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QCDLoop
One-loop scalar Feynman integrals
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The Triangle integral. More...
#include <triangle.h>
Public Member Functions | |
| Triangle () | |
| The Constructor. | |
| ~Triangle () | |
| The Destructor. | |
| void | integral (vector< TOutput > &res, TScale const &mu2, vector< TMass > const &m, vector< TScale > const &p) |
| Computes the Triangle integral. More... | |
| void | T0 (vector< TOutput > &res, TMass const (&xpi)[6], int const &massive) const |
| General case triangle integral I(p1,p2,p3;m1,m2,m3) More... | |
| void | T1 (vector< TOutput > &res, TScale const &mu2, TScale const &p) const |
| Divergent triangle integral I(0,0,p2;0,0,0) More... | |
| void | T2 (vector< TOutput > &res, TScale const &mu2, TScale const &p2, TScale const &p3) const |
| Divergent triangle integral I(0,p1,p2;0,0,0) More... | |
| void | T3 (vector< TOutput > &res, TScale const &mu2, TMass const &m, TScale const &p1, TScale const &p2) const |
| Divergent triangle integral I(0,p1,p2;0,0,m2) More... | |
| void | T4 (vector< TOutput > &res, TScale const &mu2, TMass const &m, TScale const &p2) const |
| Divergent triangle integral I(0,m2,p2;0,0,m2) More... | |
| void | T5 (vector< TOutput > &res, TScale const &mu2, TMass const &m) const |
| Divergent triangle integral I(0,m2,m2;0,0,m2) More... | |
| void | T6 (vector< TOutput > &res, TScale const &mu2, TMass const &m2, TMass const &m3, TScale const &p2) const |
| Divergent triangle integral I(m2,s,m3;0,m2,m3) More... | |
 Public Member Functions inherited from ql::Topology< TOutput, TMass, TScale > | |
| Topology (string name="None") | |
| Topology<TOutput, TMass>::Topology. More... | |
| Topology (const Topology &obj) | |
| Topology<TOutput, TMass>::Topology. More... | |
| virtual | ~Topology () |
| Topology<TOutput, TMass>::~Topology. | |
| string const & | getName () const |
| Get topology name. | |
| Public Member Functions inherited from ql::Tools< TOutput, TMass, TScale > | |
| Tools () | |
| The Constructor. | |
| ~Tools () | |
| The Destructor. | |
| bool | iszero (TMass const &psq) const |
| Check for zeros. More... | |
| TOutput | cLn (TOutput const &z, TScale const &isig) const |
| Log of complex argument with explicit sign for imag part. More... | |
| TOutput | cLn (TScale const &x, TScale const &isig) const |
| TOutput | fndd (int const &n, TOutput const &x, TScale const &iep) const |
| The fndd function. More... | |
| TOutput | Lnrat (TOutput const &x, TOutput const &y) const |
| Computes the ratio of logs. More... | |
| TOutput | Lnrat (TScale const &x, TScale const &y) const |
| TMass | ddilog (TMass const &x) const |
| Computes the dilog function for real argument. More... | |
| TOutput | denspence (TOutput const &z, TScale const &isig) const |
| Spence's function. More... | |
| TOutput | spencer (TOutput const &zrat1, TOutput const &zrat2, TScale const &ieps1, TScale const &ieps2) const |
| Spence's function for ratio arguments. More... | |
| TOutput | xspence (TOutput const (&z1)[2], TScale const (&im1)[2], TOutput const &z2, TScale const &im2) const |
| Difference of complex Spence's function. More... | |
| TOutput | cspence (TOutput const &z1, TScale const &im1, TOutput const &z2, TScale const &im2) const |
| Complex Spence's function. More... | |
| TOutput | Li2omrat (TScale const &x, TScale const &y) const |
| Expression for dilog(1-(x-i*ep)/(y-i*ep)). More... | |
| TOutput | Li2omrat (TOutput const &x, TOutput const &y, TScale const &ieps1=-1, TScale const &ieps2=-1) const |
| TOutput | Li2omx2 (TScale const &v, TScale const &w, TScale const &x, TScale const &y) const |
| Expression for dilog(1-(v-i*ep)*(w-i*ep)/(x-i*ep)/(y-i*ep)). More... | |
| TOutput | Li2omx2 (TOutput const &v, TOutput const &w, TOutput const &x, TOutput const &y, TScale const &ieps1=-1, TScale const &ieps2=-1) const |
| TOutput | cLi2omx2 (TOutput const &z1, TOutput const &z2, TScale const &ieps1=-1, TScale const &ieps2=-1) const |
| TOutput | Li2omx (TMass const &x1, TMass const &x2, TScale const &ieps1, TScale const &ieps2) const |
| Calculate Li[2](1-(x1+ieps1)*(x2+ieps2)) More... | |
| TOutput | cLi2omx3 (TOutput const &z1, TOutput const &z2, TOutput const &z3, TScale const &ieps1, TScale const &ieps2, TScale const &ieps3) const |
| Calculate Li[2](1-(z1+ieps1)*(z2+ieps2)*(z3+ieps3)) More... | |
| TOutput | L0 (TMass const &x, TMass const &y) const |
| TOutput | L1 (TMass const &x, TMass const &y) const |
| TOutput | R3int (TOutput const &p, TOutput const &s1, TOutput const &s2, TOutput const &t1) const |
| TOutput | R3int (TOutput const &p, TOutput const &s1, TOutput const &s2, TOutput const &t1, TOutput const &t2, TOutput const &t3, TOutput const &t4) const |
| TOutput | R2int (TOutput const &a, TOutput const &b, TOutput const &y0) const |
| TOutput | Rint (TOutput const &y, TOutput const &z, TScale const &ieps) const |
| void | R (TOutput &r, TOutput &d, TOutput const &q) const |
| Tools<TOutput, TMass, TScale>::R. More... | |
| TOutput | Zlogint (TOutput const &z, TScale const &ieps) const |
| Tools<TOutput, TMass, TScale>::qlZlogint. More... | |
| TOutput | ltspence (int const &i_in, TOutput const &z_in, TScale const &s) const |
| TOutput | li2series (TOutput const &z, TScale const &isig) const |
| TOutput | ltli2series (TOutput const &z1, TScale const &s) const |
| TOutput | eta2 (TOutput const &a, TOutput const &b) const |
| TOutput | eta3 (TOutput const &a, TOutput const &b, TOutput const &c) const |
| TOutput | eta5 (TOutput const &a, TOutput const &b, TOutput const &c, TOutput const &d, TOutput const &e) const |
| TOutput | xetatilde (TOutput const (&z1)[2], TScale const (&im1)[2], TOutput const &z2, TScale const &im2, TOutput const (&l1)[2]) const |
| TOutput | xeta (TOutput const (&z1)[2], TScale const (&im1)[2], TOutput const &z2, TScale const &im2, TScale const &im12, TOutput const (&l1)[2]) const |
| int | eta (TOutput const &z1, TScale const &s1, TOutput const &z2, TScale const &s2, TScale const &s12) const |
| int | etatilde (TOutput const &c1, TScale const &im1x, TOutput const &c2, TScale const &im2x) const |
| void | kfn (TOutput(&res)[3], TScale &ieps, TMass const &xpi, TMass const &xm, TMass const &xmp) const |
| The K-function. More... | |
| void | solveabc (TMass const &a, TMass const &b, TMass const &c, TOutput(&z)[2]) const |
| Solution of the quadratic equation. More... | |
| void | solveabcd (TOutput const &a, TOutput const &b, TOutput const &c, TOutput const &d, TOutput(&z)[2]) const |
| Solution of the quadratic equation passing the discriminant. More... | |
| void | solveabcd (TOutput const &a, TOutput const &b, TOutput const &c, TOutput(&z)[2]) const |
| void | ratgam (TOutput &ratp, TOutput &ratm, TScale &ieps, TMass const &p3sq, TMass const &m3sq, TMass const &m4sq) const |
| Ratio function. More... | |
| void | ratreal (TMass const &si, TMass const &ta, TMass &rat, TScale &ieps) const |
| Ratio function. More... | |
| Public Member Functions inherited from ql::LRUCache< size_t, vector< TOutput > > | |
| LRUCache (int const &size=1) | |
| LRUCache constructor. More... | |
| LRUCache (const LRUCache &obj) | |
| void | setCacheSize (int const &size) |
| Set the Cache size. | |
| int const & | getCacheSize () const |
| Get the Cache size. | |
| void | store (size_tconst &key, vector< TOutput >const &value) |
| Store the cached data. | |
| bool | get (size_tconst &key, vector< TOutput > &out) |
| Get the cached data. | |
Additional Inherited Members | |
| Public Types inherited from ql::LRUCache< size_t, vector< TOutput > > | |
| typedef std::pair< size_t, vector< TOutput > > | key_value_pair_t |
| typedef std::list< key_value_pair_t >::iterator | list_iterator_t |
| Protected Member Functions inherited from ql::Topology< TOutput, TMass, TScale > | |
| bool | checkCache (TScale const &, vector< TMass > const &, vector< TScale > const &) |
| < Check stored cached results More... | |
| void | storeCache (TScale const &, vector< TMass > const &, vector< TScale > const &) |
| Topology<TOutput, TMass>::storeCache. | |
| Protected Attributes inherited from ql::Topology< TOutput, TMass, TScale > | |
| string | _name |
| size_t | _key |
| TScale | _mu2 |
| vector< TMass > | _m |
| vector< TScale > | _p |
| vector< TOutput > | _val |
| ContainerHasher< TMass, TScale > * | _ch |
| Protected Attributes inherited from ql::Tools< TOutput, TMass, TScale > | |
| TScale | _pi |
| TScale | _pi2 |
| TScale | _pio3 |
| TScale | _pio6 |
| TScale | _pi2o3 |
| TScale | _pi2o6 |
| TScale | _pi2o12 |
| TScale | _zero |
| TScale | _half |
| TScale | _one |
| TScale | _two |
| TScale | _three |
| TScale | _four |
| TScale | _five |
| TScale | _six |
| TScale | _ten |
| TScale | _eps |
| TScale | _eps4 |
| TScale | _eps7 |
| TScale | _eps10 |
| TScale | _eps14 |
| TScale | _eps15 |
| TScale | _xloss |
| TScale | _neglig |
| TScale | _reps |
| TOutput | _2ipi |
| TOutput | _ipio2 |
| TOutput | _ipi |
| TOutput | _czero |
| TOutput | _chalf |
| TOutput | _cone |
| TOutput | _ctwo |
| TOutput | _cthree |
| TOutput | _cfour |
| TOutput | _ieps |
| TOutput | _ieps2 |
| TOutput | _ieps50 |
The Triangle integral.
Parses automatically the topology and computes the integral
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virtual |
Computes the Triangle integral.
Computes the Triangle integral defined as:
\[ I_{3}^{D}(p_1^2,p_2^2,p_3^2;m_1^2,m_2^2,m_3^2)= \frac{\mu^{4-D}}{i \pi^{D/2} r_{\Gamma}} \int d^Dl \frac{1}{(l^2-m_1^2+i \epsilon)((l+q_1)^2-m_2^2+i \epsilon)((l+q_2)^2-m_3^2+i\epsilon)} \]
where \(q_1=p_1,q_2=p_1+p_2\).
Implementation of the formulae of Denner and Dittmaier [5] and 't Hooft and Veltman [9].
| res | output object res[0,1,2] the coefficients in the Laurent series |
| mu2 | is the square of the scale mu |
| m | are the squares of the masses of the internal lines |
| p | are the four-momentum squared of the external lines |
Implements ql::Topology< TOutput, TMass, TScale >.
| void ql::Triangle< TOutput, TMass, TScale >::T0 | ( | vector< TOutput > & | res, |
| TMass const (&) | xpi[6], | ||
| int const & | massive | ||
| ) | const |
General case triangle integral I(p1,p2,p3;m1,m2,m3)
Parses finite triangle integrals. Formulae from 't Hooft and Veltman [9]
| res | output object res[0,1,2] the coefficients in the Laurent series |
| xpi | an array with masses and momenta squared. |
| void ql::Triangle< TOutput, TMass, TScale >::T1 | ( | vector< TOutput > & | res, |
| TScale const & | mu2, | ||
| TScale const & | p | ||
| ) | const |
Divergent triangle integral I(0,0,p2;0,0,0)
The integral is defined as:
\[ I_{3}^{D=4-2 \epsilon}(0,0,p^2;0,0,0)= \frac{1}{p^2} \left( \frac{1}{\epsilon^2} + \frac{1}{\epsilon} \ln \left( \frac{\mu^2}{-p^2-i \epsilon} \right) + \frac{1}{2} \ln^2 \left( \frac{\mu^2}{-p^2-i \epsilon} \right) \right) + O(\epsilon) \]
Implementation of the formulae from Beenakker et al. [3].
| res | output object res[0,1,2] the coefficients in the Laurent series |
| mu2 | is the squre of the scale mu |
| p | is the four-momentum squared of the external line |
| void ql::Triangle< TOutput, TMass, TScale >::T2 | ( | vector< TOutput > & | res, |
| TScale const & | mu2, | ||
| TScale const & | p1, | ||
| TScale const & | p2 | ||
| ) | const |
Divergent triangle integral I(0,p1,p2;0,0,0)
The integral is defined as:
\[ I_{3}^{D=4-2 \epsilon}(0,p_1^2,p_2^2;0,0,0)= \frac{1}{p_1^2-p_2^2} \left\{ \frac{1}{\epsilon} \left[ \ln \left( \frac{\mu^2}{-p_1^2-i \epsilon} \right) - \ln \left( \frac{\mu^2}{-p_2^2-i \epsilon} \right) \right] + \frac{1}{2} \left[ \ln^2 \left( \frac{\mu^2}{-p_1^2-i \epsilon} \right) - \ln^2 \left( \frac{\mu^2}{-p_2^2-i \epsilon} \right) \right] \right\} + O(\epsilon) \]
Implementation of the formulae from Beenakker et al. [3].
| res | output object res[0,1,2] the coefficients in the Laurent series |
| mu2 | is the squre of the scale mu |
| p1 | is the four-momentum squared of the external line |
| p2 | is the four-momentum squared of the external line |
| void ql::Triangle< TOutput, TMass, TScale >::T3 | ( | vector< TOutput > & | res, |
| TScale const & | mu2, | ||
| TMass const & | m, | ||
| TScale const & | p1, | ||
| TScale const & | p2 | ||
| ) | const |
Divergent triangle integral I(0,p1,p2;0,0,m2)
The integral is defined as:
\[ I_{3}^{D=4-2 \epsilon}(0,p_1^2,p_2^2;0,0,m^2)= \frac{1}{p_1^2-p_2^2} \left( \frac{\mu^2}{m^2} \right)^\epsilon \left\{ \frac{1}{\epsilon} \ln \left( \frac{m^2-p_2^2}{m^2-p_1^2} \right) + {\rm Li}_2 \left( \frac{p_1^2}{m^2} \right) - {\rm Li}_2 \left( \frac{p_2^2}{m^2} \right) + \ln^2 \left( \frac{m^2-p_1^2}{m^2} \right) - \ln^2 \left( \frac{m^2-p_2^2}{m^2} \right) \right\} + O(\epsilon) \]
Implementation of the formulae from Beenakker et al. [3].
| res | output object res[0,1,2] the coefficients in the Laurent series |
| mu2 | is the squre of the scale mu |
| m | is the square of the mass of the internal line |
| p1 | is the four-momentum squared of the external line |
| p2 | is the four-momentum squared of the external line |
| void ql::Triangle< TOutput, TMass, TScale >::T4 | ( | vector< TOutput > & | res, |
| TScale const & | mu2, | ||
| TMass const & | m, | ||
| TScale const & | p2 | ||
| ) | const |
Divergent triangle integral I(0,m2,p2;0,0,m2)
The integral is defined as:
\[ I_{3}^{D=4-2 \epsilon}(0,m^2,p_2^2;0,0,m^2)= \left( \frac{\mu^2}{m^2} \right)^\epsilon \frac{1}{p_2^2-m^2} \left[ \frac{1}{2 \epsilon^2} + \frac{1}{\epsilon} \ln \left( \frac{m^2}{m^2-p_2^2} \right) + \frac{\pi^2}{12} + \frac{1}{2} \ln^2 \left( \frac{m^2}{m^2-p_2^2} \right) - {\rm Li}_2 \left( \frac{-p_2^2}{m^2-p_2^2} \right) \right] + O(\epsilon) \]
Implementation of the formulae from Beenakker et al. [3].
| res | output object res[0,1,2] the coefficients in the Laurent series |
| mu2 | is the squre of the scale mu |
| m | is the square of the mass of the internal line |
| p | is the four-momentum squared of the external line |
| void ql::Triangle< TOutput, TMass, TScale >::T5 | ( | vector< TOutput > & | res, |
| TScale const & | mu2, | ||
| TMass const & | m | ||
| ) | const |
Divergent triangle integral I(0,m2,m2;0,0,m2)
The integral is defined as:
\[ I_{3}^{D=4-2 \epsilon}(0,m^2,m^2;0,0,m^2)= \left( \frac{\mu^2}{m^2} \right)^\epsilon \frac{1}{m^2} \left( -\frac{1}{2 \epsilon} + 1 \right) + O(\epsilon) \]
Implementation of the formulae from Beenakker et al. [3].
| res | output object res[0,1,2] the coefficients in the Laurent series |
| mu2 | is the squre of the scale mu |
| m | is the square of the mass of the internal line |
| void ql::Triangle< TOutput, TMass, TScale >::T6 | ( | vector< TOutput > & | res, |
| TScale const & | mu2, | ||
| TMass const & | m2sq, | ||
| TMass const & | m3sq, | ||
| TScale const & | p2 | ||
| ) | const |
Divergent triangle integral I(m2,s,m3;0,m2,m3)
The integral is defined as:
\[ I_{3}^{D=4-2 \epsilon}(m_2^2,s,m_3^2;0,m_2^2,m_3^2)= \frac{\Gamma(1+\epsilon)\mu^\epsilon}{2\epsilon r_\Gamma} \int_0^1 d\gamma \frac{1}{\left[ \gamma m_2^2 + (1-\gamma) m_3^2 - \gamma (1-\gamma)s - i\epsilon \right]^{1+\epsilon}} + O(\epsilon) \]
Implementation of the formulae from Beenakker et al. [3].
| res | output object res[0,1,2] the coefficients in the Laurent series |
| mu2 | is the squre of the scale mu |
| m2 | is the square of the mass of the internal line |
| m3 | is the square of the mass of the internal line |
| p2 | is the four-momentum squared of the external line |
1.8.9.1
