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Line 259: − + − + − + − + − + − + − + + − + − <math>\rho_{\lambda_i \lambda_i'} \propto \rho_{\lambda_i \lambda_i'}</math>. The same property holds for <math>\rho_{\lambda_f \lambda_f'}</math> by definition of the summation over the final states. As a result, the term characterizing the target proton's transition with the emission of the Reggeon factorizes, allowing us to drop indices for the proton states in the T matrix: − − :<math> − T_{(f)(i)} = − T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1}) − (\epsilon_\gamma \epsilon_R t)} − </math> + − + + − + − + − ::<math>=\sum_{\lambda_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho,m_X} + − \langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle+ − \langle J_X M_X \epsilon_X | V |+ − \epsilon_\gamma; J_R \lambda_R \epsilon_R; s,t \rangle+ − − :::::::<math>\times − \langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1 , \lambda_{b_1} \rangle − \langle \mathbf{q}_\rho \lambda_\rho 0| U_\omega | 1 , \lambda_\omega \rangle − \langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle − − Line 256:
Line 288: − a^J_{LS}(m_X) = a^J_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)+ − b^1_{L1}(m_{b1}) = b^1_{L1} BW_L(m_{b1};m^0_{b1},\Gamma^0_{b1})+ − c^1_{LJ}(m_{\omega}) = c^1_{LJ} BW_L(m_\omega;m^0_\omega,\Gamma^0_\omega)+ − f^L_{L0}(m_{\omega}) = f^L_{L0} BW_L(m_\rho;0,\infty)+ − The di-pion system in the decay of the <math>\omega</math> is non-resonant, so its width is entered as infinity. The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:+ Line 283:
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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 03:22, 25 October 2011
, 03:22, 25 October 2011→Angular Distribution of Two-Body Decay
Let's begin with a general amplitude for the two-body decay of a state with angular momentum quantum numbers ''J'',''m''. Specifically, we want to know the amplitude of this state for having daughter 1 with momentum direction <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.
Let's begin with a general amplitude for the two-body decay of a state with angular momentum quantum numbers ''J'',''m''. Specifically, we want to know the amplitude of this state for having daughter 1 with momentum direction <math>\Omega=(\phi,\theta)</math> in the center of mass reference frame, and helicity <math>\lambda_1</math>, while daughter 2 has direction <math>-\Omega=(\phi+\pi,\pi-\theta)</math> and helicity <math>\lambda_2</math>.
Let U be the decay operator from the initial state into the given 2-body final state. Intermediate between the at-rest initial state of quantum numbers (qn) J,m and the final plane-wave state is a basis of outgoing waves describing the outgoing 2-body state in a basis of good J,m and helicities. Insertion of the complete set of intermediate basis vectors, and summing over all intermediate J,m gives
Let U be the decay operator from the initial state into the given 2-body final state. Intermediate between the at-rest initial state of quantum numbers (qn) J,M and the final plane-wave state is a basis of outgoing waves describing the outgoing 2-body state in a basis of good J,m and helicities. Insertion of the complete set of intermediate basis vectors, and summing over all intermediate J,m gives
:<math>
:<math>
\langle \Omega \lambda_1 \lambda_2 | U | J M \rangle
\langle \Omega \lambda_1 \lambda_2 | U | J M \rangle
:<math>
:<math>
=\sum_{L,S}
=\sum_{L,S}
\left[ \sqrt{\frac{2J+1}{4\pi}} D_{m \lambda}^{J *}(\Omega) \right]
\left[ \sqrt{\frac{2J+1}{4\pi}} D_{M \lambda}^{J *}(\Omega) \right]
\left[ \sqrt{\frac{2L+1}{2J+1}}
\left[ \sqrt{\frac{2L+1}{2J+1}}
\left(\begin{array}{cc|c}
\left(\begin{array}{cc|c}
Acting on a state of good ''J,m'', the reflectivity operator has a particularly simple effect.
Acting on a state of good ''J,m'', the reflectivity operator has a particularly simple effect.
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-m} | J \; -M \rangle </math>
:<math>\mathbb{R}| J M \rangle = P(-1)^{J-M} | J \; -M \rangle </math>
where P is the intrinsic parity of the system.
where P is the intrinsic parity of the system.
The eigenstates of the reflectivity operator are formed out of states of good ''J,M'' as follows.
The eigenstates of the reflectivity operator are formed out of states of good ''J,M'' as follows.
:<math>
:<math>
\langle J M \epsilon|\mathbb{R}^{-1} V \mathbb{R}|
\langle J M \epsilon|\mathbb{R}^{-1} V \mathbb{R}|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle =
\epsilon_\gamma ; \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle =
\epsilon \epsilon_\gamma \epsilon_R \langle J M \epsilon|V|
\epsilon \epsilon_\gamma \epsilon_R \langle J M \epsilon|V|
\epsilon_\gamma ; J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
</math>
</math>
It is convenient to adopt the Gottfried Jackson frame. In particular, we boost into the reference frame of the produced resonance and orient the coordinate system such that the photon is in the +z direction and the x-axis is co-planar to the recoiling proton, thus defining xz as the production plane. A consequence of this choice is that <math>m=\lambda_\gamma-\lambda_R</math>
It is convenient to adopt the Gottfried Jackson frame. In particular, we boost into the reference frame of the produced resonance and orient the coordinate system such that the photon is in the +z direction and the x-axis is co-planar to the recoiling proton, thus defining xz as the production plane. A consequence of this choice is that <math>m=\lambda_\gamma-\lambda_R</math>
It is convenient to express above matrix element as
:<math>
\langle J M \epsilon|V|
\epsilon_\gamma ; \lambda_R \epsilon_R ; \Omega_0 \rangle
= v_{J,m,\epsilon;\epsilon_\gamma; \lambda_R,\epsilon_R}
</math>
so that the indexed coefficient ''v'' specifies the couplings together with the consequences of angular momentum and parity conservation. The function ''v'' is implicitly dependent upon the kinematical variables ''s'' and ''t''. This dependence will be made explicit in a following section, after the matrix element for the baryon vertex has been studied.
To express the initial photon linear polarization state in the reflectivity basis, we relate the linear polarization bases in the laboratory and Gottfried-Jackson coordinate systems:
To express the initial photon linear polarization state in the reflectivity basis, we relate the linear polarization bases in the laboratory and Gottfried-Jackson coordinate systems:
\end{array}\right)
\end{array}\right)
\right]
\right]
u_{L_X 1}^{X:J_X}
</math>
</math>
\end{array}\right)
\end{array}\right)
\right]
\right]
u_{L_{b_1} 1}^{b_1:1}
</math>
</math>
\end{array}\right)
\end{array}\right)
\right]
\right]
u_{L_\omega J_\rho}^{\omega:1}
</math>
</math>
\end{array}\right)
\end{array}\right)
\right]
\right]
u_{L_\rho\,0}^{\rho:J_\rho}=
</math>
</math>
:::::::::<math>
:::::::::<math>
=Y_{M_\rho}^{J_\rho}(\Omega_{\pi})
=Y_{M_\rho}^{J_\rho}(\Omega_{\pi})
u_{J_\rho\,0}^{\rho:J_\rho}
</math>
</math>
\left(\frac{q_\pi dm_\rho}{16\pi^3}\right)
\left(\frac{q_\pi dm_\rho}{16\pi^3}\right)
</math>
</math>
where <math>p_i\,</math> [<math>p_f\,</math>] is the target [recoil] nucleon momentum in the center of mass frame of the overall reaction. The explicit kinematic factors from the initial-state flux and the density of final states for each of the decays are not factored into the T matrix so that we can make sure that it explicitly respects unitarity in each partial wave. In terms of the individual decay matrix elements introduced earlier, the T matrix element can be written as
where <math>p_i\,</math> [<math>p_f\,</math>] is the target [recoil] nucleon momentum in the center of mass frame of the overall reaction. In terms of the individual decay matrix elements introduced earlier, the T matrix element can be written as
:<math>
:<math>
T_{(f)(i)} =
T_{(f)(i)} =
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f)
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b_1} \mathbf{p}_f \epsilon_f)
(\epsilon_\gamma \epsilon_R t \lambda_i)}=
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=
</math>
</math>
:::<math>
=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
=\sum_{R,\lambda_R,\epsilon_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b_1}\lambda_{b_1} 0
| U | \epsilon_\gamma; J_R \lambda_R \epsilon_R;s,t \rangle
| UV | \epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
</math>
:::::::::<math> \times
:::::::::<math> \times
\langle J_R \lambda_R \epsilon_R;s,t; \mathbf{p_f}, \lambda_f | W | \mathbf{p_i}, \lambda_i\rangle
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
To obtain the second line in the above equation, we factorized the T operator into two vertex factors ''UV'' and ''W'', and inserted between them a sum over a complete set of intermediate exchanges ''R'' represented as plane waves moving along the ''-z'' axis. The upper vertex operator has been written as ''UV'' in anticipation of its further factorization into the primary resonance production operator ''V'' and its decay operator ''U''. Polarizations of all particles are represented by their respective reflectivity quantum numbers <math>\epsilon</math>. For the nucleon, the reflectivity is a complete description of its spin state. For reactions involving higher spin baryons, it would need to be supplemented by an additional <math>|\lambda|</math> quantum number.
:<math>
T_{(f)(i)} =
\sum_{
\begin{array}{c}
\scriptstyle
R,\lambda_R,\epsilon_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho\\
\scriptstyle
X,M_X,\epsilon_X;\epsilon_i,\epsilon_f
\end{array}}
\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle \langle \mathbf{q}_\omega \lambda_\omega 0| U_{b_1} | 1 , \lambda_{b_1} \rangle
</math>
:::::::::<math>\times
\langle \mathbf{q}_\rho \lambda_\rho 0| U_\omega | 1 , \lambda_\omega \rangle \langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle
</math>
:::::::::<math>\times
\langle J_X M_X \epsilon_X | V |
\epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
:::::::::<math>\times
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
Parity conservation requires that <math>\epsilon_X=\epsilon_\gamma \epsilon_R</math> and <math>\epsilon_R=\epsilon_i\epsilon_f</math>. The last two matrix elements in the expression above for <math>T_{fi}</math> not known ''a priori'', so we parameterized them into a pair of unknown functions ''v(s,t)'' and ''w(s,t)''.
:<math>\displaystyle
v^{X,M_X,\epsilon_X}_{\lambda_R,\epsilon_R}(s,t) =
\langle J_X M_X \epsilon_X | V |
\epsilon_\gamma; \lambda_R \epsilon_R; \Omega_0\rangle
</math>
:<math>\displaystyle
w_{\lambda_R \epsilon_R; \epsilon_i}(s,t) =
\langle \lambda_R \epsilon_R; \Omega_0;\mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
</math>
:<math>
:<math>
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} \propto
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} \propto
\frac{1}{2}\sum_{\lambda_i \lambda_f \lambda_i' \lambda_f'}
\sum_{\epsilon_\gamma \epsilon_\gamma' \epsilon_i \epsilon_f \epsilon_i' \epsilon_f'}
\rho_{\lambda_i \lambda_i'}
\rho_{\epsilon_\gamma \epsilon_\gamma'}
\rho_{\lambda_f \lambda_f'}
\rho_{\epsilon_i \epsilon_i'}
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f)
\delta_{\epsilon_f \epsilon_f'}
(\epsilon_\gamma \epsilon_R t \lambda_i)}
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f)
T^*_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f')
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}
(\epsilon_\gamma \epsilon_R t \lambda_i')}
T^*_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f')
(\mathbf{k}_\gamma \epsilon_\gamma' \mathbf{p}_i \epsilon_i')}
</math>
</math>
where density matrices <math>\rho</math> represent weights of proton's states in the summations. The unpolarized target presents an initial state with both helicities equally likely, resulting in
where density matrices <math>\rho</math> represent the initial state particles' spin states. The unpolarized target presents an initial state with both reflectivities equally likely, resulting in
:<math>\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}</math>.
The remaining production and decay matrix elements can be further broken up into a product of individual decay amplitudes,
In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex: <math>\epsilon_R=\epsilon_i \epsilon_f</math>
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> survive in the sum over exchange quantum numbers. The quadratic sum expression above for the differential cross section invokes a double-sum over all of the internal quantum numbers that have been introduced, plus the spins of the initial and final nucleons. This sum is of the generic form
:<math>
:<math>
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
\sum_{X,M_X,\cdots} \cdots \sum_{\epsilon_i\epsilon_f}
| U | \epsilon_\gamma; J_R \lambda_R \epsilon_R; s,t \rangle
|w_{\lambda_R \epsilon_R; \epsilon_i \epsilon_f}|^2
</math>
</math>
Note that the measured cross section only depends on the summed
modulus squared of the ''w'' coefficients, independent of the couplings
to the individual nucleon helicity states. Because of this, the sum over nucleon helicities can be dropped, and the ''w'' factors absorbed into
the ''v'' coefficients. Thus the final expression for the differential cross section contains no reference to the quantum numbers <math>\epsilon_i, \epsilon_f</math>, nor to any ''w'' coefficients. It is important to recognize that this simplification of the cross section is only possible in the case of an unpolarized target and no recoil polarimetry.
</math>
</math>
=== Mass dependence ===
=== Mass dependence ===
Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> have already been written down above, in terms of the unknown mass-dependent factors ''a'', ''b'', ''c'', and ''f''. The mass dependence of the ''a'' factor can be written in terms of a standard relativistic Breit-Wigner resonance lineshape as follows, although often its mass dependence is determined empirically by binning in the mass of ''X'', and fitting each bin independently. In a global mass-dependent fit, the central mass and width of ''X'' are free parameters in the fit. The remaining factors ''b'', ''c'', and ''f'' are assigned standard resonance forms, with their central mass, width and partial wave matrix elements fixed to agree with the established values for these resonances.
Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> have already been written down above, in terms of the unknown mass-dependent factors ''a'', ''b'', ''c'', and ''f''. The mass dependence of the ''a'' factor can be written in terms of a standard relativistic Breit-Wigner resonance lineshape as follows, although often its mass dependence is determined empirically by binning in the mass of ''X'', and fitting each bin independently. In a global mass-dependent fit, the central mass and width of ''X'' are free parameters in the fit. The remaining factors ''b'', ''c'', and ''f'' are assigned standard resonance forms, with their central mass, width and partial wave matrix elements fixed to agree with the established values for these resonances.
:<math>
:<math>
u^{X:J}_{LS}(m_X) = u^{X:J}_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)
</math>
</math>
:<math>
:<math>
u^{b_1:1}_{L1}(m_{b1}) = u^{b_1:1}_{L1} BW_L(m_{b1};m^0_{b1},\Gamma^0_{b1})
</math>
</math>
:<math>
:<math>
u^{\omega:1}_{LJ}(m_{\omega}) = u^{\omega:1}_{LJ} BW_L(m_\omega;m^0_\omega,\Gamma^0_\omega)
</math>
</math>
:<math>
:<math>
u^{\rho:L}_{L0}(m_{\omega}) = u^{\rho:L}_{L0} BW_L(m_\rho;m^0_\rho,\Gamma^0_\rho)
</math>
</math>
The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:
:<math>
:<math>
BW_L(m; m_0,\Gamma_0)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)}
BW_L(m; m_0,\Gamma_0)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)}
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
q(m) = \sqrt{\left(\frac{m^2+m_1^2-m_2^2}{2m}\right)^2-m_1^2}
</math>
</math>
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature.
The functions <math>F_L(q)</math> are the angular momentum barrier factors that are given in the literature. The first few are listed below with <math>z=[q/(197\mathrm{MeV/c})]^2</math>
:<math>\displaystyle F_0(q)=1</math>
:<math>F_1(q)=
\sqrt{\frac{2z}{z+1}}
</math>
:<math>F_2(q)=
\sqrt{\frac{13z^2}{(z-3)^2+9z}}
</math>
:<math>F_3(q)=
\sqrt{\frac{277z^3}{z(z-15)^2+9(2z-5)^2}}
</math>
===Describing s and t dependence===
===Describing s and t dependence===
The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:
The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:
:<math>
:<math>
T_{\epsilon_\gamma} = \frac{\sqrt{-\epsilon_\gamma}}{2}
T_{\epsilon_\gamma} = \sqrt{\frac{-\epsilon_\gamma}{2}}
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
\left( T_{-1}e^{-i\alpha} - \epsilon_\gamma T_{+1}e^{i\alpha} \right)
</math>
</math>
T_{\pm 1}=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
T_{\pm 1}=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
| U | (1\; \pm 1)_{\mathrm{lab}}; J_R \lambda_R \epsilon_R;s,t \rangle
| U | (1\; \pm 1)_{\mathrm{lab}}; \lambda_R \epsilon_R;s,t \rangle
</math>
</math>