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abstract: 'The electron current tensor for the scattering of heavy photon on longitudinally polarized electron accompanied with two hard real photon is considered. The contribution of collinear and semicollinear kinematics is computed. The obtained result allows to calculate the corresponding contribution into second order radiative correction to DIS or electron–positron annihilation cross–sections with the next–to–leading accuracy.'
author:
- 'M.Konchatnij, N.P.Merenkov'
title: Current tensor with heavy photon for double hard photon emission by longitudinally polarized electron
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=-3.0cm =24.5cm =17.0cm =0.0cm =0.0cm

1.The recent polarised experiments on deep inelastic scattering \[1,2\] cover the kinematical region $y\simeq0.9$, where the electromagnetic corrections to the cross–section are extremely large . The first order QED correction have been computed in \[3,4\], and it is of the order the Born cross–section in this region. That is why the calculation of the second order QED correction becames very important for interpretation of these experiments. The first step in such calculation was done in \[5\] where one–loop corrected Compton tensor with a heavy photon had been considered. That is one of the contribution into polarized electron current tensor which appears in the second order of perturbation theory. Another contributions arise due to double hard photon emission and pair production.

Here we calculate the contribution into the polarized electron current tensor caused by two hard photon emission. We investigate double collinear kinematics and semicollinear one. This allows us to compute the corresponding second order radiative correction to different observables with the next–to–leading accuracy in the same manner as it was done, for example, for small–angle Bhabha scattering \[6\] and tagged photon cross–sections at DIS \[7\] and electron–positron annihilation \[8\].

In the Born approximation the electron current tensor with longitudinally polarized electron has a form $$\label{1Born}
L_{\mu\nu}^B = Q_{\mu\nu} + i\lambda E_{\mu\nu}\ , \ \ Q_{\mu\nu} =
-4(p_1p_2)g_{\mu\nu} + p_{1\mu}p_{2\nu} + p_{1\nu}p_{2\mu}\ ,$$ $$E_{\mu\nu} = 4\epsilon_{\mu\nu\rho\sigma}p_{1\rho}p_{2\sigma} \ ,$$ where $p_1(p_2)$ is the 4–momentum of the initial (final) electron, and $\lambda = 1 (-1)$ if the initial electron is polarized along (against) its 3–momentum direction.

In the case of single collinear photon emission the corresponding contribution into electron current tensor conserves the born structure for radiation along the scattered electron momentum direction $$\label{2 single scattered photon}
L_{\mu\nu}^{(1)f} =
\frac{\alpha}{2\pi}\biggl[\frac{1+(1+y)^2}{y}\widetilde L_0 -
\frac{2(1+y)}{y}\biggr]dyL_{\mu\nu}^B \ , \ \
y=\frac{\omega}{\varepsilon_2} \ , \ \ \widetilde L_0 =
\ln\frac{\varepsilon_2^2\theta_0^2}{m^2} \ ,$$ and gets an additional part (which is proportional to $i\lambda E{\mu\nu}$) for radiation along the initial electron momentum direction \[9\] $$\label{3 single initial photon}
L_{\mu\nu}^{(1)i} =
\frac{\alpha}{2\pi}\biggl\{\biggl[\frac{1+(1-x)^2}{x}L_0 -
\frac{2(1-x}{x}\biggr]L_{\mu\nu}^B -2xi\lambda E_{\mu\nu}\biggr\}dx \ ,
\  \ \ x=\frac{\omega}{\varepsilon_1}
\ , \ \  L_0 = \ln\frac{\varepsilon_1^2\theta_0^2}{m^2} \ .$$ In Eqs.(2) and (3) $\omega$ is the photon energy, $\varepsilon_1
(\varepsilon_2)$ is the energy of initial (final) electron, $m$ is the electron mass, and parameter $\theta_0$ defines the angular phase space of hard colinear photon. Index $i(f)$ labels initial (final) electron state.

Looking at Eq.(3) we see that the additional part does not contribute in main logarithmic approximation and has not infrared divergence. Another words, the born structure of the electron current tensor in the case of longitudinally polarized electron is disturbed only in the next–to–leading approximation due to radiation just by initial polarized electron.

In general, the contribution into current tensor $L_{\mu\nu}$ due to emission of $n$ collinear photons can be written as follows: $$\label{4 n photons}
L_{\mu\nu}^{(n)} =
\biggl(\frac{\alpha}{2\pi^2}\biggr)^n[I^{(n)}L_{\mu\nu}^B +
K^{(n)}i\lambda E_{\mu\nu}]\prod_{i=1}^n\frac{d^3k_i}{\omega_i} \ ,$$ where the quantity $K^{(n)}$ equals to zero if (and only if) all $n$ collinear photons are emitted along the scattered (unpolarized) electron momentum direction. The first term in the right side of Eq.(4) had been obtained in \[10\] with the next–to–leading accuracy. Our goal is to find also the second one with the same accuracy.

2.We use the covariant method of calculation and start from the general expression for polarized current tensor which arises due to two hard photon emission $$\label{5 tensor}
L_{\mu\nu}^{(2)} = \biggl(\frac{\alpha}{4\pi}\biggr)^2\frac{d^3k_1d^3k_2}
{\omega_1\omega_2}Sp(\hat p_2+m)Q_{\mu}^{\lambda\rho}(\hat
p_1+m)(1-\gamma_5\hat P)(Q_{\nu}^{\lambda\rho})^+ \ ,$$ where $P$ is the polarization 4–vector of initial electron. The quantity $Q_{\mu}^{\lambda\rho}$ reads $$Q_{\mu}^{\lambda\rho} = \gamma_{\mu}\frac{\hat \Delta +m}{\Delta^2-m^2}
\gamma_{\rho}\frac{\hat p_1-\hat k_1+m}{-2p_1k_1}\gamma_{\lambda} +
\gamma_{\rho}\frac{\hat p_2-\hat k_2+m}{2p_2k_2}\gamma_{\mu}
\frac{\hat p_1-\hat k_1+m}{-2p_1k_1}\gamma_{\lambda} +$$ $$\label{6 general spur}
\gamma_{\rho}\frac{\hat p_2=\hat k_2+m}{2p_2k_2}\gamma_{\lambda}
\frac{\hat\Sigma + m}{\Sigma^2-m^2}\gamma_{\mu} + (1\leftrightarrow 2) \
, \ \ \Delta = p_1-k_1-k_2 \ , \ \ \Sigma = p_2+k_1+k_2 \ .$$

For the important case of the longitudinally polarized electron in the frame of the choosen accuracy we can write the polarization vector in the form $$\label{7 polarization vector}
P = \frac{\lambda}{m}\biggl(p_1-\frac{m^2k}{p_1k}\biggr) \ ,$$ where $\lambda$ is the doubled electron helicity, and 4–vector $k$ has a components $(\varepsilon_1,-\vec p_1), \ k^2 = m^2.$ It is easy to see that $$P^2 = -1+O(m^4/\varepsilon^4) \ , \ \ Pp_1 = 0 \ .$$ Note that for calculation in the leading approximation we can neglect with the second term in the right side of Eq.(7) as it was done in \[5\].

There are four collinear regions in the case of double photon emission: $(\vec k_1,\vec k_2 \parallel \vec p_1); \ (\vec k_1,\vec k_2 \parallel
\vec p_2); \ (\vec k_1 \parallel \vec p_1, \vec k_2 \parallel \vec p_2) $ and $(\vec k_1 \parallel \vec p_2, \vec k_2 \parallel \vec p_1).$ The strightforward calculation in the region $(\vec k_1, \vec k_2 \parallel
\vec p_1$) when both hard collinear photons are emitted by the initial–state polarized electron gives $$\frac{m^4}{4}I_{ii}^{(2)} =
\frac{1+y^2}{2x_1x_2\eta_1\eta_2} +
\frac{1}{d\eta_1}\biggl[-(1-x_2)+2y\Bigl(1-\frac{x_1}{x_2}\Bigr) +
\frac{1-x_1}{x_1x_2}((1-y)(x_1-x_2)-2y)\biggr] - \frac{y\eta_2}{d^2\eta_1}
+$$ $$\label{8 I region ii}
\frac{2}{d\eta_1^2}\Bigl(x_2+\frac{2y(1-x_1)}{x_2}\Bigr) -
\frac{4y}{d^2\eta_1} + \frac{(1-y)(2y+x_1x_2)}{x_1x_2d\eta_1\eta_2} +
\frac{4y}{d^2\eta_1}\Bigl(\frac{1}{\eta_1}+\frac{1}{\eta_2}\Bigr) +
(1\leftrightarrow 2)\ ,$$

$$\frac{m^4}{4}K_{ii}^{(2)} =
\frac{2}{d\eta_1^2}\Bigl(1-x_2-x_1x_2+\frac{2yx_1^2}{x_2}\Bigr) +
\frac{1}{d\eta_1\eta_2}\Bigl(3-3y+2y^2+\frac{4x_2^2}{x_1}\Bigl) +
\frac{2}{d^2\eta_1}(3y-x_1^2-x_2^2) +$$ $$\label{9 K region ii}
+\frac{2y\eta_2}{d^2\eta_1^2}
+
\frac{4}{d^2\eta_1}\Bigr(\frac{1}{\eta_1}+\frac{1}{\eta_2}\Bigr)[(1-y)^2-
x_1x_2] + (1\leftrightarrow 2)\ , \ \ y=1-x_1-x_2 \ .$$ When writing Eqs.(8),(9) we used the following notations $$2p_1k_{1,2} =
m^2\eta_{1,2}\ , \ \ \Delta^2-m^2 = m^2d \ , \ \ x_{1,2}=
\frac{\omega_{1,2}}{\varepsilon_1} \ .$$

In the region $(\vec k_1, \vec k_2 \parallel \vec p_2)$ when both hard colinear photons are emitted by the final–state unpolarized electron we have $$\label{10 region ff}
K_{ff}^{(2)} = 0 \ , \ \ I_{ff}^{(2)} = I_{ii}^{(2)}(x_{1,2}\rightarrow
-y_{1,2}\ , \eta_{1,2}\rightarrow -\sigma_{1,2} \ , \ d\rightarrow \sigma
\ , \ y\rightarrow \eta=1+y_1+y_2) \ ,$$ where $$y_{1,2} =
\frac{\omega_{1,2}}{\varepsilon_2}\ , \ 2p_2k_{1,2} = m^2\sigma_{1,2} \ ,
\ \Sigma^2-m^2=m^2\sigma \ .$$

In accordance with the quasireal electron method \[9\] we can express the electron current tensor in the region $(\vec k_1\parallel\vec p_1,
\vec k_2\parallel\vec p_2)$ as a production of the radiation probability of the collinear photon with the energy $\omega_2$ by the scattered electron (which is the coefficient at $L_{\mu\nu}^B$ in the right side of Eq.(2) with $y=y_2$) and electron current tensor due to single photon emission by the initial electron as given by Eq.(3) with $x=x_1$. Therefore, the contribution of the regions $(\vec k_1\parallel\vec p_1,
\vec k_2\parallel\vec p_2)$ and $(\vec k_2\parallel\vec p_1,
\vec k_1\parallel\vec p_2)$ reads $$\label{11 regions if}
L_{\mu\nu}^{(2)if} =
\biggl(\frac{\alpha}{2\pi}\biggr)^2\Bigl[\frac{1+(1+y_2)^2}{y_2}\widetilde
L_0 -
\frac{2(1+y_2)}{y_2}\Bigr]\Bigl\{\Bigl[\frac{1+(1-x_1)^2}{x_1}L_0-\frac
{2(1-x_1)}{x_1}\Bigr]L_{\mu\nu}^B -$$ $$2x_1i\lambda E_{\mu\nu}\Bigr\}dy_2dx_1 +(1\leftrightarrow 2)\ .$$

In order to derive the corresponding contributions in the regions $(\vec k_1, \vec k_2\parallel\vec p_1) $ and $(\vec k_1, \vec k_2\parallel
\vec p_2) $ we have to perform the angular integration in Eq.(4) using Eqs. (8) and (9). Moreover, we can also integrate over the energy fraction $x_1 \ (y_1)$ in the region $(\vec k_1, \vec k_2\parallel\vec p_1) $  ($(\vec k_1, \vec k_2\parallel\vec p_2) $) at fixed value of the quantity $x_1+x_2=1-y \ (y_1+y_2=\eta-1)$ because of 4–momentum of the heavy photon which interacts with hadronic part of the amplitude depends on $1-y
\ (\eta-1)$ in this case.

The expressions (8) and (9) for $I_{ii}^{(2)}$ and $K_{ii}^{(2)}$ are suitable for the calculations with a power accuracy (up to terms of the order $m^2/\varepsilon_1^2$). But here we restrict ourselves with the logarithmic accuracy and therefore can omitt terms proportional to $1/d\eta_1\eta_2, \ 1/d^2\eta_1, \ 1/d^2\eta_1^2 $ and $1/d^2\eta_1\eta_2$ in the right sides of Eqs.(8) and (9). In this approximation the integration of the quantity $I_{ii}^{(2)}$ leads to (see\[10\]) $$\label{12 itegration I_ii}
\int\frac{d^3k_1d^3k_2}{\omega_1\omega_2}\frac{I_{ii}^{(2)}}{m^4} =
\pi^2\biggl[\frac{1}{2}L_0^2A(y,\delta) + L_0B(y,\delta)\biggr]dy \ ,$$ $$\label{13 leading part I for ii}
A=4\frac{1+y^2}{1-y}\ln\frac{1-y-\delta}{\delta}+(1+y)\ln y -2(1-y) \ ,$$ $$\label{14 next-to-leading part I for ii}
B=3(1-y) + \frac{3+y^2}{2(1-y})\ln^2y -\frac{2(1+y)^2}{1-y}\ln\frac
{1-y-\delta}{\delta} \ ,$$ where $\delta<<1$ is the infrared cut for the energy fraction of each photon. Analogously, the integration of the quantity $K_{ii}^{(2)}$ reads $$\label{15 K for ii}
\int\frac{d^3k_1d^3k_2}{\omega_1\omega_2}\frac{K_{ii}^{(2)}}{m^4} =
\pi^2L_0C(y,\delta)dy \ , \ C = 2(1-y)\biggl[2-\ln y - 2\ln\frac
{1-y}{\delta}\biggr]dy \ .$$ By using the Eqs.(12),(15) together with Eq.(4) we obtain $$\label{16 total ii}
L_{\mu\nu}^{(2)ii} =
\biggl(\frac{\alpha}{2\pi}\biggr)^2\Bigl[\Bigr(\frac{1}{2}L_0^2A(y,\delta)
+L_0B(y,\delta)\Bigr)L_{\mu\nu}^B + C(y,\delta)L_0i\lambda
E_{\mu\nu}\Bigr]dy$$ for the contribution of the region $(\vec k_1, \vec k_2 \parallel\vec p_1)$ into the current tensor of longitudinally polarized electron. In some applications the quantyity $y$ kept fixed (for example, for calculation of the tagged photon cross–sections). In this case we can write $\ln((1-y)/\delta)$ instead of $\ln((1-y-\delta)/\delta)$ in expressions for the quantities $A$ and $B$.

The corresponding contribution of the region $(\vec k_1, \vec k_2 \parallel \vec p_2)$ can be written as follows $$\label{17 total for ff} L_{\mu\nu}^{(2)ff} =
\biggl(\frac{\alpha}{2\pi}\biggr)^2\Bigl[(\frac{1}{2}\widetilde
L_0^2\widetilde A(\eta,\delta') + \widetilde
L_0\widetilde B(\eta,\delta')\Bigr]L_{\mu\nu}^B d\eta \ ,  \delta' =
\frac{\delta\varepsilon_1}{\varepsilon_2}\ ,$$ where $$\label{18 leading part for ff}
\widetilde A = 4\frac{1+\eta^2}{1-\eta}\ln\frac{\eta-1-\delta'}{\delta'}
-(1+\eta)\ln\eta -2(\eta-1) \ ,$$ $$\label{19 next-to-leading part for ff}
\widetilde B = 3(\eta-1) + \frac{3+\eta^2}{2(\eta-1)}\ln^2\eta
-2\frac{(1+\eta)^2}{\eta-1}\ln\frac{\eta-1-\delta'}{\delta'} \ .$$ Note that the quantities $\widetilde A$ and $\widetilde B$ can be reconstructed from the quantities $A$ and $B$ by the rule $$\widetilde A(\eta,\delta') = - A(\eta,-\delta') \ , \ \ \widetilde
B(\eta,\delta) = - B(\eta,-\delta') \ .$$

As we saw above (Eq.(3)) the additional part to the Born structure of polarized electron current tensor due to single collinear photon emission has neither colinear (does not contain large logarithm) nor infrared (is finite in the limit $x$ go to zero) singularities. But these singularities appear in the corresponding contribution due to double collinear photon emission (Eqs.(11),(16)). Nevertheless, the additional part never conribute in the leading aproximation.

The infrared parameter $\delta$ must eliminate in any physical application if photons are unobserved. Such elimination takes place because of contributions due to double virtual and soft photon emission as well as virtual and soft correction to single hard photon emission. The last contribution have been consider recenty \[5\] inside approximation $m^2=0$ which describes the large–angle photon radiation. If we put $m^2=0$ in our calculations than we will leave only with the born–like structure in Eqs.(3),(11) and (16). Moreover, quantities $B$ and $\widetilde B$ in Eqs.(16) and (17) will be changed in this approximation. We see consequently, that it needs to keep finite the electron mass to be correct inside next–to–leading approximation in any physical application with unobserved photons (for example in classical deep inelastic scattering). We conclude therefore that the results of work \[5\] have to be improved for such kind applications.

3.Let us consider double hard photon emission in semicollinear regions $\vec k_1 \parallel \vec p_1$ or $\vec p_2$ and $\vec k_2$ is arbitrary. In this situation we can use the quasireal electron method for longitudinally polarized initial electron \[9\]. In accordance with this method the contribution of the region $\vec k_1 \parallel \vec p_2$ into electron current tensor is defined by its born–like structure $L_{\mu\nu}^{^{\gamma}}$ as follows $$\label{20 semicollinear f}
L_{\mu\nu}(\vec k_1\parallel\vec p_2) =
\frac{\alpha^2}{8\pi^3}\frac{d^3k_2}{\omega_2}\frac{dy_1}{1+y_1}
\Bigl[\frac{1+(1+y_1)^2}{y_1}\widetilde L_0 - \frac{2(1+y_1)}{y_1}\Bigr]
L_{\mu\nu}^{^{\gamma}}(p_1,p_2(1+y_1),k_2) \ ,$$ where for large–angle emission tensor $L_{\mu\nu}^{^{\gamma}}$ we can use the approximation $m^2=0$ \[3,5,11\] $$\label{21 born large angle}
L_{\mu\nu}^{^{\gamma}}(p_1,p_2,k_2) = 4(B_{\mu\nu} + i\lambda
E_{\mu\nu}^{^{\gamma}}) \ ,$$ $$B_{\mu\nu} = \frac{1}{st}[(s+u)^2+(t+u)^2]\widetilde g_{\mu\nu} +
\frac{4q^2}{st}(\tilde p_{1\mu}\tilde p_{2\nu} + \tilde p_{1\nu}p_{2\mu})
\ ,$$ $$E_{\mu\nu}^{^{\gamma}} =
\frac{2\epsilon_{\mu\nu\rho\sigma}}{st}[(u+t)p_{1\rho}q_{\sigma}+(u+s)
p_{2\rho}q_{\sigma}] \ , \ \ \widetilde g_{\mu\nu} =
g_{\mu\nu}-\frac{q_{\mu}q_{\nu}}{q^2} \ ,$$ $$\tilde p_{\mu} = p - \frac{(pq)q_{\mu}}{q^2}\ , \ u=-2p_1p_2\ , \
s=2p_2k_2 \ , \ t=-2p_1k_2 \ , \ q=p_2+k_2-p_1 \ .$$

As above, the emission of collinear photon by the initial electron disturbs the Born structure of the electron current tensor just in the same manner as it done in Eq.(11) $$\label{22 semicollinear i}
L_{\mu\nu}(\vec k_1\parallel\vec p_1) =
\frac{\alpha^2}{8\pi^3}\frac{d^3k_2}{\omega_2}\frac{dx_1}{1-x_1}\biggl\{
\bigl[\frac{1+(1-x_1)^2}{x_1} L_0 - \frac{2(1-x_1)}{x_1}\bigr]
L_{\mu\nu}^{^{\gamma}}(p_1(1-x_1),p_2,k_2) -$$ $$2x_1i\lambda E_{\mu\nu}^{^{\gamma}}(p_1(1-x_1),p_2,k_2)\biggr\}$$

Formulae (20) and (22) are derived by us independently on quasireal electron method starting from the general expression for current tensor as given by Eqs.(5), (6) and (7).

When calculating the radiative corrections to polarized DIS cross–section we have to integrate over all phase space of photons. At this case the angular cut parameter $\theta_0$ is unphysical and must vanish in the sum of contributions of double collinear and semicollinear regions. At taken accuracy this fact goes to cancellation terms of the type $L_0\ln\theta_0^2$, and that can be verifyied by separation of $\ln\theta_0^2$ at integration of $L_{\mu\nu}^{^{\gamma}}(p_1(1-x_1),p_2,k_2)$ in the limit $\vec k_2\parallel \vec p_1$: $$\label{23 elimination theta}
\int\frac{d^3k_2}{\omega_2}L_{\mu\nu}^{^{\gamma}}(p_1(1-x_1),p_2,k_2\approx
x_2p_1) = -
2\pi\ln\theta_0^2dx_2\frac{y^2+(1-x_1)^2}{(1-x_1)x_2}L_{\mu\nu}^B \ .$$ Taking into account that at fixed $x_1+x_2=1-y$ $$\label{24}
\int dx_1dx_2\frac{[1+(1-x_1)^2][y^2+(1-x_1)^2]}{x_1x_2(1-x_1)^2} =
A(y,\delta)dy \ ,$$ we carry conviction that the terms of the type $L_0\ln\theta_0^2$ indeed vanish in the sum of contributions due to colinear kinematics and semicollinear one. Analogous cancellation take place of course for radiation in the final state.

Note in conclusion that the electron current tensor has an universal character. It can be used for calculation of cross–sections in different processes including most interesting DIS and $e^+e^-$–annihilation into hadrons. To obtain the corresponding cross–sections we have to multiply the electron current tensor by the hadron one. Just hadron tensor carry important information about hadronic structure and fragmentation functions \[12\], and the study of radiative corrections to electron current tensor is necessary for interpretation experimental data in terms of these hadronic functions.

Authors thank A.B.Arbuzov and I.V. Akushevich for discussion. This work supported in part (N.P.M.) by INTAS grant 93–1867 ext and Ukranian DFFD by grant N 24/379.\
\

1.  SMC., D. Adams et al., Phys.Rev. [**D 56**]{} (1997) 5330.

2.  HERMES, K. Acherstaff et al., Phys.Lett. [**B 404**]{} (1997) 383.

3.  T.V. Kukhto and N.M. Shumeiko, Nucl.Phys.[**B 219**]{} (1983) 412.

4.  I.V. Akushevich and N.M. Shumeiko, J.Phys. [**G 20**]{} (1994) 513.

5.  I.V. Akushevich, A.B. Arbuzov and E.A. Kuraev, Phys.Lett. [**B 432**]{} (1998) 222.

6.  A.B. Arbuzov et al., Nucl.Phys. [**B 485**]{} (1997) 457; Phys.Lett.[**B 399**]{} (1997) 312; N.P. Merenkov, JETP [**112**]{} (1997) 400.

7.  H. Anlauf, A.B. Arbuzov, E.A. Kuraev and N.P. Merenkov, HEP–PH/9711333.

8.  A.B. Arbuzov, E.A. Kuraev, N.P. Merenkov and L. Trentadue, (to be published in JHEP).

9.  V.N. Baier, V.S. Fadin V.A. Khoze, Nucl.Phys. [**B 65**]{} (1973) 381; V.N.Baier, V.S. Fadin V.A. Khoze and E.A. Kuraev, Phys.Rep. [**78**]{} (1981) 293.

10. N.P. Merenkov, Sov.J.Nucl.Phys. [**48**]{} (1988) 1073.

11. E.A. Kuraev, N.P. Merenkov and V.S. Fadin, Yad.Fiz. [**45**]{} (1987) 782.

12. X. Ji, Phys.Rev. [**D 49**]{} (1994) 114; R.L. Jaffe and X. Ji, Phys.Rev.Lett. [**67**]{} (1991) 552.
