PHYSICS

A. N. TAVKHELIDZE and V. K. FEDYANIN

APPROXIMATE EQUATIONS FOR THE AMPLITUDE OF PHOTON SCATTERING ON NUCLEONS

(Presented by Academician N. N. Bogolyubov, 20 XI 1957)

The study of the process of photon scattering on nucleons can yield important information about the mesonic structure of the nucleon. In the present work, on the basis of dispersion relations for Compton scattering ((^{1})), approximate equations have been obtained for the physical amplitudes.*

1. Kinematic study of the amplitude

Let us denote the initial momenta of the nucleon and photon by (p) and (k), and the final ones by (p') and (k'). From the vectors (p) and (k), (p') and (k'), taking into account the conservation laws for energy-momentum (p+k=p'+k'), one can construct two independent scalar products (\nu) and (\nu_1):**

[
\nu=(p+p')\cdot k,\qquad \nu_1=k\cdot k'.
\tag{1}
]

From the relativistic invariance of the process, the amplitude can be represented in the form

[
\hat R=\sum_i \sum_{\nu,\mu=0}^{3}\Omega_i(\nu,\nu_1)\,\bar u(p')\,\hat R_{\mu\nu}^{\,i}\,u(p)\,e'\mu e\nu,
\tag{2}
]

where (e_\nu, e'\mu) are the polarization vectors of the initial and final photon; (\bar u(p')), (u(p)) are spinors characterizing the nucleon in the final and initial states; (\Omega_i(\nu,\nu_1)) are invariant functions possessing only an isotopic structure; (\hat R) are operators containing the spin structure of the amplitude of the process. The summation is over all independent structures.}^{\,i

From the conditions of relativistic and gauge invariance one can find the number of independent structures (\hat R_{\mu\nu}^{\,i}) and obtain their explicit expression ((^{3})). In the case of a pseudoscalar meson field the number of independent structures is equal to 10. If one takes into account the invariance of the amplitude with respect to time reversal, this number is reduced to 6. As 6 independent structures we choose the following expressions (\left(\hat R_i=\sum_{\mu,\nu} e'\mu \hat R e_\nu\right)):}^{\,i

[
\hat R_1=
\frac{1}{p\cdot k\,p\cdot k'}
\left{
e\cdot e'\,p\cdot k\,p\cdot k'
+\left(e\cdot p'\,e'\cdot p\,p\cdot k
-e'\cdot p'\,e\cdot p\,p'\cdot k\right)
\right};
]

[
\hat R_2=
\frac{(k\cdot k')^{1/2}}{p\cdot k\,p\cdot k'}
\left{
\hat e'\left(e\cdot p'\,p\cdot k-e\cdot p\,p'\cdot k\right)
+\hat e\left(e'\cdot p\,p'\cdot k'-e'\cdot p'\,p\cdot k'\right)
\right};
]

[
\hat R_3=
\frac{1}{(k\cdot k')^{1/2}}
\left{
\hat e'\,(\hat k+\hat k')\,\hat e
-\hat e\,(\hat k+\hat k')\,\hat e'
\right};
]

[
\hat R_4=
\frac{1}{p\cdot k\,p\cdot k'}
\left{
\hat e'\cdot \hat k'\left(e\cdot p'\,p\cdot k-e\cdot p\,p'\cdot k\right)
+\hat e\,\hat k\left(e'\cdot p'\,p\cdot k'-e'\cdot p\,p'\cdot k'\right)
\right};
]

* Many interesting relations for Compton scattering are contained in the works ((^{2})).

** In the system (p+p'=0), the photon energy is (E=\nu/2p_0), and the recoil momentum is (\mathbf p^2=\nu_1/2).

[
\hat R_5=\frac{(k\cdot k')^{1/2}}{p\cdot k\,p\cdot k'}
{\hat e'(\hat k+\hat k')\,\hat e\,p\cdot k'
+\hat e(\hat k+\hat k')\,\hat e'\,p\cdot k}-2\hat R_2;
]

[
\hat R_6=\frac{(k\cdot k')^{-1/2}}{p\cdot k\,p\cdot k'}
{2(\hat k+\hat k')(e\cdot p'\,e'\cdot p\,p\cdot k
+e'\cdot p'\,e\cdot p\,p\cdot k')
]
[
-2p\cdot k\,p'\cdot k[\hat e'(e\cdot p'+e\cdot p)
+\hat e(e'\cdot p+e'\cdot p')]}+\hat R_3;
]

[
a\cdot b=a_0b_0-\mathbf{ab},\qquad \hat a=a_0\gamma_0-\mathbf{a}\boldsymbol{\gamma}.
\tag{3}
]

The isotopic dependence of (\Omega_i) is obvious:

[
\Omega_i=\Omega_i^1+\Omega_i^2\tau_3
=\Omega_i^{(p)}\frac{1+\tau_3}{2}
+\Omega_i^{(n)}\frac{1-\tau_3}{2},
\tag{4}
]

where (\Omega_i^{(p)}) describes scattering on a proton, and (\Omega_i^{(n)}), scattering on a neutron.

Let us establish certain symmetry properties of the functions (\Omega_i(\nu,\nu_1)). With the aid of the (S)-matrix formalism, the matrix element of Compton scattering can be written in the form ({}^{1})

[
\langle \gamma'|s|\gamma\rangle
=\frac{i}{\sqrt{4k_0k_0'}}
\int e^{i(k'x-ky)}
\left\langle \Phi_{p'}\left|
\frac{\delta j_\nu(y)}{\delta A_\mu(x)}
\right|\Phi_p\right\rangle dx\,dy,
\tag{5}
]

[
\langle \gamma'|s|\gamma\rangle
=i\frac{(2\pi)^4}{\sqrt{4k_0k_0'}}
\delta(p+k-p'-k')\,R,
]

where (A_\mu(x)) is the electromagnetic-field operator; (j_\mu(x)=-i\dfrac{\delta S}{\delta A_\mu(x)}S^+); (|\Phi_p\rangle) is the state vector of the nucleon.

From (5) one can establish that

[
R(p,k,p',k')=R^*(p',-k,p,-k').
\tag{6}
]

Substituting (3) into (2) and taking (6) into account, we obtain the following important property of (\Omega_i(\nu,\nu_1)):

[
\Omega_i(\nu,\nu_1)=\Omega_i^*(-\nu,\nu_1),
\tag{7}
]

or

[
\operatorname{Re}\Omega_i(\nu,\nu_1)=\operatorname{Re}\Omega_i(-\nu,\nu_1),
\qquad
\operatorname{Im}\Omega_i(\nu,\nu_1)=-\operatorname{Im}\Omega_i(-\nu,\nu_1).
]

2. Dispersion relations for the relativistic amplitudes (\Omega_i). Using the analyticity property of the amplitude (R) in the upper half-plane of the variable (\nu) ({}^{4}), we have

[
\operatorname{Re}\Omega_i(\nu,\nu_1)
=\frac{1}{\pi}P\int_{-\infty}^{\infty}
\frac{\operatorname{Im}\Omega_i(\nu',\nu_1)}{\nu'-\nu}\,d\nu'.
\tag{8}
]

The region of negative (\nu) in (8) can be eliminated with the aid of (7). The method of excluding the region (0<\nu<2M\mu+\mu^2-\nu_1) for the amplitude (R) is given in work ({}^{1}). In this region the Hermitian part of the amplitude (R), (D), is written in the form

[
D=\sum_i \bar u(p')\hat R_i u(p)\Omega_i^0
=-\bar u(p'){4(\mu^2M+\mu\varepsilon_p)\hat R_1+
]

[
+(2\mu^2M+\mu\varepsilon_p)\hat R_4'
+\frac{1}{4\nu_1^{1/2}}(2\hat\mu M+e\tau_p)^2\hat R_5
+\frac{\nu_1^{1/2}}{2}\mu^2\hat R_6}u(p),
]

[
\tau_p=\frac{1+\tau_3}{2},\qquad
\tau_n=\frac{1-\tau_3}{2},\qquad
\hat\mu=\mu_p'\tau_p+\mu_n'\tau_n,
\tag{9}
]

where (\mu_p') and (\mu_n') are the anomalous magnetic moments of the proton and neutron.

Taking (7) and (9) into account, expression (8) is written as follows:

[
\operatorname{Re}\Omega_i(\nu,\nu_1)
=\Omega_i^0+\frac{1}{\pi}P
\int_{2M\mu+\mu^2-\nu_1}^{\infty}
\left(\frac{1}{\nu'-\nu}+\frac{1}{\nu'+\nu}\right)
\operatorname{Im}\Omega_i(\nu',\nu_1)\,d\nu'.
\tag{10}
]

1. Dispersion relations for physical amplitudes. The derivation of dispersion relations for the relativistic amplitudes is an intermediate stage. In order to obtain dispersion relations for the physical amplitudes (M_i(\nu,\nu_1)), we represent the Compton-scattering amplitude, expanded in three-dimensional structures (\hat r_i) (all calculations below are carried out in the center-of-mass system):

[
\hat R=\sum_{i=1}^{6} M_i(\nu,\nu_1)\,\hat r_i,
\tag{11}
]

where
[
\hat r_1=i\left[\vec\sigma(\mathbf e\times\mathbf n)\,\mathbf e'!\cdot!\mathbf n
-\vec\sigma(\mathbf e'\times\mathbf n')\,\mathbf e!\cdot!\mathbf n'\right],
\quad
\hat r_2=\mathbf e!\cdot!\mathbf n\,\mathbf e'!\cdot!\mathbf n,
\quad
\hat r_3=i\vec\sigma(\mathbf e'\times\mathbf n')\times(\mathbf e\times\mathbf n),
]
[
\hat r_4=i\vec\sigma(\mathbf n'\times\mathbf n)\,\mathbf e'!\cdot!\mathbf e,
\quad
\hat r_5=i\vec\sigma(\mathbf e\times\mathbf e'),
\quad
\hat r_6=\mathbf e!\cdot!\mathbf e',
\quad
\mathbf n=\mathbf k/|\mathbf k|,
\quad
\mathbf n'=\mathbf k'/|\mathbf k'|.
]

Substituting (3) into (2) and comparing with (11), we establish the relation between (M_i) and (\Omega_i):

[
M_i(\nu,\nu_1)=\sum_{j=1}^{6} c_{ij}(\nu,\nu_1)\,\Omega_j(\nu,\nu_1).
\tag{12}
]

One can also obtain the inverse relations:

[
\Omega_i(\nu,\nu_1)=\sum_{j=1}^{6} b_{ij}(\nu,\nu_1)\,M_j(\nu,\nu_1).
\tag{13}
]

Taking into account (10), (12), and (13), one may write the dispersion relations for the physical amplitudes (M_i):

[
\operatorname{Re} M_i(\nu,\nu_1)
=
\sum_j c_{ij}(\nu,\nu_1)\,\Omega_j^0
+
]

[
+\sum_{j,k}\frac{1}{\pi}P
\int_{2M\mu+\mu^2-\nu_1}^{\infty}
d\nu'\left(\frac{1}{\nu'-\nu}+\frac{1}{\nu'--\nu}\right)
c_{ij}(\nu,\nu_1)b_{jk}(\nu',\nu_1)\operatorname{Im}M_k .
\tag{14}
]

Because of the unwieldiness of the matrices (c_{ik}), (b_{ik}), formulas (14) are not written out explicitly. However, in the energy region where terms of order ((k_0/M)^2) may be neglected, the matrices (c_{ik}) and (b_{ik}) are simplified, and, retaining the leading terms of the expansion, we obtain approximate equations for the physical amplitudes, which are not presented here for lack of space.

1. The unitarity condition. The dispersion relations (14) relate the Hermitian and anti-Hermitian parts of the reaction amplitude. The unitarity condition (S^{+}S=1), written in the one-meson approximation, allows us to express the anti-Hermitian part of Compton scattering through photoproduction amplitudes. As a result, (14) acquire the meaning of equations. From (S^{+}S=1) and (S=1+R), in the one-meson approximation we have

[
\langle\gamma'|R+R^{+}|\gamma\rangle
=
-\langle\gamma'|R^{+}|\pi\rangle\langle\pi|R|\gamma\rangle;
\tag{15}
]

(|\pi\rangle) characterizes the nucleon and meson in the intermediate state with momenta (p'',q) and other quantum numbers.

Taking into account (5) and the definition of the photoproduction amplitude ({}^{(5)})

[
\langle\pi|R|\gamma\rangle
=
i\frac{(2\pi)^4}{\sqrt{4k_0q_0}}\,
\delta(p+k-p''-q)\,T,
\tag{16}
]

where
[
T=i\vec\sigma!\cdot!\mathbf e\,F_1
+\vec\sigma!\cdot!\mathbf m\,\vec\sigma(\mathbf n\times\mathbf e)\,F_2
+i\vec\sigma!\cdot!\mathbf n\,\mathbf m!\cdot!\mathbf e\,F_3
+i\vec\sigma!\cdot!\mathbf m\,\mathbf m!\cdot!\mathbf e\,F_4,
\quad
\mathbf m=\mathbf q/|\mathbf q|,
]
after simple calculations we obtain the relation between (\operatorname{Im}M_i) and (F_i):

[
\operatorname{Im}M_2
=
(4\pi)^{-2}\lambda
\int d\Omega
\left{
\frac{m_x}{n'x}F
+\beta F_{21}
+\frac{m_x}{n'x}F
+\beta F_{31}
-F_{22}
+\frac{n_z'^2m_y^2-m_x^2}{n_z'^2}F_{23}
+
\right.
]

[
\left.
+(\alpha n_z-\beta m_z)F_{32}
+\alpha\left[
F_{14}+F_{41}+n'zF
+\mathbf n'!\cdot!\mathbf m\,F_{34}
+\mathbf n!\cdot!\mathbf m\,F_{43}
+F_{44}
\right]
\right};
]

[
\operatorname{Im}M_6
=
(4\pi)^{-2}\lambda
\int d\Omega
\left{
F_{11}
-\mathbf n!\cdot!\mathbf m\,F_{12}
-\mathbf n'!\cdot!\mathbf m\,F_{21}
+n'zF
+
\right.
]

[
\begin{aligned}
&+m_y^2 [F_{14}+F_{41}+n'z F+n'}+n_z'F_{32zF}\,\mathbf n'!\cdot!\mathbf m\,F_{34
+\mathbf n!\cdot!\mathbf m\,F_{43}+F_{44}]};\
&n_x'^2\operatorname{Im}M_1-n_z'\operatorname{Im}M_3+\operatorname{Im}M_5
=(4\pi)^{-2}\lambda\int d\Omega{-F_{11}+\mathbf n!\cdot!\mathbf m\,F_{12}
+\mathbf n'!\cdot!\mathbf m\,F_{21}-\
&\qquad\qquad -m_x^2F_{14}-m_y^2F_{41}-n_z'F_{22}+n_x'm_x\beta F_{23}
-n_z'm_y^2F_{32}+n_x'm_xF_{24}+n_x'm_xm_y^2F_{34}};\
&\operatorname{Im}M_3-n_z'\operatorname{Im}M_5
=(4\pi)^{-2}\lambda\int d\Omega{n_z'F_{11}-\mathbf n'!\cdot!\mathbf m\,F_{12}
-\mathbf n!\cdot!\mathbf m\,F_{21}+\
&\qquad\qquad +n_z'm_y^2F_{14}-n_x'm_x\beta F_{41}+F_{22}+m_y^2F_{23}
+n_x'^2\beta^2F_{32}-n_x'^2\beta F_{31}-n_x'^2\beta m_y^2F_{34}};\
&2\operatorname{Im}M_1+n_z'\operatorname{Im}M_3+\operatorname{Im}M_5
=(4\pi)^{-2}\lambda\int d\Omega\left{-F_{11}
+\left(\beta-m_x\frac{n_z'}{n_x'}\right)F_{12}-\right.\
&\qquad\qquad -\left(\beta n_z'-\frac{m_x}{n_x'}\right)F_{21}
-m_x\frac{n_z'}{n_x'}F_{13}-\beta n_z'F_{31}
-\mathbf n'!\cdot!\mathbf m\,\frac{m_x}{n_x'}F_{14}
-\mathbf n!\cdot!\mathbf m\,\beta F_{41}+n_z'F_{22}+\
&\qquad\qquad +\left(n_z'm_y^2+\mathbf n!\cdot!\mathbf m\,\frac{m_x}{n_x'}\right)F_{23}
+\left(n_z'm_y^2+\beta\,\mathbf n'!\cdot!\mathbf m\right)F_{32}+\
&\qquad\qquad \left.+\frac{m_x}{n_x'}F_{24}+\beta F_{42}
+\alpha n_x'^2F_{33}+\alpha n_x'^2\left(\beta F_{34}+\frac{m_x}{n_x'}F_{43}\right)\right};\
&\operatorname{Im}(M_3+M_4)
=(4\pi)^{-2}\lambda\int d\Omega\left{-\frac{m_x}{n_x'}F_{12}
-\beta F_{21}+F_{22}+m_y^2(F_{23}+F_{32})+\right.\
&\qquad\qquad \left.+m_y^2F_{33}+m_y^2\left(\beta F_{34}+\frac{m_x}{n_x'}F_{43}\right)\right}.
\end{aligned}
\tag{17}
]

Here (n_x'=\sin\theta,\ n_z'=\cos\theta=\mathbf n!\cdot!\mathbf n',\ m_x=\sin\theta'\cos\varphi',\ m_y=\sin\theta'\sin\varphi',)
(m_z=\mathbf n!\cdot!\mathbf m=\cos\theta',\ \mathbf n'!\cdot!\mathbf m=\cos\theta''=\cos\theta\cos\theta'+\sin\theta\sin\theta'\cos\varphi',\ d\Omega=)
[
=\sin\theta'\,d\theta'\,d\varphi',\quad
\beta=\frac{(\mathbf m\times\mathbf n')y}{n_x'},\quad
\alpha=\frac{1}{n_x'}\left[m_xm_z+\frac{n_z'}{n_x'}(m_y^2-m_x^2)\right],\quad
F=
]
[
=F_i^{+}(\mathbf n'!\cdot!\mathbf m)F_k(\mathbf n!\cdot!\mathbf m),\quad
\lambda=\frac{\omega E_k}{W}.
]

In conclusion, we express our deep gratitude to Academician N. N. Bogolyubov and A. A. Logunov for valuable discussions and constant interest in the work.

Joint Institute
for Nuclear Research

Received
14 XI 1957

References Cited

1. N. N. Bogolyubov, D. V. Shirkov, DAN, 113, No. 3 (1957).
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3. M. Kawaguchi, N. Mugibayashi, Progr. Theor. Phys., 8, 2 (1952).
4. A. A. Logunov, P. S. Isaev, Nucl. Phys. (in press).
5. A. A. Logunov, L. D. Solov’ev, A. N. Tavkhelidze, Nucl. Phys., 4, 497 (1957).