Modeling photon radiation in soft hadronic collisions ^††thanks: Presented at “Diffraction and Low-$x$ 2022”, Corigliano Calabro (Italy), September 24-30, 2022.

Within the parton model radiation of a heavy photon of mass $M$ (Drell-Yan) in the target rest frame, based on the factorization theorem, has the form,

	$\displaystyle\frac{d^{4}\sigma}{dM^{2}dx_{F}dk_{T}^{2}}=\frac{\alpha_{em}}{3\pi M^{2}}\frac{x_{1}}{x_{1}+x_{2}}$		$\displaystyle\int\limits_{x_{1}}^{1}\frac{d\alpha}{\alpha^{2}}\sum\limits_{f}Z_{f}^{2}\left\{q_{f}\left(\frac{x_{1}}{\alpha}\right)+\bar{q}_{f}\left(\frac{x_{1}}{\alpha}\right)\right\}$
	$\displaystyle\times\frac{d\sigma(q_{f}N\to\gamma^{*}X)}{d\ln{\alpha}d^{2}k_{T}},$				(1)

with the standart notations, $\alpha=p_{+}^{\gamma}/p_{+}^{q}$; $x_{1}x_{2}=M^{2}/s$; $x_{1}-x_{2}=x_{F}$.

The hard perturbative scale is imposed by the large invariant mass $M$ of the photon (dilepton). The sum of the (anti)quark distribution function in (1) is given by the well measured proton stricture function $F_{2}(x,M^{2})$. The parton distribution functions in the colliding hadrons are illustrated by a parton comb in Fig. 1.

Extrapolation of expression (1) to the soft regime is a challenge, since involves unknown nonperturbative effects. That can be done only within models. For the quark distribution function we rely on the popular and successful quark-gluon string model (QGSM) [4, 5], or a similar dual parton model [6, 7]. Both models assume Regge behavior at the end-points $x\to 1$ or $x\to 0$, of the quark distribution functions, and a simple, but ad hoc, interpolation at medium $x$. We skip the simple, but lengthy expressions. The details can be found e.g. in [5].

The last factor in the radiation cross section (1) $d\sigma(q_{f}N\to\gamma X)/d\ln{\alpha}/d^{2}k_{T}$ is calculated at the soft scale within the color dipole phenomenology [8]-[13], adjusted to precise data on DIS from HERA,

	$\displaystyle\frac{d\sigma(qN\to\gamma X)}{d\ln{\alpha}d^{2}k_{T}}$	$\displaystyle=$	$\displaystyle\frac{1}{2\pi)^{2}}\int d^{2}r_{1}d^{2}r_{2}\,\exp[i\vec{k}_{T}(\vec{r}_{1}-\vec{r}_{2})]$		(2)
		$\displaystyle\times$	$\displaystyle\Psi^{*}_{\gamma q}(\alpha,\vec{r}_{1})\Psi_{\gamma q}(\alpha,\vec{r}_{2})\,\sigma_{\gamma}(\vec{r}_{1},\vec{r}_{2},\alpha),$

where

$$\sigma_{\gamma}(\vec{r}_{1},\vec{r}_{2},\alpha)={1\over 2}\left\{\sigma_{\bar{q}q}(\alpha r_{1})+\sigma_{\bar{q}q}(\alpha r_{2})-\sigma_{\bar{q}q}[\alpha(\vec{r}_{1}-\vec{r}_{2})]\right\}.$$

(3)

The quark-photon distribution function reads,

$$\Psi_{\gamma q}(\alpha,\vec{r})=\frac{\sqrt{\alpha_{em}}}{2\pi}\,\chi_{f}\hat{O}\chi_{i}K_{0}(\alpha m_{q}r),$$

(4)

and

$$\hat{O}=\vec{e^{*}}\left\{im_{q}\alpha^{2}\left[\vec{n}\times\vec{\sigma}\right]+\alpha\left[\vec{\sigma}\times\vec{\nabla}\right]-i(2-\alpha)\vec{\nabla}\right\}.$$

(5)

The $\bar{q}q$ dipole-nucleon cross section $\sigma_{\bar{q}q}(r)$ in (3) has been parametrized and fitted to DIS and photoproduction data from NMC and HERA. The details can be found in [10].

Combining the QGSM distribution functions with the cross section (2) results in the radiation cross section, which is parameter free (we do not fit the data to be explained), either in the shape of $k_{T}$ distribution, or in the absolute values. We assumed a primordial transverse momentum distribution of the incoming quarks to have a Gaussian shape with $\sqrt{\langle q_{T}^{2}\rangle}=0.35\,\mbox{GeV}$. Correspondingly the radiated photon acquires additional transverse momentum $\vec{k}^{\prime}_{T}=\alpha\vec{q}_{T}$.

The results of calculations are compared with data on the radiative cross section of $\pi^{+}p\to\gamma+X$ from the NA22 experiment at $E_{lab}=250\,\mbox{GeV}$ in Fig. 2, and from WA91/WA83 experiments at $E_{lab}=280\,\mbox{GeV}$ in Fig. 3.

We see no sizable deviation from data at small $k_{T}$, i.e. no anomalous enhancement of soft photons.

At somewhat higher energy $E_{lab}=450\,\mbox{GeV}$ [17] our calculations depicted by solid curve in Fig. 4, apparently overestimate the data of the WA102 experiment.

However, the experiment had specific cuts, namely, events with number of charge tracks $N_{ch}>8$, were excluded. To calculate the multiplicity distribution we assume the Poisson distribution of the number of unitary cut Pomerons, and employed the result of QGSM [18]. So we obtained a suppression factor

$$\delta=\frac{\sum\limits_{N_{ch}=0}^{8}}{\sum\limits_{N_{ch}=0}^{\infty}}=0.39.$$

(6)

Dashed curve, which incorporate this factor well agrees with data.

• •

  The observed enhancement of low-$k_{T}$ photons in comparison with incorrect calculations, should not be treated as a puzzle.

• •

  The parton model description of photon radiation is extrapolated to to the soft scale regime. The (anti)quark distribution functions are evaluated within the popular quark-gluon string model, based of Regge phenomenology.

• •

  Soft photon bremsstrahlung by projectile quarks is calculated within the color-dipole model. The quark-antiquark dipole cross section is fitted to DIS and soft photo-production data in a wide range of transverse dipole separations and energies.

Acknowledgements: This work of B.Z.K. and I.K.P. was supported in part by grant (Chile) ANID PIA/APOYO AFB220004.
The work of M.K. was supported by the project of the International Mobility of Researchers - MSCA IF IV at CTU in Prague CZ.02.2.69/0.0/0.0/20_079/0017983, Czech Republic.