Abstract
Full Text
GENERAL CALCULATION OF MATRIX ELEMENTS FOR POLARIZED VECTOR PARTICLES
A. A. Bogush, A. I. Bol’sun
(Presented by Academician V. A. Fock, 12 XII 1963)
In paper \((^1)\), on the basis of the general method of projection operators in the theory of elementary particles \((^2)\), compact matrix expressions were found for 10-dimensional wave functions describing the states of a vector meson \((m \ne 0)\) with a given value of the 4-momentum \(p=(\mathbf p, ip_0)\) and spin projections equal to \(\pm 1\). With their aid, in \((^1)\) a covariant method was developed for the direct calculation of matrix elements of longitudinally polarized particles with spin 1.
Below a generalization of the indicated method is given for the case of arbitrary spin states of a vector particle, and on this basis a general calculation of matrix elements of vector particles is carried out. As an illustration, the differential cross sections are calculated for the scattering of a polarized vector meson by a scalar particle and by a Coulomb center.
In accordance with \((^2)\), the projection dyad matrices \(\Lambda^{(r)}(p)\), which determine the states of a vector particle with 4-momentum \(p\) and spin projection \(r=\pm 1, 0\), have the form (see \((^1)\)) \((\hbar=c=1)\)
\[ \Lambda^{(\pm)}(p)=\mp \frac{1}{4m^2}\sigma_s(\sigma_s \pm 1)\hat p(\hat p \mp im)= \]
\[ =\mp \frac{1}{2m^2}\bigl(\hat e^{(\mp)}\bigr)^2\bigl(\hat e^{(\pm)}\bigr)^2\hat p(\hat p \mp im) =\psi^{(\pm 1)}(\pm p)\cdot \bar\psi^{(\pm 1)}(\pm p); \tag{1} \]
\[ \Lambda^{(0)}(p)=\pm \frac{1}{2m^2}(\sigma_s^2-1)\hat p(\hat p \mp im) =\psi^{(0)}(\pm p)\cdot \bar\psi^{(0)}(\pm p), \tag{2} \]
where \((\psi_1\cdot \bar\psi_2)_{mn}=\psi_{1m}\bar\psi_{2n}\), \(\bar\psi=\psi^*\eta\), \(\eta=2\beta_4^2-1\). Here \(\bar p=p_\mu\beta_\mu\), \(\beta_\mu\) are \(10\times 10\) Duffin–Kemmer matrices, and
\[ \sigma_s=\frac{1}{m}\delta_{\mu\nu\rho\sigma}p_\mu s_\nu\beta_\rho\beta_\sigma =\hat e^{(-)}\hat e^{(+)}-\hat e^{(+)}\hat e^{(-)} \tag{3} \]
is the spin-projection operator in the covariant notation of \((^1)\). The spin vector
\[ s=(\mathbf s,is_0)=\left(\frac{p_0}{m}\frac{\mathbf p}{|\mathbf p|},\, i\frac{|\mathbf p|}{m}\right) \]
and the circular vectors \(e^{(\pm)}\) are connected by the relations (see the case of the electromagnetic field \((^3)\))
\[ \bigl(e^{(\pm)}\bigr)^2=0,\qquad e^{(+)}e^{(-)}=1,\qquad e^{(\pm)}s=0,\qquad s^2=1. \]
Following \((^1)\), after simple transformations, from \((1)\) and \((2)\) one can obtain a general expression for the wave function of a vector meson in the form
\[ \psi^{(r)}(p)=\frac{1}{m\sqrt{2}}(\hat p-im)a_\mu^{(r)}\varepsilon^{\mu 1}, \tag{4} \]
where \(\varepsilon^{AB}\) are \(10\times 10\) matrices with a single matrix element different from zero and equal to 1 at the intersection of the \(A\)-th row and the \(B\)-th column;
\[ \begin{aligned} a^{(r)}&=e^{(-)} &&\text{for } r=+1,\\ a^{(r)}&=e^{(+)} &&\text{for } r=-1,\\ a^{(r)}&=s &&\text{for } r=0. \end{aligned} \tag{5} \]
Table 1
Matrix elements for polarized vector particles \((a_1=a_1^{(r_1)},\ a_2=a_2^{(r_2)})\)
| \(Q\) | \(M_{p_1\to p_2}^{r_1\to r_2}=\dfrac{1}{2m^2}\operatorname{Sp}\{Q(\hat p_1-im)(\hat a_2\hat a_1-a_1a_2)P(\hat p_2-im)\}\) |
|---|---|
| \(I=\bar P+P\) | \(\dfrac{1}{2m^2}\{(p_1a_2)(p_2a_1)-(a_1a_2)[(p_1p_2)-m^2]\}\) |
| \(\beta_\mu\) | \(\dfrac{i}{2m}\{(a_1a_2)(p_{1\mu}+p_{2\mu})-(p_1a_2)a_{1\mu}-(p_2a_1)a_{2\mu}\}\) |
| \(\beta_\mu\beta_\nu\) | \(\dfrac{1}{2m^2}\{(p_1a_2)p_{2\mu}a_{1\nu}+(p_2a_1)a_{2\mu}p_{1\nu}-(p_1p_2)a_{2\mu}a_{1\nu}\) \(\quad-(a_1a_2)(p_{2\mu}p_{1\nu}-m^2\delta_{\mu\nu})-m^2a_{1\mu}a_{2\nu}\}\) |
| \(\beta_\mu\beta_\nu\beta_\rho\) | \(\dfrac{i}{2m}\{(a_1a_2)(p_{2\mu}\delta_{\nu\rho}+p_{1\rho}\delta_{\mu\nu})-(p_1a_2)a_{1\rho}\delta_{\mu\nu}\) \(\quad-(p_2a_1)a_{2\mu}\delta_{\nu\rho}-p_{2\mu}a_{1\nu}a_{2\rho}+a_{2\mu}a_{1\nu}p_{2\rho}-a_{1\mu}a_{2\nu}p_{1\rho}+p_{1\mu}a_{2\nu}a_{1\rho}\}\) |
| \(\beta_\mu\beta_\nu\beta_\rho\beta_\sigma\) | \(\dfrac{1}{2m^2}\{(a_1a_2)(m^2\delta_{\mu\nu}\delta_{\rho\sigma}-p_{2\mu}p_{1\sigma}\delta_{\nu\rho})\) \(\quad+[(p_1a_2)p_{2\mu}a_{1\sigma}+(p_2a_1)a_{2\mu}p_{1\sigma}-(p_1p_2)a_{2\mu}a_{1\sigma}]\delta_{\nu\rho}\) \(\quad+(p_{2\mu}a_{2\rho}-a_{2\mu}p_{2\rho})a_{1\nu}p_{1\sigma}+(a_{2\mu}p_{2\rho}-p_{2\mu}a_{2\rho})p_{1\nu}a_{1\sigma}\) \(\quad-m^2(a_{1\mu}a_{2\nu}\delta_{\rho\sigma}+a_{2\sigma}a_{1\rho}\delta_{\mu\nu}-a_{2\nu}a_{1\rho}\delta_{\mu\sigma})\}\) |
| \(\eta_5\beta_\mu\beta_\nu\beta_\rho\) | \(\dfrac{i}{2m}\{(a_1a_2)(p_{2\mu}\delta_{\nu\rho}-p_{1\rho}\delta_{\mu\nu})+(p_1a_2)a_{1\rho}\delta_{\mu\nu}\) \(\quad-(p_2a_1)a_{2\mu}\delta_{\nu\rho}-p_{2\mu}a_{1\nu}a_{2\rho}+a_{2\mu}a_{1\nu}p_{2\rho}\) \(\quad+a_{1\mu}a_{2\nu}p_{1\rho}-p_{1\mu}a_{2\nu}a_{1\rho}\}\) |
| \(\eta_5\beta_\mu\beta_\nu\) | \(\dfrac{1}{2m^2}\{(p_1a_2)p_{2\mu}a_{1\nu}+(p_2a_1)a_{2\mu}p_{1\nu}-(p_1p_2)a_{2\mu}a_{1\nu}\) \(\quad-(a_1a_2)(p_{2\mu}p_{1\nu}+m^2\delta_{\mu\nu})+m^2a_{1\mu}a_{2\nu}\}\) |
| \(\eta_5\beta_\mu\) | \(\dfrac{i}{2m}\{(a_1a_2)(p_{2\mu}-p_{1\mu})+(p_1a_2)a_{1\mu}-(p_2a_1)a_{2\mu}\}\) |
| \(\eta_5=\bar P-P\) | \(\dfrac{1}{2m^2}\{(p_1a_2)(p_2a_1)-(a_1a_2)[(p_1p_2)+m^2]\}\) |
Represent the matrix element \(M_{p_1\to p_2}^{r_1\to r_2}\), which connects the initial state \(\psi^{(r_1)}(p_1)\) and the final state \(\psi^{(r_2)}(p_2)\) of the vector particle, in the form
\[ M_{p_1\to p_2}^{r_1\to r_2} = \bar\psi^{(r_2)}(p_2)Q\psi^{(r_1)}(p_1) = \operatorname{Sp}\{Q\psi^{(r_1)}(p_1)\bar\psi^{(r_2)}(p_2)\}, \tag{6} \]
where \(Q\) is the vertex operator determining the character of the interaction. Then, using the representation of the wave functions in the form (4) and taking into account the relations
\[ e^{(\pm)*}=e^{(\mp)},\qquad e_4^{(\pm)*}=-e_4^{(\mp)},\qquad S^*=S,\qquad S_4^*=-S_4, \]
\[ \varepsilon^{ik}\eta=\varepsilon^{ik},\qquad \varepsilon^{i4}\eta=-\varepsilon^{i4},\qquad \varepsilon^{\mu\nu}=(\delta_{\mu\nu}-\beta_\mu\beta_\nu)(3-\beta^2) \]
after simple transformations we obtain (cf. (1))
\[ M_{p_1\to p_2}^{r_1\to r_2} = \frac{1}{2m^2}\operatorname{Sp}\{Q(\hat p-im)[\hat a_2^{(r_2')}\hat a_1^{(r_1)}-a_1^{(r_1)}a_2^{(r_2)}](3-\beta^2)(\hat p_2-im)\}, \tag{7} \]
where \(r_2'=-r_2\) for \(r_2=\pm1\); \(r_2'=r_2\) for \(r_2=0\).
Table 1 collects expressions for the matrix elements for \(Q\) taken in the form
\[ Q=1,\ \beta_\mu,\ \beta_\mu\beta_\nu,\ \beta_\mu\beta_\nu\beta_\rho,\ \beta_\mu\beta_\nu\beta_\rho\beta_\sigma, \tag{8} \]
\[ \eta_5\beta_\mu\beta_\nu\beta_\rho,\ \eta_5\beta_\mu\beta_\nu,\ \eta_5\beta_\mu,\ \eta_5, \]
where the matrix \(\eta_5=P-\mathbf{P}\) is the representation of the operator of total spatial reflection in the 10-dimensional space, while the operators \(P=3-\beta^2\) and \(\mathbf{P}=\beta^2-2\) \((\beta_\mu \mathbf{P}-P\beta_\mu=0)\) are projection operators that single out, respectively, the vector (4-dimensional) and tensor (6-dimensional) parts of the 10-dimensional Duffin—Kemmer space \((^4)\).
The calculation of matrix elements is carried out with the aid of the general formulas obtained in \((^4)\) for traces of products of the matrices \(\beta_\mu\) on \(P\) and \(\mathbf{P}\). Let us note that the set of vertex operators \(Q\) (8), for which the matrix elements have been calculated, covers all linearly independent elements of the basis of the Duffin—Kemmer matrix algebra (cf. \((^5)\)). Therefore Table 1 makes it possible at once to write down the expression for the matrix element of any process of interaction of vector particles in the form of a linear combination of the tabulated data and thus to unify and simplify to a considerable extent the calculations of scattering cross sections of polarized vector particles.
As an example, let us consider the scattering of a polarized vector particle by a scalar particle. Let \(\varphi(p'_1)\) and \(\varphi(p'_2)\), \(\psi^{(r_1)}(p_1)\) and \(\psi^{(r_2)}(p_2)\) be the wave functions of the initial and final states of the scalar and vector particles, respectively. Then the matrix element \(M\) entering the general formula for the differential cross section
\[ \frac{d\sigma}{d\Omega} = \frac{4e^4m^2m'^2}{p_{10}p_{20}p'_{20}p'_{20}}\, \frac{f_2^2}{y} \left| \frac{\partial (p_{20}+p'_{20})}{\partial |{\bf p}_2|} \right|^{-1} |M|^2 \tag{9} \]
can be written in the form
\[ M=\operatorname{Sp}\{\beta_\mu^{5}\varphi(p'_1)\bar{\varphi}(p'_2)\}\, \operatorname{Sp}\{\beta_\mu^{(10)}\psi^{r_1}(p_1)\bar{\psi}^{(r_2)}(p_2)\}, \]
where \(\beta_\mu^{(5)}\) and \(\beta_\mu^{(10)}\) are the \(5\times5\) and \(10\times10\) Duffin—Kemmer matrices. Using the data of Table 1 and taking into account that, according to \((^6)\),
\[ \operatorname{Sp}\{\beta_\mu^{(5)}\varphi(p'_1)\bar{\varphi}(p'_2)\} = \frac{i}{2m'}(p'_{1\mu}+p'_{2\mu}), \]
we immediately obtain
\[ M= \frac{1}{4mm'} \{(a_1a_2)(p'_1+p'_2)(p_1+p_2) -(a_1p'_1+a_1p'_2)(a_2p_1) -(a_2p'_1+a_2p'_2)(a_1p_2)\}. \tag{10} \]
For the purpose of further simplification, let us pass to the center-of-mass system. In addition, let us suppose that the circular vectors \(e^{(\pm)}\) are taken to be three-dimensional, i.e. \(e_4^{(\pm)}=0\). Then, using the permissible arbitrariness in the choice of mutually orthogonal unit vectors \(\mathbf e_1\) and \(\mathbf e_2\), orthogonal to the direction of motion of the particle and related to \(e^{(\pm)}\) by the relations
\[ e^{(\pm)}=\frac{1}{\sqrt2}(\mathbf e_1\pm i\mathbf e_2), \]
we shall assume that the vectors \(\mathbf e_1\) before and after scattering lie in the scattering plane of the particles, while the vectors \(\mathbf e_2\) are orthogonal to this plane. In this case, considering all possible cases of initial and final polarizations of the vector particle, i.e. taking different combinations of the vectors \(a_1^{(r_1)}\) and \(a_2^{(r_2)}\) (see (5)) in the general formula (9), we obtain the following expressions for the matrix elements determining the differential cross sections:
\[ M^{++}=M^{--}=\frac{1}{2mm'}[p_0p'_0+\mathbf p^2](\cos\theta+1), \tag{11} \]
\[ M^{+-}=M^{-+}=\frac{1}{2mm'}p_0p'_0(\cos\theta-1), \tag{12} \]
\[ M^{\pm0}=M^{0\pm} = \frac{1}{2mm'} \left\{ \frac{1}{m\sqrt2}\left[(p_0^2+m^2)p'_0+p_0\mathbf p^2\right] \right\}\sin\theta, \tag{13} \]
\[ M^{00}= \frac{1}{2mm'}[2p_0p'_0\cos\theta+\mathbf p^2(\cos\theta+1)]. \tag{14} \]
Hence, for unpolarized particles we shall have
\[ |M|^2=\frac{1}{3}\sum |M^{r_1 r_2}|^2 =\left(\frac{p_0 p_0'}{m m'}\right)^2 \left(1+\frac{1}{6}\frac{\mathbf{p}^4}{m^2 p_0^2}\sin^2\theta\right) + \]
\[ +\frac{1}{m'^2}\left(\frac{\mathbf{p}}{m}\right)^2 \left[ (2p_0p_0'+\mathbf{p}^2) \left(\cos^2\frac{\theta}{2} +\frac{1}{6}\frac{\mathbf{p}^2}{m^2}\sin^2\theta\right) -\frac{1}{12}\mathbf{p}^2\sin^2\theta \right]. \tag{15} \]
Using the formulas obtained, and putting \(m'\to\infty\), we immediately find the differential cross sections for scattering of a vector particle by a Coulomb center
\[ \frac{d\sigma^{\pm\pm}}{d\Omega} =\sigma_R(\cos^2\theta+1) \tag{16} \]
\[ \frac{d\sigma^{\pm 0}}{d\Omega} = \frac{d\sigma^{0\pm}}{d\Omega} = \sigma_R \left(\frac{p_0^2+m^2}{2mp_0}\right)^2 \sin^2\theta, \tag{17} \]
\[ \frac{d\sigma^{00}}{d\Omega} = \sigma_R\cos^2\theta, \tag{18} \]
\[ \frac{d\sigma}{d\Omega} = \sigma_R \left( 1+\frac{1}{6}\frac{\mathbf{p}^4}{m^2p_0^2}\sin^2\theta \right), \tag{19} \]
where
\[ \sigma_R=\frac{1}{4}e^4 \frac{p_0^2}{\mathbf{p}^4\sin^4(\theta/2)}, \]
\(\theta\) is the scattering angle.
Formulas (16) and (18) coincide with those given in \({}^{(7)}\), whereas for case (17) in paper \({}^{(7)}\) the factor \(\sin^2\theta\) was omitted.
In conclusion, we note that the proposed method of calculation may prove useful in calculations of processes involving the deuteron, regarded as a particle with spin 1.
The authors express their deep gratitude to Prof. F. I. Fedorov for guidance and useful advice.
Institute of Physics
Academy of Sciences of the USSR
Received
10 XII 1963
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