1 //-----------------------------------------------------------------------
2 // File and Version Information:
4 // Copyright Information: See EvtGen/COPYRIGHT
9 // d Gamma / _ _ _2 mb _2 mb
10 // ---------- = 12 Gamma | (1+x-z)(z-x-p ) -- W + (1-z+p ) -- W
14 // + [x(z-x)-p ] -- (W + 2mb W + mb W ) |
20 // x = ------ , p = --- , z = ------ , x = 1-x
24 // the triple differential decay rate according to
28 // Software developed for the BaBar Detector at the SLAC B-Factory.
33 //-----------------------------------------------------------------------
34 //-----------------------
35 // This Class's Header --
36 //-----------------------
37 #include "EvtGenBase/EvtPatches.hh"
38 #include "EvtGenBase/EvtConst.hh"
39 #include "EvtGenModels/EvtVubdGamma.hh"
40 #include "EvtGenBase/EvtDiLog.hh"
51 EvtVubdGamma::EvtVubdGamma(const double &alphas)
55 // the range for the delta distribution in p2 is from _epsilon1 to
56 // _epsilon2. It was checked with the single differential formulae
57 // in the paper that these values are small enough to imitate p2 = 0
58 // for the regular terms.
59 // The ()* distributions, however need further treatment. In order to
60 // generate the correct spectrum in z a threshold need to be computed
61 // from the desired value of the coupling alphas. The idea is that
62 // for z=1 p2=0 is not allowed and therefore the part of dGamma proportional
63 // to delta(p2) should go to 0 for z->1.
64 // Using equation (3.1) and (3.2) it is possible to find the correct value
65 // for log(_epsilon3) from this requirement.
70 double lne3 = 9./16.-2*EvtConst::pi*EvtConst::pi/3.+6*EvtConst::pi/4/alphas; if ( lne3 > 0 )
71 lne3 = -7./4. - sqrt(lne3);
74 _epsilon3 = exp(lne3);
84 EvtVubdGamma::~EvtVubdGamma( )
92 double EvtVubdGamma::getdGdxdzdp(const double &x, const double &z, const double &p2)
98 if ( x < 0 || x > 1 || z < xb || z > (1+xb) )
101 double p2min = (0>z-1.?0:z-1.);
102 double p2max = (1.-x)*(z-1.+x);
104 if (p2 < p2min || p2 > p2max)
107 // // check the phase space
112 if ( p2 >_epsilon1 && p2< _epsilon2) {
114 double W1 = getW1delta(x,z);
115 double W4plus5 = getW4plus5delta(x,z);
117 dG = 12. * delta(p2,p2min,p2max) * ((1.+xb-z) * (z-xb) * W1
118 + xb*(z-xb) * (W4plus5));
122 double W1 = getW1nodelta(x,z,p2);
123 double W2 = getW2nodelta(x,z,p2);
124 double W3 = getW3nodelta(x,z,p2);
125 double W4 = getW4nodelta(x,z,p2);
126 double W5 = getW5nodelta(x,z,p2);
128 dG = 12. * ((1.+xb-z) * (z-xb-p2) * W1
130 + (xb*(z-xb)-p2) * (W3+W4+W5));
135 double EvtVubdGamma::delta(const double &x, const double &xmin, const double &xmax)
137 if ( xmin > 0 || xmax < 0 ) return 0.;
138 if ( _epsilon1 < x && x < _epsilon2 ) return 1./(_epsilon2-_epsilon1);
142 double EvtVubdGamma::getW1delta(const double &, const double &z)
147 if (z == 1) lz = -1.;
148 else lz = log(z)/(1.-z);
150 // ddilog_(&z) is actually the dilog of (1-z) in maple,
151 // also in Neuberts paper the limit dilog(1) = pi^2/6 is used
152 // this corresponds to maple's dilog(0), so
153 // I take ddilog_(&mz) where mz=1-z in order to satisfy Neubert's definition
154 // and to compare with Maple the argument in maple should be (1-mz) ...
156 double dl = 4.*EvtDiLog::DiLog(mz) + 4.*pow(EvtConst::pi,2)/3.;
158 double w = -(8.*pow(log(z),2) - 10.*log(z) + 2.*lz + dl + 5.)
159 + (8.*log(z)-7.)*log(_epsilon3) - 2.*pow(log(_epsilon3),2);
161 return (1. + w*_alphas/3./EvtConst::pi);
164 double EvtVubdGamma::getW1nodelta(const double &, const double &z, const double &p2)
168 double t2 = 1.-4.*p2/z2;
172 if ( p2 > _epsilon2 )
173 w += 4./p2*(log((1.+t)/(1.-t))/t + log(p2/z2))
174 + 1. - (8.-z)*(2.-z)/z2/t2
175 + ((2.-z)/2./z+(8.-z)*(2.-z)/2./z2/t2)*log((1.+t)/(1.-t))/t;
176 if ( p2 > _epsilon3 )
177 w += (8.*log(z)-7.)/p2 - 4.*log(p2)/p2;
179 return w*_alphas/3./EvtConst::pi;
182 double EvtVubdGamma::getW2nodelta(const double &, const double &z, const double &p2)
186 double t2 = 1.-4.*p2/z2;
188 double w11 = (32.-8.*z+z2)/4./z/t2;
191 if ( p2 > _epsilon2 )
192 w -= (z*t2/8. + (4.-z)/4. + w11/2.)*log((1.+t)/(1.-t))/t;
193 if ( p2 > _epsilon2 )
194 w += (8.-z)/4. + w11;
196 return (w*_alphas/3./EvtConst::pi);
199 double EvtVubdGamma::getW3nodelta(const double &, const double &z, const double &p2)
202 double t2 = 1.-4.*p2/z2;
208 if ( p2 > _epsilon2 )
209 w += (z*t2/16. + 5.*(4.-z)/16. - (64.+56.*z-7.*z2)/16./z/t2
210 + 3.*(12.-z)/16./t4) * log((1.+t)/(1.-t))/t;
211 if ( p2 > _epsilon2 )
212 w += -(8.-3.*z)/8. + (32.+22.*z-3.*z2)/4./z/t2 - 3.*(12.-z)/8./t4;
214 return (w*_alphas/3./EvtConst::pi);
217 double EvtVubdGamma::getW4nodelta(const double &, const double &z, const double &p2)
220 double t2 = 1.-4.*p2/z2;
226 if ( p2 > _epsilon2 )
227 w -= ((8.-3.*z)/4./z - (22.-3.*z)/2./z/t2 + 3.*(12.-z)/4./z/t4)
228 * log((1.+t)/(1.-t))/t;
229 if ( p2 > _epsilon2 )
230 w += -1. - (32.-5.*z)/2./z/t2 + 3.*(12.-z)/2./z/t4 ;
232 return w*_alphas/3./EvtConst::pi;
235 double EvtVubdGamma::getW4plus5delta(const double &, const double &z)
243 w = 2.*log(z)/(1.-z);
245 return (w*_alphas/3./EvtConst::pi);
248 double EvtVubdGamma::getW5nodelta(const double &, const double &z, const double &p2)
251 double t2 = 1.-4.*p2/z2;
256 if ( p2 > _epsilon2 )
257 w += (1./4./z - (2.-z)/2./z2/t2 + 3.*(12.-z)/4./z2/t4)
258 * log((1.+t)/(1.-t))/t;
259 if ( p2 > _epsilon2 )
260 w += -(8.+z)/2./z2/t2 - 3.*(12.-z)/2./z2/t4;
262 return (w*_alphas/3./EvtConst::pi);