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0001 #ifndef METOOLS_Explicit_C_Spinor_H 0002 #define METOOLS_Explicit_C_Spinor_H 0003 0004 #include "METOOLS/Currents/C_Vector.H" 0005 #include "ATOOLS/Phys/Spinor.H" 0006 #include "ATOOLS/Org/STL_Tools.H" 0007 0008 #include <vector> 0009 0010 namespace METOOLS { 0011 0012 template <class Scalar> 0013 class CSpinor: public CObject { 0014 public: 0015 0016 typedef std::complex<Scalar> SComplex; 0017 0018 protected: 0019 0020 static double s_accu; 0021 0022 int m_r, m_b, m_on; 0023 0024 SComplex m_u[4]; 0025 0026 static ATOOLS::AutoDelete_Vector<CSpinor> s_objects; 0027 0028 template <class _Scalar> friend std::ostream & 0029 operator<<(std::ostream &ostr,const CSpinor<_Scalar> &s); 0030 0031 public: 0032 0033 static CSpinor *New(); 0034 static CSpinor *New(const CSpinor &s); 0035 static CSpinor *New(const int r,const int b, 0036 const int cr=0,const int ca=0, 0037 const size_t &h=0,const size_t &s=0, 0038 const int on=3); 0039 0040 CObject* Copy() const; 0041 0042 void Delete(); 0043 0044 bool IsZero() const; 0045 0046 inline CSpinor(const int r=1,const int b=1, 0047 const int cr=0,const int ca=0, 0048 const size_t &h=0,const size_t &s=0, 0049 const int on=3): 0050 m_r(r), m_b(b), m_on(on) 0051 { m_u[0]=m_u[1]=m_u[2]=m_u[3]=Scalar(0.0); 0052 m_h=h; m_s=s; m_c[0]=cr; m_c[1]=ca; } 0053 inline CSpinor(const int &r,const int &b, 0054 const SComplex &u1,const SComplex &u2, 0055 const SComplex &u3,const SComplex &u4, 0056 const int cr=0,const int ca=0, 0057 const size_t &h=0,const size_t &s=0, 0058 const int on=3): 0059 m_r(r), m_b(b), m_on(on) 0060 { m_u[0]=u1; m_u[1]=u2; m_u[2]=u3; m_u[3]=u4; 0061 m_h=h; m_s=s; m_c[0]=cr; m_c[1]=ca; } 0062 inline CSpinor(const int &r,const int &b,const int &h, 0063 const ATOOLS::Vec4<Scalar> &p, 0064 const int cr=0,const int ca=0, 0065 const size_t &hh=0,const size_t &s=0, 0066 const Scalar &m2=-1.0,const int ms=1): 0067 m_r(r), m_b(b) 0068 { m_h=hh; m_s=s; m_c[0]=cr; m_c[1]=ca; Construct(h,p,m2,ms); } 0069 0070 // member functions 0071 void Add(const CObject *c); 0072 void Divide(const double &d); 0073 void Multiply(const Complex &c); 0074 void Invert(); 0075 0076 void Construct(const int h,const ATOOLS::Vec4<Scalar> &p, 0077 Scalar m2=-1.0,const int ms=1); 0078 bool SetOn(); 0079 0080 CSpinor CConj() const; 0081 0082 SComplex operator*(const CSpinor &s) const; 0083 0084 CSpinor operator*(const Scalar &d) const; 0085 CSpinor operator*(const SComplex &c) const; 0086 CSpinor operator/(const Scalar &d) const; 0087 CSpinor operator/(const SComplex &c) const; 0088 0089 CSpinor operator*=(const Scalar &d); 0090 CSpinor operator*=(const SComplex &c); 0091 CSpinor operator/=(const Scalar &d); 0092 CSpinor operator/=(const SComplex &c); 0093 0094 CSpinor operator+(const CSpinor &s) const; 0095 CSpinor operator-(const CSpinor &s) const; 0096 0097 CSpinor operator+=(const CSpinor &s); 0098 CSpinor operator-=(const CSpinor &s); 0099 0100 bool operator==(const CSpinor &s) const; 0101 0102 // inline functions 0103 inline SComplex &operator[](const size_t &i) 0104 { return m_u[i]; } 0105 inline const SComplex &operator[](const size_t &i) const 0106 { return m_u[i]; } 0107 0108 inline int R() const { return m_r; } 0109 inline int B() const { return m_b; } 0110 0111 inline int On() const { return m_on; } 0112 0113 inline CSpinor operator-() const 0114 { return CSpinor(m_r,m_b,-m_u[0],-m_u[1],-m_u[2],-m_u[3], 0115 m_c[0],m_c[1],m_h,m_s,m_on); } 0116 0117 inline CSpinor Bar() const 0118 { return CSpinor(m_r,-m_b,std::conj(m_u[2]),std::conj(m_u[3]), 0119 std::conj(m_u[0]),std::conj(m_u[1]),m_c[0],m_c[1], 0120 m_h,m_s,(m_on&1)<<1|(m_on&2)>>1); } 0121 0122 inline static void SetAccuracy(const double &accu) 0123 { s_accu=accu; } 0124 inline static void ResetAccuracy() 0125 { s_accu=1.0e-12; } 0126 0127 inline static double Accuracy() { return s_accu; } 0128 0129 };// end of class CSpinor 0130 /*! 0131 \class CSpinor 0132 \brief Class representing a Dirac spinor. 0133 0134 We employ \f$\gamma\f$-matrices in the Weyl representation, i.e. 0135 \f[ 0136 \gamma_\mu=\left(\begin{array}{cc}0&\sigma_\mu\\ 0137 \bar{\sigma}_\mu&0\end{array}\right) 0138 \f] 0139 where \f$\sigma_\mu=\left(1,-\vec{\sigma}\right)\f$ and 0140 \f$\bar{\sigma}_\mu=\left(1,\vec{\sigma}\right)\f$. 0141 The pauli matrices \f$\sigma^i\f$ are given by 0142 \f[ 0143 \sigma^1=\left(\begin{array}{cc}0&1\\1&0\end{array}\right)\;,\quad 0144 \sigma^2=\left(\begin{array}{cc}0&-i\\i&0\end{array}\right)\;,\quad 0145 \sigma^3=\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)\;. 0146 \f] 0147 In this basis 0148 \f[ 0149 \gamma^5=\left(\begin{array}{cc}-1&0\\0&1\end{array}\right) 0150 \f] 0151 The dirac equation thus yields the Eigenspinor problem 0152 \f[ 0153 0=\left(p^\mu\gamma_\mu-m\right)u\\ 0154 =\left(\begin{array}{cccc} 0155 -m&0&p^0-p^3&-p^1+ip^2\\0&-m&-p^1-ip^2&p^0+p^3\\ 0156 p^0+p^3&p^1-ip^2&-m&0\\p^1+ip^2&p^0-p^3&0&-m\end{array}\right)u 0157 \f] 0158 Employing \f$p^\pm=p^0\pm p^3\f$ and \f$p_\perp=p^1+ip^2\f$ 0159 this can be rewritten to give 0160 \f[ 0161 0=\left(\begin{array}{cccc} 0162 -m&0&p^-&-p_\perp^*\\0&-m&-p_\perp&p^+\\ 0163 p^+&p_\perp^*&-m&0\\p_\perp&p^-&0&-m\end{array}\right)u 0164 \f] 0165 The Eigenvalues are \f$\lambda=m\pm\sqrt{p^2}\f$. Eigenspinors 0166 can be found by firstly constructing solutions for \f$m=0\f$. 0167 One possible set of such solutions is 0168 \f[ 0169 u_+(p,0)=v_-(p,0)=\left(\begin{array}{c} 0170 0\\\chi_+(p)\end{array}\right)\;,\quad 0171 u_-(p,0)=v_+(p,0)=\left(\begin{array}{c} 0172 \chi_-(p)\\0\end{array}\right)\;. 0173 \f] 0174 where 0175 \f[ 0176 \chi_+(p)=\frac{1}{\sqrt{p^+}}\left(\begin{array}{c} 0177 p^+\\p_\perp\end{array}\right)=\left(\begin{array}{c} 0178 \sqrt{p^+}\\\sqrt{p^-}e^{i\phi_p}\end{array}\right)\;,\quad 0179 \chi_-(p)=\frac{e^{i\pi}}{\sqrt{p^+}}\left(\begin{array}{c} 0180 -p_\perp^*\\p^+\end{array}\right)=\left(\begin{array}{c} 0181 \sqrt{p^-}e^{-i\phi_p}\\-\sqrt{p^+}\end{array}\right)\;. 0182 \f] 0183 The spinors \f$\chi_\pm(p)\f$ are normalized to \f$2\,p_0\f$. 0184 Defining \f$\bar p=|\vec p|\f$ and 0185 \f$\hat p=(\,{\rm sgn}(p_0)\,\bar p,\vec p\;)\f$, possible solutions 0186 for \f$m\neq 0\f$ are [Nucl. Phys. B274 (1986) 1-32] 0187 \f[ 0188 u_+(p,m)=\frac{1}{\sqrt{2\,\bar p}}\left(\begin{array}{r} 0189 \sqrt{p_0-\bar p}\;\chi_+(\hat p)\\ 0190 \sqrt{p_0+\bar p}\;\chi_+(\hat p)\end{array}\right)\;,\quad 0191 v_-(p,m)=\frac{1}{\sqrt{2\,\bar p}}\left(\begin{array}{r} 0192 -\sqrt{p_0-\bar p}\;\chi_+(\hat p)\\ 0193 \sqrt{p_0+\bar p}\;\chi_+(\hat p)\end{array}\right)\;, 0194 \f] 0195 \f[ 0196 u_-(p,m)=\frac{1}{\sqrt{2\,\bar p}}\left(\begin{array}{r} 0197 \sqrt{p_0+\bar p}\;\chi_-(\hat p)\\ 0198 \sqrt{p_0-\bar p}\;\chi_-(\hat p)\end{array}\right)\;,\quad 0199 v_+(p,m)=\frac{1}{\sqrt{2\,\bar p}}\left(\begin{array}{r} 0200 \sqrt{p_0+\bar p}\;\chi_-(\hat p)\\ 0201 -\sqrt{p_0-\bar p}\;\chi_-(\hat p)\end{array}\right)\;. 0202 \f] 0203 These Eigenspinors are orthogonal and satisfy the relations 0204 \f$\bar u_\lambda(p,m)u_\lambda(p,m)=2m\f$ 0205 and \f$\bar v_\lambda(p,m)v_\lambda(p,m)=-2m\f$. 0206 In the above spinor basis the color-ordered quark-quark-gluon vertex reads 0207 \f[ 0208 \frac{i}{\sqrt{2}}\bar{u}\gamma^\mu v\\= 0209 \frac{i}{\sqrt{2}}\left(\bar{u}_0,\bar{u}_1,\bar{u}_2,\bar{u}_3\right) 0210 \left(\begin{array}{c}\left(\begin{array}{cc}0&1\\1&0\end{array}\right)\\ 0211 \left(\begin{array}{cc}0&\sigma^1\\-\sigma^1&0\end{array}\right)\\ 0212 \left(\begin{array}{cc}0&\sigma^2\\-\sigma^2&0\end{array}\right)\\ 0213 \left(\begin{array}{cc}0&\sigma^3\\-\sigma^3&0\end{array}\right) 0214 \end{array}\right) 0215 \left(\begin{array}{c}v_0\\v_1\\v_2\\v_3\end{array}\right)\\= 0216 \frac{i}{\sqrt{2}}\left(\begin{array}{c} 0217 \bar{u}_0v_2+\bar{u}_1v_3+\bar{u}_2v_0+\bar{u}_3v_1\\ 0218 \bar{u}_0v_3+\bar{u}_1v_2-\bar{u}_2v_1-\bar{u}_3v_0\\ 0219 -i\left(\bar{u}_0v_3-\bar{u}_1v_2-\bar{u}_2v_1+\bar{u}_3v_0\right)\\ 0220 \bar{u}_0v_2-\bar{u}_1v_3-\bar{u}_2v_0+\bar{u}_3v_1\end{array}\right) 0221 \f] 0222 The color-ordered quark-gluon-quark vertices read 0223 \f[ 0224 \frac{i}{\sqrt{2}}j^\mu\gamma_\mu v\\= 0225 \frac{i}{\sqrt{2}}\left(\begin{array}{cccc} 0226 0&0&j^-&-j_\perp^*\\0&0&-j_\perp&j^+\\ 0227 j^+&j_\perp^*&0&0\\j_\perp&j^-&0&0\end{array}\right) 0228 \left(\begin{array}{c}v_0\\v_1\\v_2\\v_3\end{array}\right)\\= 0229 \frac{i}{\sqrt{2}}\left(\begin{array}{c} 0230 j^-v_2-j_\perp^*v_3\\-j_\perp v_2+j^+v_3\\ 0231 j^+v_0+j_\perp^*v_1\\j_\perp v_0+j^-v_1\end{array}\right) 0232 \f] 0233 and 0234 \f[ 0235 \frac{i}{\sqrt{2}}\bar{u}j^\mu\gamma_\mu\\= 0236 \frac{i}{\sqrt{2}}\left(\bar{u}_0,\bar{u}_1,\bar{u}_2,\bar{u}_3\right) 0237 \left(\begin{array}{cccc}0&0&j^-&-j_\perp^*\\0&0&-j_\perp&j^+\\ 0238 j^+&j_\perp^*&0&0\\j_\perp&j^-&0&0\end{array}\right)\\= 0239 \frac{i}{\sqrt{2}}\left( 0240 \bar{u}_2j^++\bar{u}_3j_\perp,\bar{u}_2j_\perp^*+\bar{u}_3j^-, 0241 \bar{u}_0j^--\bar{u}_1j_\perp,-\bar{u}_0j_\perp^*+\bar{u}_1j^+\right) 0242 \f] 0243 */ 0244 0245 template <class Scalar> 0246 std::ostream &operator<<(std::ostream &ostr,const CSpinor<Scalar> &s); 0247 0248 }// end of namespace BCF 0249 0250 #define DDSpinor METOOLS::CSpinor<double> 0251 #define QDSpinor METOOLS::CSpinor<long double> 0252 0253 #endif
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