ( 4d0 * ln1mx ) / 3d0 ) * pgq ) - ( 4d0 * x ) / 3d0 ) 2 + 4d0 * CF**2 * ( - 2.5d0 - ( 3d0 * ln1mx + ln1mx**2 ) 3 * pgq - lnx**2 * ( 1d0 - x / 2d0 ) - ( 7d0 * x ) / 2d0 4 - 2d0 * ln1mx * x + lnx * ( 2d0 + ( 7d0 * x ) / 2d0 ) ) 5 + 4d0 * CA * CF * ( 3.111111111111111d0 + pgq * ( 0.5 d0 6 + ( 11d0 * ln1mx ) / 3d0 + ln1mx**2 - 2d0 * ln1mx * lnx 7 + lnx**2 / 2d0 - pi**2 / 6d0 ) + pgqmx * S2x 8 + ( 65d0 * x ) / 18d0 + 2d0 * ln1mx * x + ( 44d0 * x**2 ) 9 / 9d0 + lnx**2 * ( 4d0 + x ) - lnx * ( 12d0 + 5d0 * x 1 + ( 8d0 * x**2 ) / 3d0 ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..The regular piece of the 2-loop gg splitting function P_gg^(1) * FUNCTION X1GGA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X1GGB and X1GGC * COMMON / P1GSOFT / A2G * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pgg = ( 1d0 / ( 1d0 - x ) + 1d0 / x - 2d0 + x * ( 1d0 - x ) ) pggmx = ( 1d0 / ( 1d0 + x ) - 1d0 / x - 2d0 - x * ( 1d0 + x ) ) S2x = S2(x) DM = 1D0/(1D0-X) * ggg1 = 2d0 * CF * NF * ( - 16d0 + 4d0 / ( 3d0 * x ) + 8d0 * x 1 + ( 20d0 * x**2 ) / 3d0 - lnx**2 * ( 2d0 + 2d0 * x ) 2 - lnx * ( 6d0 + 10d0 * x ) ) 3 + 2d0 * CA * NF * ( 2d0 - ( 20d0 * pgg ) / 9d0 - 2d0 * x 4 - ( 4d0 * lnx * ( 1d0 + x ) ) / 3d0 + ( 26d0 * ( 5 - ( 1d0 / x ) + x**2 ) ) / 9d0 ) 6 + 4d0 * CA**2 * ( pgg * ( 7.444444444444445d0 7 - 4d0 * ln1mx * lnx + lnx**2 - pi**2 / 3d0 ) 8 + 2d0 * pggmx * S2x + ( 27d0 * ( 1d0 - x ) ) / 2d0 9 + 4d0 * lnx**2 * ( 1d0 + x ) + ( 67d0 * ( - ( 1d0 / x ) 1 + x**2 ) ) / 9d0 - lnx * ( 8.333333333333334d0 2 - ( 11d0 * x ) / 3d0 + ( 44d0 * x**2 ) / 3d0 ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2G = - 40D0/9D0*ca*nf + 268D0/9D0*ca**2 - 8D0*zeta2*ca**2 * GGG1L = DM * A2G * * ...The regular piece of the coefficient function * X1GGA = GGG1 - GGG1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X1GGB (Y) IMPLICIT REAL*8 (A - Z) * COMMON / P1GSOFT / A2G * X1GGB = A2G/(1D0-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X1GGC (Y, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...The coefficient of delta(1-x) * P1DELT = , - 2D0*cf*nf , - 8D0/3D0*ca*nf , + 32D0/3D0*ca**2 , + 12D0*zeta3*ca**2 * * ...The soft (`+'-distribution) part of the splitting function * A2G = - 40D0/9D0*ca*nf + 268D0/9D0*ca**2 - 8D0*zeta2*ca**2 * X1GGC = DLOG (1D0-Y) * A2G + P1DELT * RETURN END * * ===================================================================== * * * ..The 1-loop gluon->quark splitting functions P_qg^(0) * FUNCTION X0QGA (X, NF) * IMPLICIT REAL*8 (A - Z) INTEGER NF * X0QGA = 2D0 * NF * ( 1. - 2. * X + 2. * X**2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 1-loop quark->gluon splitting functions P_gq^(0) * FUNCTION X0GQA (X) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0GQA = 4D0 * CF * ( - 1. + 0.5 * X + 1D0 / X ) * RETURN END * * --------------------------------------------------------------------- * * * ..The regular piece of the 1-loop gg splitting function P_gg^(0) * FUNCTION X0GGA (X) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0GGA = 4D0*CA * ( - 2. + X - X**2 + 1D0/X ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X0GGB (X) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0GGB = 4D0 * CA / ( 1D0 - X ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X0GGC (X,NF) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * INTEGER NF * X0GGC = 4D0*CA * LOG (1D0-X) - 2D0/3D0 * NF + 11D0/3D0 * CA * RETURN END * * =================================================================av== function S2(x) * implicit none * include "../commons/consts.h" ** * Input Variables * double precision x ** * Internal Variables * double precision lnx double precision ddilog!,wgplg ** * Output Variables * double precision S2 * lnx = dlog(x) * S2 = - 2d0 * ddilog(-x) + lnx**2 / 2d0 1 - 2d0 * lnx * log(1d0+x) - pi**2 / 6d0 c S2 = - 2d0 * wgplg(1,1,-x) + lnx**2 / 2d0 c 1 - 2d0 * lnx * log(1d0+x) - pi**2 / 6d0 * return end * ************************************************************************ * * Time-like splitting functions for the fragmentation function * evolution. * Notice that the one-loop time-like splitting functions coincide * with the space-like ones. * *********************************************************************** * * ..This is the regular 2-loop piece for P_NS^+. * FUNCTION X1NSPTA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X2NSBT and X2NSCT * COMMON / P1SOFTT / A2T * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) DM = 1D0/(1D0-X) * gqq1 = 2d0 * CF * NF * ( ( - 1.1111111111111112d0 - ( 2d0 * lnx ) 1 / 3d0 ) * pqq - ( 4d0 * ( 1d0 - x ) ) / 3d0 ) 2 + 4d0 * CA * CF * ( ( 3.7222222222222223d0 + ( 11d0 * lnx ) 3 / 6d0 + lnx**2 / 2d0 - pi**2 / 6d0 ) * pqq 4 + ( 20d0 * ( 1d0 - x ) ) / 3d0 + lnx * ( 1d0 + x ) ) 5 + 4d0 * CF**2 * ( ( ( 3d0 * lnx ) / 2d0 + 2d0 * ln1mx 6 * lnx - 2d0 * lnx**2 ) * pqq - 5d0 * ( 1d0 - x ) 7 + ( lnx**2 * ( 1d0 + x ) ) / 2d0 8 - lnx * ( 3.5d0 + ( 3d0 * x ) / 2d0 ) ) 9 + 4d0 * CF * ( CF - CA / 2d0 ) * ( 2d0 * pqqmx * S2x 1 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2T = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * GQQ1L = DM * A2T * * ...The regular piece of the coefficient function * X1NSPTA = GQQ1 - GQQ1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the regular 2-loop piece for P_NS^-. * FUNCTION X1NSMTA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X2NSBT and X2NSCT * COMMON / P1SOFTT / A2T * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) DM = 1D0/(1D0-X) * gqq1 = 2d0 * CF * NF * ( ( - 1.1111111111111112d0 - ( 2d0 * lnx ) 1 / 3d0 ) * pqq - ( 4d0 * ( 1d0 - x ) ) / 3d0 ) 2 + 4d0 * CA * CF * ( ( 3.7222222222222223d0 + ( 11d0 * lnx ) 3 / 6d0 + lnx**2 / 2d0 - pi**2 / 6d0 ) * pqq 4 + ( 20d0 * ( 1d0 - x ) ) / 3d0 + lnx * ( 1d0 + x) ) 5 + 4d0 * CF**2 * ( ( ( 3d0 * lnx ) / 2d0 + 2d0 * ln1mx 6 * lnx - 2d0 * lnx**2 ) * pqq - 5d0 * ( 1d0 - x ) 7 + ( lnx**2 * ( 1d0 + x ) ) / 2d0 8 - lnx * ( 3.5d0 + ( 3d0 * x ) / 2d0 ) ) 9 - 4d0 * CF * ( CF - CA / 2d0 ) * ( 2d0 * pqqmx * S2x 1 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2T = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * GQQ1L = DM * A2T * * ...The regular piece of the coefficient function * X1NSMTA = GQQ1 - GQQ1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X1NSTB (Y) IMPLICIT REAL*8 (A - Z) * COMMON / P1SOFTT / A2T * X1NSTB = A2T/(1D0-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X1NSTC (Y, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...The coefficient of delta(1-x) * P1DELT = & - 1D0/3D0*cf*nf & + 3D0/2D0*cf**2 & + 17D0/6D0*ca*cf & + 24D0*zeta3*cf**2 & - 12D0*zeta3*ca*cf & - 8D0/3D0*zeta2*cf*nf & - 12D0*zeta2*cf**2 & + 44D0/3D0*zeta2*ca*cf * * ...The soft (`+'-distribution) part of the splitting function * A2T = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * X1NSTC = LOG (1D0-Y) * A2T + P1DELT * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop pure-singlet time-like splitting functions P_ps^(1) * FUNCTION X1PSTA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * INTEGER NF * * ...some abbreviations * DX = 1D0/X LNX = DLOG(X) HR200 = LNX * LNX / 2D0 * * ...The splitting function in terms of the harmonic polylogs * X1PSTA = & + nf*cf * ( - 32D0 + 16D0*x + 224D0/9D0*x**2 - 80D0/9D0* & dx - 20D0*LNX - 36D0*LNX*x - 32D0/3D0*LNX*x**2 + & 8D0*HR200 + 8D0*HR200*x ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop gluon->quark time-like splitting functions P_qg^(1) * FUNCTION X1QGTA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqg = x**2 + ( 1d0 - x )**2 pqgmx = x**2 + ( 1d0 + x )**2 S1x = - ddilog(1d0-x) S2x = S2(x) * X1QGTA = ( NF**2 * ( - 8d0 / 3d0 - ( 16d0 / 9d0 1 + 8d0 * lnx / 3d0 + 8d0 * ln1mx / 3d0 ) * pqg ) 2 + 2d0 * CF * NF * ( - 2d0 + 3d0 * x 3 + ( - 7d0 + 8d0 * x ) * lnx - 4d0 * ln1mx 4 + ( 1d0 - 2d0 * x ) * lnx**2 5 + ( - 2d0 * ( lnx + ln1mx )**2 - 2d0 * ( ln1mx - lnx ) 6 + 16d0 * S1x + 2d0 * pi**2 - 10d0 ) * pqg ) 7 + 2d0 * CA * NF * ( - 152d0 / 9d0 + 166d0 * x / 9d0 8 - 40d0 / 9d0 / x + ( - 4d0 / 3d0 - 76d0 * x / 3d0 ) * lnx 9 + 4d0 * ln1mx + ( 2d0 + 8d0 * x ) * lnx**2 1 + ( 8d0 * lnx * ln1mx - lnx**2 - 4d0 * lnx / 3d0 2 + 10d0 * ln1mx / 3d0 + 2d0 * ln1mx**2 - 16d0 * S1x 3 - 7d0 * pi**2 / 3d0 + 178d0 / 9d0 ) * pqg 4 + 2d0 * pqgmx * S2x ) ) / 2d0 / NF * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop quark->gluon time-like splitting functions P_gq^(1) * FUNCTION X1GQTA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pgq = ( 1d0 + ( 1d0 - x )**2 ) / x pgqmx = - ( 1d0 + ( 1d0 + x )**2 ) / x S1x = - ddilog(1d0-x) S2x = S2(x) * X1GQTA = 2d0 * NF * ( 4d0 * CF**2 * ( - 1d0 / 2d0 + 9d0 * x 1 / 2d0 + ( - 8d0 + x / 2d0 ) * lnx + 2d0 * x * ln1mx 2 + ( 1d0 - x / 2d0 ) * lnx**2 + ( ln1mx**2 3 + 4d0 * lnx * ln1mx - 8d0 * S1x 4 - 4d0 * pi**2 / 3d0 ) * pgq ) 5 + 4d0 * CF * CA * ( 62d0 / 9d0 - 35d0 * x / 18d0 6 - 44d0 * x**2 / 9d0 7 + ( 2d0 + 12d0 * x + 8d0 * x**2 / 3d0 ) * lnx 8 - 2d0 * x * ln1mx - ( 4d0 + x ) * lnx**2 + pgqmx * S2x 9 + ( - 2d0 * lnx * ln1mx - 3d0 * lnx - 3d0 * lnx**2 / 2d0 1 - ln1mx**2 + 8d0 * S1x + 7d0 * pi**2 / 6d0 2 + 17d0 / 18d0 ) * pgq ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..The regular piece of the 2-loop gg time-like splitting function P_gg^(1) * FUNCTION X1GGTA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X1GGB and X1GGC * COMMON / P1GSOFTT / A2GT * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pgg = ( 1d0 / ( 1d0 - x ) + 1d0 / x - 2d0 + x * ( 1d0 - x ) ) pggmx = ( 1d0 / ( 1d0 + x ) - 1d0 / x - 2d0 - x * ( 1d0 + x ) ) S2x = S2(x) DM = 1D0/(1D0-X) * ggg1 = 2d0 * CF * NF * ( - 4d0 + 12d0 * x - 164d0 * x**2 / 9d0 1 + ( 10d0 + 14d0 * x + 16d0 * x**2 / 3d0 + 16d0 / 3d0 / x ) 2 * lnx + 92d0 / 9d0 / x + 2d0 * ( 1d0 + x ) * lnx**2 ) 3 + 2d0 * CA * NF * ( 2d0 - 2d0 * x 4 + 26d0 * ( x**2 - 1d0 / x ) / 9d0 5 - 4d0 * ( 1d0 + x ) * lnx / 3d0 6 - ( 20d0 / 9d0 + 8d0 * lnx / 3d0 ) * pgg ) 7 + 4d0 * CA * CA * ( 27d0 * ( 1d0 - x ) / 2d0 8 + 67d0 * ( x**2 - 1d0 / x ) / 9d0 9 + ( 11d0 / 3d0 - 25d0 * x / 3d0 - 44d0 / 3d0 / x ) * lnx 1 - 4d0 * ( 1d0 + x ) * lnx**2 2 + ( 4d0 * lnx * ln1mx - 3d0 * lnx**2 + 22d0 * lnx / 3d0 3 - 2d0 * zeta2 + 67d0 / 9d0 ) * pgg + 2d0 * pggmx * S2x ) * * ...The soft (`+'-distribution) part of the splitting function * A2GT = - 40D0/9D0*ca*nf + 268D0/9D0*ca**2 - 8D0*zeta2*ca**2 * GGG1L = DM * A2GT * * ...The regular piece of the coefficient function * X1GGTA = GGG1 - GGG1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X1GGTB (Y) IMPLICIT REAL*8 (A - Z) * COMMON / P1GSOFTT / A2GT * X1GGTB = A2GT/(1D0-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X1GGTC (Y, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...The coefficient of delta(1-x) * P1DELT = , - 2D0*cf*nf , - 8D0/3D0*ca*nf , + 32D0/3D0*ca**2 , + 12D0*zeta3*ca**2 * * ...The soft (`+'-distribution) part of the splitting function * A2GT = - 40D0/9D0*ca*nf + 268D0/9D0*ca**2 - 8D0*zeta2*ca**2 * X1GGTC = DLOG (1D0-Y) * A2GT + P1DELT * RETURN END * * ===================================================================== * * * ..File: xpij2pt.f * * __ * ..The parametrized 3-loop MS singlet splitting functions P^(2)T * for the evolution of unpolarized singlet fragmentation densities * at mu_r = mu_f. The expansion parameter is alpha_s/(4 pi). * ..The distributions (in the mathematical sense) are given as in eq. * (B.27) of Floratos, Kounnas, Lacaze: Nucl. Phys. B192 (1981) 417. * The name-endings A, B, and C of the functions below correspond to * the kernel superscripts [2], [3], and [1] in that equation. * * ..The relative accuracy of these parametrisations, as well as of * the convolution results, is better than one part in thousand. * ..The coefficients of 1/(1-x)_+, (ln x)/x and 1/x are exact (up * to a truncation of irrational coefficients). Furthermore all * coefficients written as fractions (e.g., 160./27.D0) are exact. * The other terms at x < 1 have fitted to the exact results for x * between 10^-6 and 1 - 10^-6. The coefficient of delta(1-x) of * P_gg^(2) have been very slightly adjusted using the low moments. * * ..References: S. Moch and A. Vogt, * Phys. Lett. B659 (2008) 290, arXiv:0708.3899 * A. Almasy, S. Moch and A. Vogt, * arXiv:1107.nnnn * * ===================================================================== * * ..The (regular) pure-singlet splitting functions P_ps^(2)T. * P_qq^(2)T is obtained by adding the non-singlet quantity P_NS+^(2)T * A parametrization of the latter is provided in the file xpns2pt.f. * FUNCTION P2PSTA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2PST1 = - 256./(9.D0*Y)* DL**3 - 128./(9.D0*Y)* DL**2 & + 324.07/Y* DL + 479.87/Y & - 5.926* DL1**3 - 9.751* DL1**2 - 8.65* DL1 - 106.65 & - 848.97* Y + 368.79* Y**2 - 61.284* Y**3 & + 96.171* DL*DL1 + 656.49* DL + 425.14* DL**2 & + 47.322* DL**3 + 9.072* DL**4 P2PST2 = - 128./(81.D0*Y) + 1.778* DL1**2 + 16.611* DL1 + 87.795 & - 57.688* Y - 41.827* Y**2 + 25.628* Y**3 - 7.9934* Y**4 & - 2.1031* DL*DL1 + 57.713* DL + 9.1682* DL**2 & - 1.9* DL**3 + 0.019122* DL**4 & + 26.294* Y*DL - 7.8645* Y*DL**3 * P2PSTA = (1.-Y) * NF * ( P2PST1 + NF * P2PST2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The quark-gluon splitting functions P_qg^(2)T. * The nf^3 part is exact up to a truncation of zeta_2 * FUNCTION P2QGTA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2QG1 = - 64./Y* DL**3 - 64./Y* DL**2 + 675.83/Y* DL + 1141.7/Y & + 100./27.D0* DL1**4 + 350./9.D0* DL1**3 & + 263.07* DL1**2 + 693.84* DL1 + 603.71 & - 882.48* Y + 4723.2* Y**2 - 4745.8* Y**3 - 175.28* Y**4 & + 1864.* DL + 1512.* DL**2 + 361.28* DL**3 & + 42.328* DL**4 - 1809.4* DL*DL1 - 107.59* Y*DL*DL1 & - 885.5* Y*DL**4 P2QG2 = - 32./(27.D0*Y)* DL**2 - 3.1752/Y* DL - 2.8986/Y & - 100./27.D0* DL1**3 - 35.446* DL1**2 - 103.609* DL1 & - 113.81 + 341.26* Y - 853.35* Y**2 + 492.1* Y**3 & + 14.803* Y**4 + 619.75* DL + 255.62* DL**2 & + 21.569* DL**3 + 966.96* DL*DL1 - 1.593*DL*DL1**2 & - 333.8* Y*DL**3 - 709.1* Y*DL*DL1 P2QG3 = 4./9.D0* (4. + 6.* (DL + DL1) & + (1. - 2.*Y + 2.*Y**2) * ( 3.8696 + 4.* (DL + DL1) & + 3.* (DL + DL1)**2 ) ) * P2QGTA = ( P2QG1 + NF * P2QG2 + NF**2 * P2QG3 ) / 2D0 * RETURN END * * --------------------------------------------------------------------- * * * ..The gluon-quark splitting functions P_gq^(2)T * FUNCTION P2GQTA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2GQ0 = 400./81.D0* DL1**4 + 520./27.D0* DL1**3 & - 220.13* DL1**2 - 152.60* DL1 + 272.85 - 7188.7* Y & + 5693.2* Y**2 + 146.98* Y**3 + 128.19* Y**4 & - 30.062* DL**4 - 126.38* DL**3 - 0.71252* DL**2 & + 4.4136* DL - 1300.6* DL*DL1 - 71.23* DL*DL1**2 & + 543.8* Y*DL**3 & + 256./Y* DL**4 + 3712./(3.D0*Y)* DL**3 & + 1001.89/Y* DL**2 + 4776.5/Y* DL + 5803.7/Y P2GQ1 = 80./81.D0* DL1**3 + 1040./81.D0* DL1**2 - 16.914* DL1 & - 871.3 + 790.13* Y - 241.23* Y**2 + 43.252* Y**3 & - 48.600 * DL**3 - 343.1* DL**2 - 492.* DL & + 55.048* DL*DL1 - 4.3465* Y*DL**3 & + 6.0041/Y + 141.93/Y* DL & + 2912./(27.D0*Y)* DL**2 + 1280./(81.D0*Y)*DL**3 * P2GQTA = 2D0 * NF * ( P2GQ0 + NF * P2GQ1 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The regular piece of the gluon-gluon splitting function P_gg^(2) * FUNCTION P2GGTA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) * P2GGTA0 = 576./Y* DL**4 + 3168./Y* DL**3 + 3651.1/Y* DL**2 & + 10233./Y* DL + 14214.4/Y - 3590.1* DL1 - 28489. & + 7469.* Y + 30421.* Y**2 - 53017.* Y**3 + 19556.* Y**4 & + 191.99* DL**4 + 3281.7* DL**3 + 13528.* DL**2 & + 12258.* DL - 186.4* DL*DL1 - 21328.* DL**2*DL1 & + 5685.8* Y*DL**3 P2GGTA1 = + 448./(9.D0*Y)* DL**3 + 2368./(9.D0*Y)* DL**2 & - 5.47/Y* DL - 804.13/Y + 248.95 + 319.97* DL1 & + 260.6* Y + 272.79* Y**2 + 2133.2* Y**3 - 926.87* Y**4 & + 4.9934* DL + 482.94* DL**2 + 155.10* DL**3 & + 18.085* DL**4 + 485.18* Y*DL**3 + 1266.5* DL*DL1 & - 29.709* DL**2*DL1 + 87.771* DL*DL1**2 P2GGTA2 = 32./(27.D0*Y)* DL**2 + 368./(81.D0*Y)* DL & + 472./(243.D0*Y) & - 77.190 + 153.27* Y - 106.03* Y**2 + 11.995* Y**3 & - 5.0372* DL**3 - 44.8* DL**2 - 69.712* DL & - 115.01* DL*DL1 + 96.522* Y*DL*DL1 - 62.908* DL**2*DL1 * P2GGTA = P2GGTA0 + NF * ( P2GGTA1 + NF * P2GGTA2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The singular piece of the gluon-gluon splitting function P_gg^(2)T * (identical to the spacelike case) * FUNCTION P2GGTB (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * P2GGTB = (2643.521 - NF * 412.172 - NF**2 * 16./9.D0) / ( 1.-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..The 'local' piece of the gluon-gluon splitting function P_gg^(2) * (as in the spacelike case, up to the smaller delta(1-x) adjustment) * FUNCTION P2GGTC (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * DL1 = LOG (1.-Y) * P2GGTC = 2643.521 * DL1 + 4425.448 + 0.003 , - NF * ( 412.172 * DL1 + 528.720 - 0.001 ) , + NF**2 * ( - 16./9.D0 * DL1 + 6.4630 - 0.0002) * RETURN END * * * =================================================================aaa= * ..File: xpns2tp.f * * __ * ..The parametrized 3-loop MS non-singlet splitting functions P^(2)T * for the evolution of unpolarized fragmentation densities at * mu_r = mu_f. The expansion parameter is alpha_s/(4 pi). * * ..The distributions (in the mathematical sense) are given as in eq. * (B.26) of Floratos, Kounnas, Lacaze: Nucl. Phys. B192 (1981) 417. * The name-endings A, B, and C of the functions below correspond to * the kernel superscripts [2], [3], and [1] in that equation. * * ..The relative accuracy of these parametrizations, as well as of * the convolution results, is better than one part in thousand. * * ..References: A. Mitov, S. Moch and A. Vogt, * Phys. Lett. B638 (2006) 61, hep-ph/0604053 (exact res.) * A. Almasy, S. Moch and A. Vogt, * arXiv:1107.nnnn (this code) * * ..Some (parts) of these functions are identical to the spacelike * results of S. Moch, J. Vermaseren and A. Vogt, * Nucl. Phys. B688 (2004) 101, hep-ph/040319 * * ===================================================================== * * * ..This is the regular piece of P2_NS+. The rational coefficients are * exact, the rest has been fitted for x between 10^-6 and 1 - 10^-6. * The N_f^2 part is exact is identical to the spacelike case. * FUNCTION P2NSPTA (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) D81 = 1./81.D0 * P2NSPTA = 1658.7 - 707.67* DL1 + 1327.5* DL - 56.907* DL*DL1 & - 189.37* DL**2 - 519.37* DL1*DL**2 - 352./9.D0* DL**3 & + 128./81.D0* DL**4 - 4249.4* Y - 559.1* DL1*DL*Y & - 1075.3* Y**2 + 593.9* Y**3 & + NF * (64./27.D0* DL**3 - 176./81.D0* DL**2 - 168.89* DL & - 198.10 + 466.29* Y + 181.18* Y**2 - 31.84* Y**3 & + 5120./81.D0* DL1 - 50.758* DL*DL1 + 28.551* DL**2*DL1 & - 39.113* Y*DL + 85.72* Y*DL*DL1 - 23.102* Y*DL**2*DL1) & + NF**2 * ( 32.* Y*DL/(1.-Y) * (3.* DL + 10.) + 64. & + (48.* DL**2 + 352.* DL + 384.) * (1.-Y) ) * D81 * RETURN END * * --------------------------------------------------------------------- * * * ..This is the regular piece of P2_NS-. The rational coefficients are * exact, the rest has been fitted for x between 10^-6 and 1 - 10^-6. * The N_f^2 part is exact (and identical to that of P2_NS+). * FUNCTION P2NSMTA (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) D81 = 1./81.D0 * P2NSMTA = - 140./81.D0* DL**4 - 1024./27.D0* DL**3 & - 38.298* DL**2 + 1625.5* DL - 707.94* DL1 + 1981.3 & - 4885.7* Y - 577.42* Y**2 + 407.89* Y**3 & + 1905.4* DL**2*DL1 + 1969.5* Y*DL**2*DL1 & + 4563.2* DL*DL1 - 34.683* Y*DL**4 & - 5140.6* Y*DL*DL1 - 437.03* Y*DL**3 & + NF * ( 128./81.D0* DL**3 - 784./81.D0* DL**2 & - 188.99* DL - 217.84 + 511.92* Y + 209.19* Y**2 & - 85.786* Y**3 + 5120./81.D0* DL1 + 71.428* DL*DL1 & + 30.554* DL**2*DL1 + 92.453* Y*DL - 23.722* Y*DL*DL1 & - 18.975* Y*DL**2*DL1 ) , + NF**2 * ( 32.* Y*DL/(1.-Y) * (3.* DL + 10.) + 64. , + (48.* DL**2 + 352.* DL + 384.) * (1.-Y) ) * D81 * RETURN END * * * --------------------------------------------------------------------- * * * ..This is the singular piece of both P2_NS+ and P2_NS-. It is exact * up to the truncation of the irrational coefficients, and identical * to the spacelike result. * FUNCTION P2NSTB (Y, NF) IMPLICIT REAL*8 (A-Z) INTEGER NF * P2NSTB = ( 1174.898 - NF * 183.187 - NF**2 * 64./81.D0 ) / (1.-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece of P2_NS+. Coefficients of delta(1-x) have * been shifted (minimally) relative to the exact (truncated) values. * FUNCTION P2NSPTC (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL1 = LOG (1.-Y) * P2NSPTC = 1174.898 * DL1 + 1295.624 + 0.001 , - NF * ( 183.187 * DL1 + 173.938 - 0.003) , + NF**2 * ( - 64./81.D0 * DL1 + 1.13067 ) * RETURN END * * * --------------------------------------------------------------------- * * ..This is the 'local' piece of P2_NS-. Coefficients of delta(1-x) have * been shifted (minimally) relative to the exact (truncated) values. * FUNCTION P2NSMTC (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL1 = LOG (1.-Y) * P2NSMTC = 1174.898 * DL1 + 1295.624 - 0.002 , - NF * ( 183.187 * DL1 + 173.938 - 0.0004 ) , + NF**2 * ( - 64./81.D0 * DL1 + 1.13067 ) * RETURN END * * * --------------------------------------------------------------------- * * * ..This is P2_NSS, the difference of P2_NSV and P2_NS-. It is identical * to the spacelike result. * FUNCTION P2NSSTA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * D27 = 1./27.D0 DL = LOG (Y) Y1 = 1.- Y DL1 = LOG (Y1) * P2NSSA = Y1* ( 151.49 + 44.51 * Y - 43.12 * Y**2 + 4.820 * Y**3 ) 1 + 40.*D27 * DL**4 - 80.*D27 * DL**3 + 6.892 * DL**2 2 + 178.04 * DL + DL*DL1 * ( - 173.1 + 46.18 * DL ) 4 + Y1*DL1 * ( - 163.9 / Y - 7.208 * Y ) * P2NSSTA = NF * P2NSSA * RETURN END * ************************************************************************ * * Space-like polarized splitting functions. * References: * - Vogt website: http://www.liv.ac.uk/~avogt/split.html * - hep-ph/9603366 * *********************************************************************** * * ..This is the regular 1-loop piece. * FUNCTION X0NSPA (X) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0NSPA = - 2D0 * CF * ( 1D0 + X ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X0NSPB (Y) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0NSPB = 4D0 * CF / ( 1D0 - Y ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X0NSPC (Y) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0NSPC = 4D0 * CF * LOG ( 1D0 - Y ) + 3D0 * CF * RETURN END * * ===================================================================== * * * ..The 1-loop gluon->quark splitting functions P_qg^(0) * FUNCTION X0QGPA (X, NF) * IMPLICIT REAL*8 (A - Z) * INTEGER NF * X0QGPA = 2D0 * NF * ( 2D0 * X - 1D0 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 1-loop quark->gluon splitting functions P_gq^(0) * FUNCTION X0GQPA (X) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0GQPA = 2D0 * CF * ( 2D0 - X ) * RETURN END * * --------------------------------------------------------------------- * * * ..The regular piece of the 1-loop gg splitting function P_gg^(0) * FUNCTION X0GGPA (X) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0GGPA = 4D0 * CA * ( - 2D0 * X + 1D0 ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X0GGPB (X) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * X0GGPB = 4D0 * CA / ( 1D0 - X ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X0GGPC (X,NF) IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * INTEGER NF * X0GGPC = 4D0 * CA * LOG ( 1D0 - X ) 1 - 2D0 / 3D0 * NF + 11D0 / 3D0 * CA * RETURN END * * ===================================================================== * * * ..This is the regular 2-loop piece for P_NS^-. * FUNCTION X1NSMPA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X2NSB and X2NSC * COMMON / P1SOFTP / A2P * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) DM = 1d0 / ( 1d0 - x ) * gqq1 = 2d0 * CF * NF * ( ( - 1.1111111111111112d0 - ( 2d0 * lnx ) 1 / 3d0 ) * pqq - ( 4d0 * ( 1d0 - x ) ) / 3d0 ) 2 + 4d0 * CA * CF * ( ( 3.7222222222222223d0 + ( 11d0 * lnx ) 3 / 6d0 + lnx**2 / 2d0 - pi**2 / 6d0 ) * pqq 4 + ( 20d0 * ( 1d0 - x ) ) / 3d0 + lnx * ( 1d0 + x) ) 5 + 4d0 * CF**2 * ( ( ( - 3d0 * lnx ) / 2d0 - 2d0 * ln1mx 6 * lnx ) * pqq - 5d0 * ( 1d0 - x ) - ( lnx**2 * ( 1d0 7 + x ) ) / 2d0 - lnx * ( 1.5d0 + ( 7d0 * x ) / 2d0 ) ) 8 + 4d0 * CF * ( CF - CA / 2d0 ) * ( 2d0 * pqqmx * S2x 9 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2P = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * GQQ1L = DM * A2P * * ...The regular piece of the coefficient function * X1NSMPA = GQQ1 - GQQ1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the regular 2-loop piece for P_NS^+. * FUNCTION X1NSPPA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X2NSB and X2NSC * COMMON / P1SOFTP / A2P * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) DM = 1d0 / ( 1d0 - x ) * gqq1 = 2d0 * CF * NF * ( ( - 1.1111111111111112d0 - ( 2d0 * lnx ) 1 / 3d0 ) * pqq - ( 4d0 * ( 1d0 - x ) ) / 3d0 ) 2 + 4d0 * CA * CF * ( ( 3.7222222222222223d0 + ( 11d0 * lnx ) 3 / 6d0 + lnx**2 / 2d0 - pi**2 / 6d0 ) * pqq 4 + ( 20d0 * ( 1d0 - x ) ) / 3d0 + lnx * ( 1d0 + x) ) 5 + 4d0 * CF**2 * ( ( ( - 3d0 * lnx ) / 2d0 - 2d0 * ln1mx 6 * lnx ) * pqq - 5d0 * ( 1d0 - x ) - ( lnx**2 * ( 1d0 7 + x ) ) / 2d0 - lnx * ( 1.5d0 + ( 7d0 * x ) / 2d0 ) ) 8 - 4d0 * CF * ( CF - CA / 2d0 ) * ( 2d0 * pqqmx * S2x 9 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2P = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * GQQ1L = DM * A2P * * ...The regular piece of the coefficient function * X1NSPPA = GQQ1 - GQQ1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION X1NSPB (Y) IMPLICIT REAL*8 (A - Z) * COMMON / P1SOFTP / A2P * X1NSPB = A2P / ( 1D0 - Y ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X1NSPC (Y, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...The coefficient of delta(1-x) * P1DELT = & - 1D0/3D0*cf*nf & + 3D0/2D0*cf**2 & + 17D0/6D0*ca*cf & + 24D0*zeta3*cf**2 & - 12D0*zeta3*ca*cf & - 8D0/3D0*zeta2*cf*nf & - 12D0*zeta2*cf**2 & + 44D0/3D0*zeta2*ca*cf * * ...The soft (`+'-distribution) part of the splitting function * A2 = - 40D0/9D0*cf*nf + 268D0/9D0*ca*cf - 8D0*zeta2*ca*cf * X1NSPC = LOG (1D0-Y) * A2 + P1DELT * RETURN END * * --------------------------------------------------------------------- * * ..The 2-loop pure-singlet splitting functions P_ps^(1) * function X1PSPA(x,nf) * implicit none * include "../commons/ColorFactors.h" ** * Input Variables * integer nf double precision x ** * Internal Variables * double precision lnx,lnx2 ** * Output Variables * double precision X1PSPA * * Abbreviation * lnx = dlog(x) lnx2 = lnx * lnx * * ...The splitting function in terms of the harmonic polylogs * X1PSPA = 8d0 * CF * TR * nf * ( ( 1d0 - x ) 1 - ( 1d0 - 3d0 * x ) * lnx - ( 1d0 + x ) * lnx2 ) * return end * * --------------------------------------------------------------------- * * * ..The 2-loop gluon->quark splitting function DeltaP_qg^(1) * FUNCTION X1QGPA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * lnx = dlog(x) ln1mx = dlog(1d0 - x) dpqg = 2d0 * x - 1d0 dpqgmx = - 2d0 * x - 1d0 S2x = S2(x) * X1QGPA = 4d0 * TR * nf * ( CF * ( - 22d0 + 27d0 * x - 9d0 * lnx 1 + 8d0 * ( 1d0 - x ) * ln1mx + dpqg * ( 2d0 * ln1mx**2 3 - 4d0 * ln1mx * lnx + lnx**2 - 4d0 * zeta2 ) ) 3 + CA * ( ( 24d0 - 22d0 * x ) 4 - 8d0 * ( 1d0 - x ) * ln1mx + ( 2d0 + 16d0 * x ) * lnx 5 - 2d0 * ( ln1mx**2 - zeta2 ) * dpqg 6 - ( 2d0 * S2x - 3d0 * lnx**2 ) * dpqgmx ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop quark->gluon splitting function DeltaP_gq^(1) * FUNCTION X1GQPA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) dpgq = 2d0 - x dpgqmx = 2d0 + x S2x = S2(x) * X1GQPA = 4d0 * ( CF * TR * nf * (- 4d0 / 9d0 * ( x + 4d0 ) 1 - 4d0 / 3d0 * dpgq * ln1mx ) 2 + CF * CF * ( - 1d0 / 2d0 - ( 4d0 - x ) * lnx / 2d0 3 - dpgqmx * ln1mx + ( - 4d0 - ln1mx**2 + lnx**2 / 2d0 ) 4 * dpgq ) 5 + CF * CA * ( ( 4d0 - 13d0 * x ) * lnx + ( 10d0 + x ) 6 * ln1mx / 3d0 + ( 41d0 + 35d0 * x ) / 9d0 7 + ( - 2d0 * S2x + 3d0 * lnx**2 ) * dpgqmx / 2d0 8 + ( ln1mx**2 - 2d0 * ln1mx * lnx - zeta2 ) * dpgq ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..2-loop gluon-gluon splitting function DeltaP_gg^(1), regular part * FUNCTION X1GGPA (X, NF) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ..The soft coefficient for use in X1GGB and X1GGC * COMMON / P1GSOFTP / A2GP * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) dpgg = 1d0 / ( 1d0 - x ) - 2d0 * x + 1d0 dpggmx = 1d0 / ( 1d0 + x ) + 2d0 * x + 1d0 S2x = S2(x) DM = 1d0 / ( 1d0 - x ) * ggg1 = 4d0 * ( - CA * TR * nf * ( 4d0 * ( 1d0 - x ) 1 + 4d0 / 3d0 * ( 1d0 + x ) * lnx + 20d0 / 9d0 * dpgg ) 2 - CF * TR * nf * ( 10d0 * ( 1d0 - x ) 3 + 2d0 * ( 5d0 - x ) * lnx + 2d0 * ( 1d0 + x ) * lnx**2 ) 4 + CA * CA * ( ( 29d0 - 67d0 * x ) * lnx / 3d0 5 - 19d0 * ( 1d0 - x ) / 2d0 + 4d0 * ( 1d0 + x ) * lnx**2 4 - 2d0 * S2x * dpggmx + ( 67d0 / 9d0 - 4d0 * ln1mx * lnx 5 + lnx**2 - 2d0 * zeta2 ) * dpgg ) ) * * ...The soft (`+'-distribution) part of the splitting function * A2GP = - 80d0 / 9d0 * CA * TR * nf + 268d0 / 9d0 * CA**2 1 - 8d0 * zeta2 * CA**2 * GGG1L = DM * A2GP * * ...The regular piece of the coefficient function * X1GGPA = GGG1 - GGG1L * RETURN END * * --------------------------------------------------------------------- * * ..The singular (soft) piece of DeltaP_gg^(1) * FUNCTION X1GGPB (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * COMMON / P1GSOFTP / A2GP * X1GGPB = A2GP/(1.D0-Y) * RETURN END * * --------------------------------------------------------------------- * * ..The 'local' piece of DeltaP_gg^(1) * FUNCTION X1GGPC (Y, NF) IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * INTEGER NF * * ...The coefficient of delta(1-x) * P1GDELT = , - 4d0 * CF * TR * nf , - 16d0 / 3d0 * CA * TR * nf , + 32d0 / 3d0 * CA**2 , + 12d0 * zeta3 * CA**2 * * ...The soft (`+'-distribution) part of the splitting function * A2GP = - 80d0 / 9d0 * CA * TR * nf + 268d0 / 9d0 * CA**2 1 - 8d0 * zeta2 * CA**2 * X1GGPC = DLOG (1d0-Y) * A2GP + P1GDELT * RETURN END * * ===================================================================== * * * ..The pure-singlet splitting function DeltaP_ps^(2). * The quark-quark splitting function DeltaP_qq^(2) is obtained by * adding the unpolarized quantity P_ns^(2)- given in hep-ph/0403192. FUNCTION P2PSPA (x, nf) * IMPLICIT REAL*8 (A-Z) INTEGER nf * L = log (x) L1 = log (1.d0-x) * P2ps1 = - 344./27.d0 * L**4 - (90.9198 + 81.50* x)* L**3 , - (368.6 - 349.9* x)* L*L - (739.0 - 232.57* L1)* L , - 1362.6 + 1617.4 * x - 674.8 * x*x + 167.41 * x**3 , - 204.76 * L1 - 12.61 * L1*L1 - 6.541 * L1**3 P2ps2 = (1.1741 - 0.8253* x)* L**3 + (13.287 + 10.657* x)* L*L , + 45.482 * L + 49.13 - 30.77 * x - 4.307 * x*x , - 0.5094 *x**3 + 9.517 * L1 + 1.7805 * L1*L1 * P2PSPA = (1.-x) * nf * ( P2ps1 + nf * P2ps2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The gluon->quark splitting function DeltaP_qg^(2). * FUNCTION P2QGPA (x, nf) * IMPLICIT REAL*8 (A-Z) INTEGER nf * L = log (x) L1 = log (1.d0-x) * P2qg1 = - 151./3.d0 * L**4 - (385.64 + 73.30* x)* L**3 , - (894.8 - 1145.3* x)* L*L - (1461.2 - 825.4* L1)* L , - 2972.4 + 4672.* x - 1221.6 * x*x - 18.0 * x**3 , + 278.32* L1 - 90.26* L1*L1 - 5.30* L1**3 + 3.784*L1**4 P2qg2 = 16./9.d0 * L**4 + (30.739 + 10.186* x) * L**3 , + (196.96 + 179.1* x)* L*L + (526.3 - 47.30* L1)* L , + 499.65 - 432.18 * x - 141.63 * x*x - 11.34 * x**3 , - 6.256 * L1 + 7.32 * L1*L1 + 0.7374 * L1**3 * P2QGPA = nf * ( P2qg1 + nf * P2qg2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The quark->gluon splitting functions DeltaP_gq^(2). P2gq2 is exact. * FUNCTION P2GQPA (x, nf) * IMPLICIT REAL*8 (A-Z) INTEGER nf * L = log (x) L1 = log (1.d0-x) * P2gq0 = 11512./81.d0* L**4 + (888.003 + 175.1* x)* L**3 , + (2140. - 850.7* x)* L*L + (4046.6 - 1424.8* L1)* L , + 6159. - 3825.9 * x + 1942.* x*x - 742.1 * x**3 , + 1843.7* L1 + 451.55* L1*L1 + 59.3* L1**3 + 5.143* L1**4 P2gq1 = - 128./27.d0 * L**4 - (39.3872 + 30.023*x)* L**3 , - (202.46 + 126.53* x)* L*L - (308.98 + 16.18* L1)* L , - 301.07 - 296.0 * x + 406.13 * x*x - 101.62 * x**3 , - 171.78* L1 - 47.86 * L1*L1 - 4.963 * L1**3 P2gq2 = 16./27.d0 * ( - 12. + 10.* x + ( 8.+ 2.*x)* L1 , + (6.- 3.*x)* L1*L1 ) * P2GQPA = P2gq0 + nf * (P2gq1 + nf * P2gq2) * RETURN END * * --------------------------------------------------------------------- * * * ..The gluon-gluon splitting function DeltaP_gg^(2), regular piece. * FUNCTION P2GGPA (x, nf) * IMPLICIT REAL*8 (A-Z) INTEGER NF * L = log (x) L1 = log (1.d0-x) * P2ggA0 = 504.d0 * L**4 + (3777.5 + 1167.* x)* L**3 , + (10902. - 863.* x)* L*L + (23091. - 12292.* L1)* L , + 30988. - 39925.* x + 13447.* x*x - 4576.* x**3 , - 13247.* (1.-x)*L1 + 3801.* L1 P2ggA1 = - 766./27.d0 * L**4 - (357.798 - 131.* x)* L**3 , - (1877.2 - 613.1* x)* L*L - (3524. + 7932.* L1)* L , - 1173.5 + 2648.6 * x - 2160.8 * x*x + 1251.7 * x**3 , - 6746.* (1.-x)*L1 - 295.7* L1 P2ggA2 = - 1.1809 * L**3 - (6.679 - 15.764* x)* L*L , - (13.29 + 16.944* L1) * L - 16.606 + 32.905 * x , - 18.30 * x*x + 2.637 * x**3 - 0.210 * L1 * P2GGPA = P2ggA0 + nf * ( P2ggA1 + nf * P2ggA2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The gluon-gluon splitting function DeltaP_gg^(2), singular piece. * FUNCTION P2GGPB (x, nf) * IMPLICIT REAL*8 (A-Z) INTEGER nf * P2GGPB = ( 2643.521 - nf * (412.172 + nf * 16./9.D0 ) ) 1 / (1.d0-x) * RETURN END * * --------------------------------------------------------------------- * * * ..The gluon-gluon splitting function DeltaP_gg^(2), `local' piece. * FUNCTION P2GGPC (x, nf) * IMPLICIT REAL*8 (A-Z) INTEGER nf * L1 = log (1.d0-x) * P2GGPC = 2643.521 * L1 + 4425.448 + 2.314 , - nf * ( 412.172 * L1 + 528.720 - 0.184 ) , - nf*nf * ( 16./9.D0 * L1 - 6.4630 + 0.0023 ) * RETURN END * * ===================================================================== * * FUNCTION P2NSMPA (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) D81 = 1./81.D0 * P2NSMPA = 1641.1 - 3135.* Y + 243.6 * Y**2 - 522.1 * Y**3 , + 128.*D81 * DL**4 + 2400.*D81 * DL**3 , + 294.9 * DL**2 + 1258.* DL , + 714.1 * DL1 + DL*DL1 * (563.9 + 256.8 * DL) , + NF * ( -197.0 + 381.1 * Y + 72.94 * Y**2 + 44.79 * Y**3 , - 192.*D81 * DL**3 - 2608.*D81 * DL**2 - 152.6 * DL , - 5120.*D81 * DL1 - 56.66 * DL*DL1 - 1.497 * Y*DL**3 ) , + NF**2 * ( 32.* Y*DL/(1.-Y) * (3.* DL + 10.) + 64. , + (48.* DL**2 + 352.* DL + 384.) * (1.-Y) ) * D81 * RETURN END * * --------------------------------------------------------------------- * * FUNCTION P2NSPPA (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL = LOG (Y) DL1 = LOG (1.-Y) D81 = 1./81.D0 * P2NSPPA = 1860.2 - 3505.* Y + 297.0 * Y**2 - 433.2 * Y**3 , + 116.*D81 * DL**4 + 2880.*D81 * DL**3 , + 399.2 * DL**2 + 1465.2 * DL , + 714.1 * DL1 + DL*DL1 * (684.0 + 251.2 * DL) , + NF * ( -216.62 + 406.5 * Y + 77.89 * Y**2 + 34.76 * Y**3 , - 256.*D81 * DL**3 - 3216.*D81 * DL**2 - 172.69 * DL , - 5120.*D81 * DL1 - 65.43 * DL*DL1 - 1.136 * Y*DL**3 ) , + NF**2 * ( 32.* Y*DL/(1.-Y) * (3.* DL + 10.) + 64. , + (48.* DL**2 + 352.* DL + 384.) * (1.-Y) ) * D81 * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular piece of both P2_NS+ and P2_NS-. It is exact * up to the truncation of the irrational coefficients. * FUNCTION P2NSPB (Y, NF) IMPLICIT REAL*8 (A-Z) INTEGER NF * P2NSPB = ( 1174.898 - NF * 183.187 - NF**2 * 64./81.D0 ) / (1.-Y) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece of P2_NS+. The coefficients of delta(1-x) * have been partly shifted relative to the exact (truncated) values. * FUNCTION P2NSMPC (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL1 = LOG (1.-Y) * P2NSMPC = 1174.898 * DL1 + 1295.624 - 0.24 , - NF * ( 183.187 * DL1 + 173.938 - 0.011 ) , + NF**2 * ( - 64./81.D0 * DL1 + 1.13067 ) * RETURN END * * * --------------------------------------------------------------------- * * * ..This is the 'local' piece of P2_NS-. The coefficients of delta(1-x) * have been partly shifted relative to the exact (truncated) values. * FUNCTION P2NSPPC (Y, NF) IMPLICIT REAL*8 (A - Z) INTEGER NF * DL1 = LOG (1.-Y) * P2NSPPC = 1174.898 * DL1 + 1295.624 - 0.154 , - NF * ( 183.187 * DL1 + 173.938 - 0.005 ) , + NF**2 * ( - 64./81.D0 * DL1 + 1.13067 ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is P2_NSS, the difference of P2_NSV and P2_NS-. * (For the polarized case, this contributions has not been calculated * yet. We include the function but we set it to zero.) * FUNCTION P2NSSPA (Y, NF) * IMPLICIT REAL*8 (A-Z) INTEGER NF * P2NSSPA = 0D0 * RETURN END * *********************************************************************** * * O(alphas alpha) splitting functions for the NLO QED evolution * *********************************************************************** * * * ..This is the regular 2-loop piece for P_NS^+. * FUNCTION X1NSPA_ASA (X) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) * X1NSPA_ASA = 8d0 * CF * ( ( ( - 3d0 * lnx ) / 2d0 - 2d0 * ln1mx 1 * lnx ) * pqq - 5d0 * ( 1d0 - x ) - ( lnx**2 * ( 1d0 2 + x ) ) / 2d0 - lnx * ( 1.5d0 + ( 7d0 * x ) / 2d0 ) ) 3 + 8d0 * CF * ( 2d0 * pqqmx * S2x 4 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the regular 2-loop piece for P_NS^-. * FUNCTION X1NSMA_ASA (X) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x pqqmx = 2d0 / ( 1d0 + x ) - 1d0 + x S2x = S2(x) * X1NSMA_ASA = 8d0 * CF * ( ( ( - 3d0 * lnx ) / 2d0 - 2d0 * ln1mx 1 * lnx ) * pqq - 5d0 * ( 1d0 - x ) - ( lnx**2 * ( 1d0 2 + x ) ) / 2d0 - lnx * ( 1.5d0 + ( 7d0 * x ) / 2d0 ) ) 3 - 8d0 * CF * ( 2d0 * pqqmx * S2x 4 + 4d0 * ( 1d0 - x ) + 2d0 * lnx * ( 1d0 + x ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X1NSC_ASA () * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * * ...The coefficient of delta(1-x) * X1NSC_ASA = 2d0 * cf * ( & + 3D0/2D0 & + 24D0*zeta3 & - 12D0*zeta2 ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop gluon->quark splitting functions P_qgamma^(1,1) * FUNCTION X1QGAMA_ASA (X) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" include "../commons/ColorFactors.h" * lnx = dlog(x) ln1mx = dlog(1d0 - x) pqg = x**2 + ( 1d0 - x )**2 * X1QGAMA_ASA = 2d0 * CF * ( 4d0 + 4d0 * ln1mx + ( 10d0 - 4d0 1 * ( ln1mx - lnx ) + 2d0 * ( - ln1mx + lnx )**2 2 - 2d0 * pi**2 / 3d0 ) * pqg - lnx * ( 1d0 - 4d0 * x ) 3 - lnx**2 * ( 1d0 - 2d0 * x ) - 9d0 * x ) * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop quark->gluon splitting functions P_gq^(1,1) * FUNCTION X1GAMQA_ASA (X) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) pgq = ( 1d0 + ( 1d0 - x )**2 ) / x * X1GAMQA_ASA = 4d0 * CF * ( - 2.5d0 - ( 3d0 * ln1mx + ln1mx**2 ) 1 * pgq - lnx**2 * ( 1d0 - x / 2d0 ) - ( 7d0 * x ) / 2d0 2 - 2d0 * ln1mx * x + lnx * ( 2d0 + ( 7d0 * x ) / 2d0 ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..The regular piece of the 2-loop gg splitting function P_ggamma^(1,1) * FUNCTION X1GGAMA_ASA (X) * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * * ...some abbreviations * lnx = dlog(x) ln1mx = dlog(1d0 - x) * X1GGAMA_ASA = 4d0 * CF *( - 16d0 + 4d0 / ( 3d0 * x ) + 8d0 * x 1 + ( 20d0 * x**2 ) / 3d0 - lnx**2 * ( 2d0 + 2d0 * x ) 2 - lnx * ( 6d0 + 10d0 * x ) ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION X1GAMGAMC_ASA () * IMPLICIT REAL*8 (A - Z) * include "../commons/ColorFactors.h" * * ...The coefficient of delta(1-x) * X1GAMGAMC_ASA = - 4D0 * CF * RETURN END * *********************************************************************** * * O(alpha^2) splitting functions for the NLO QED evolution * *********************************************************************** * * * ..The 2-loop pure-singlet splitting functions P_ps^(0,2) * FUNCTION X1PSA_AA (X) * IMPLICIT REAL*8 (A - Z) * * ...some abbreviations * DX = 1D0/X LNX = DLOG(X) HR200 = LNX * LNX / 2D0 * * ...The splitting function in terms of the harmonic polylogs * X1PSA_AA = & ( - 8D0 + 24D0*x - 224D0/9D0*x**2 + 80D0/9D0* & dx + 4D0*LNX + 20D0*LNX*x + 32D0/3D0*LNX*x**2 - & 8D0*HR200 - 8D0*HR200*x ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece of P_gammagamma^(0,2). * FUNCTION X1GAMGAMC_AA () * IMPLICIT REAL*8 (A - Z) * * ...The coefficient of delta(1-x) * X1GAMGAMC_AA = - 4D0 * RETURN END * * --------------------------------------------------------------------- * * * ..The 2-loop remainder term to the quark->gluon splitting functions P_gammaq^(0,2) * FUNCTION REM_X1GAMQA_AA (X) * IMPLICIT REAL*8 (A - Z) * * ...some abbreviations * ln1mx = dlog(1d0 - x) pgq = ( 1d0 + ( 1d0 - x )**2 ) / x * REM_X1GAMQA_AA = 4d0 * ( - 4d0 * x / 3d0 1 - pgq * ( 20d0 / 9d0 + 4d0 * ln1mx / 3d0 ) ) * RETURN END * * ===================================================================== * * * ..This is the remainder term to the regular 2-loop piece for P_NS^\pm(0,2). * FUNCTION REM_X1NSA_AA (X) * IMPLICIT REAL*8 (A - Z) * * ..The soft coefficient for use in X2NSB and X2NSC * COMMON / REM_P1SOFT_AA / A2 * * ...some abbreviations * lnx = dlog(x) pqq = 2d0 / ( 1d0 - x ) - 1d0 - x DM = 1d0 / ( 1d0 - x ) * gqq1 = 4d0 * ( ( - 1.1111111111111112d0 - ( 2d0 * lnx ) 1 / 3d0 ) * pqq - ( 4d0 * ( 1d0 - x ) ) / 3d0 ) * * ...The soft (`+'-distribution) part of the splitting function * A2 = - 80D0/9D0 * GQQ1L = DM * A2 * * ...The regular piece of the coefficient function * REM_X1NSA_AA = GQQ1 - GQQ1L * RETURN END * * --------------------------------------------------------------------- * * * ..This is the singular (soft) piece. * FUNCTION REM_X1NSB_AA (Y) IMPLICIT REAL*8 (A - Z) * COMMON / REM_P1SOFT_AA / A2 * REM_X1NSB_AA = A2 / ( 1d0 - Y ) * RETURN END * * --------------------------------------------------------------------- * * * ..This is the 'local' piece. * FUNCTION REM_X1NSC_AA (Y) * IMPLICIT REAL*8 (A - Z) * include "../commons/consts.h" * * ...The coefficient of delta(1-x) * P1DELT = & - 2D0/3D0 & - 16D0/3D0*zeta2 * * ...The soft (`+'-distribution) part of the splitting function * A2 = - 80D0/9D0 * REM_X1NSC_AA = LOG (1D0-Y) * A2 + P1DELT * RETURN END