# The Low theorem for diffractive bremsstrahlung and the soft photon puzzle
††thanks: Presented at “Diffraction and Low-$x$ 2022”, Corigliano Calabro
(Italy), September 24-30, 2022.

B. Z. Kopeliovich  I. K. Potashnikova  Ivan Schmidt  Departamento de Física,
Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso,
Chile

###### Abstract

The anomalous excess of small-$k_{T}$ photons radiated along with multi-hadron
production, is challenging the physics community over four decades, but no
solution has been proposed so far. We argue that the problem is rooted in the
comparison with incorrect calculations, based on the so-called “bremsstrahlung
model” (BM). It is believed to be an extension of the Low theorem from the
$2\to 2+\gamma$ process to radiative multi-particle production $2\to
n+\gamma$, where either initial, or final charged hadrons participate in
radiation. We demonstrate that this breaks down unitarity of the $S$-matrix,
thus contradicting the optical theorem.

##  1 Low theorem revisited

The process considered in the Low’s paper [1] is hadronic $2\to 2$ scattering,
supplemented with soft photon radiation,

$h_{1}+h_{2}\to h_{1}^{\prime}+h_{2}^{\prime}+\gamma,$ (1)

with corresponding 4-momenta, $p_{1},\ p_{2},\ p_{1}^{\prime},\
p_{2}^{\prime}$ and $k$. The process is depicted in Fig. 1, where the short-
range hadronic interaction is illustrated by the blob $T(s,t)$, with $s$ and
$t$ which are center-of-mass energy and 4-momentum transfer squared,
respectively.

Figure 1: Graphical representation of the process (1) with external (upper
graphs) and intrinsic (bottom) radiation.

##  2 Time scales

The process (1) is characterized by two time scales:

(i) One is the coherence length of photon radiation [2] (defined by Low [1] as
“the distance a particle can move with energy imbalance”),

$l_{c}^{\gamma}=\frac{2E_{1}x_{1}(1-x_{1})}{k_{T}^{2}+x_{1}m_{h}^{2}}.$ (2)

Here $k_{T}$ is the photon transverse momentum; $E_{1}$ is the energy of
$h_{1}$ in the rest frame of $h_{2}$. The fractional light-cone momentum
$x_{1}$ of the hadron $h_{1}$, carried by the photon,

$x_{1}=\frac{k^{\gamma}_{+}}{p^{h_{1}}_{+}}$ (3)

is invariant relative to longitudinal Lorentz boosts. However the hadron and
photon energies, $E_{1}$ and $E_{\gamma}$ respectively, are not invariant, so
they should be taken in Eq. (2) within the same reference frame, as well as
$l_{c}^{\gamma}$;

(ii) At the same time, the range of strong interactions, responsible for the
scattering in (1) is assumed to be short in comparison with $l_{c}^{\gamma}$.
Typical range of strong interaction in the target rest frame is given by the
inverse pion mass $l_{h}\sim 1/m_{\pi}\sim 1\,\mbox{fm}$. In the same frame

$l_{c}^{\gamma}=\frac{2\omega}{k_{T}^{2}+2\omega m_{h}^{3}/s},$ (4)

At high energies $l_{c}^{\gamma}$ depends on photon rapidity and usually is
pretty long, e.g. at the mid-rapidity,
$l_{c}^{\gamma}=\sqrt{s}/(2k_{T}m_{h})$. Even at large $k_{T}\sim
1\,\mbox{GeV}$ and energy of the LHC range, say $\sqrt{s}=8\,\mbox{TeV}$,
$l_{c}^{\gamma}\approx 800\,\mbox{fm}$.

Thus, we arrive at the main condition of the Low theorem,

$l_{c}^{\gamma}\gg l_{h}.$ (5)

Such a significant distinction between the radiation and interaction length
scales allows to classify radiation as external and internal, respectively.

##  3 ”External” vs ”Internal” radiation

External bremsstrahlung is related to the radiation by the incoming and
outgoing hadrons $h_{1}$ and $h_{1}^{\prime}$, respectively (upper graphs in
Fig. 1), while radiation from the interaction blob (bottom graph) we classify
as internal. The amplitude of the process (1) is proportional to
$e^{\mu}M_{\mu}$, where $e^{\mu}$ is the photon polarization vector, and
$M_{\mu}=M_{\mu}^{ext}+M_{\mu}^{int}$. The first term is infra-red divergent
and reads [1],

$M_{\mu}^{ext}=\left(\frac{p_{1\mu}^{\prime}}{p_{1}^{\prime}k}-\frac{p_{1\mu}}{p_{1}k}\right)\,T(s,t).$
(6)

Like in [1], for the sake of simplicity, we assume that $h_{1}$ is spin-less
and $h_{2}$ is electrically neutral. Moreover, we focus on the high-energy
behavior of the process (1), where the energy dependence of $T(s,t)$ is very
weak and can be neglected. Indeed, the extra term with the derivative $dT/ds$
included in [1], steeply vanishes with $s$. If $T(s)\propto s^{\epsilon}$,
$dT/ds\propto s^{\epsilon-1}$ and can be safely ignored. So expression (6) is
full and contains no other terms.

The external radiation term is enhanced at small $k$ due to the divergent pre-
factor in parentheses in Eq. (6). It comes from the propagators of the hadrons
$h_{1}^{\prime}$ and $h_{1}$ before and after interaction respectively.

The second term $M_{\mu}^{int}$ is related to $M_{\mu}^{ext}$ by charge
conservation $k^{\mu}M_{\mu}=0$. So it is finite at $k\to 0$, i.e. suppressed
in comparison with external radiation. This is the key observation of the
Low’s paper.

The Low theorem can be also treated as a formal proof of the Landau-
Pomeranchuk principle [2], which states that any variation of the electric
current within a finite distance does not affect the spectrum of radiation at
much longer coherence length, Eq. (2). Important is to keep the incoming
($l<-l_{c}^{\gamma}$) and outgoing ($l>l_{c}^{\gamma}$) currents unaffected by
the current variations on shorter length scales. This means that only
extrinsic radiation from initial and final hadrons $h_{1}$, $h_{1}^{\prime}$
matters.

##  4 Photon production in inelastic collisions

The so-called bremsstrahlung model (BM) [3], pretending to extend the Low
theorem, proven only for radiation in elastic scattering, to inelastic
collisions with multi-particle production as is depicted in Fig. 2, strictly
contradicts data on radiative multi-particle production (see [4]).

Figure 2:  Multi-hadron production and photon radiation in inelastic
collision, as is assumed within the BM.

Like in Fig. 1, the photon is assumed to be radiated by participating charge
particles, either the incoming, or outgoing. Similar to Eq. (6), each
radiation acquires an infra-red divergent Feynman propagator. Such an
“extension” of the Low theorem [3] is unjustified, and strictly contradicts
the optical theorem, as demonstrated below.

##  5 Unitarity relation

The inelastic process of multi-particle production, depicted in Fig. 2, is
connected to the imaginary part of the forward elastic amplitude
$f^{h_{1}h_{2}}_{el}$, like is illustrated in Fig. 3, due to unitarity of the
$S$-matrix, $S^{\dagger}S=I$.

Figure 3:  Unitarity relation between elastic and inelastic collisions.

The amplitude $f^{h_{1}h_{2}}_{el}$ is nearly imaginary at high energies, and
the small elastic cross section in the r.h.s. is neglected.

The essence of the unitarity relation is the connection between the inelastic
cross section of production of a final state (including production and
propagation, confinement effects, mutual interaction, etc.) with the forward
elastic amplitude. Due to unitarity, it makes no difference how to cut the
elastic amplitude, i.e. at which stage of time development of the final state
the cross section is calculated. E.g. it can be done at the early stage of
multi-gluon radiation, or later, when gluons fragment to quark-antiquark
pairs, or at the final stage of hadronization with multiple production of
hadrons. Any of these cuts satisfy the unitarity relation, due to conservation
of the total probability summed over all final states.

However, photon radiation at some later stage of final state formation, like
is illustrated in Fig. 2, breaks down the $S$-matrix unitarity. Indeed,
production of the states before, or after photon radiation in Fig. 2, have
quite different probabilities.

Moreover, one can make a unitarity cut through the radiated gluons, at the
early stage, when no charged hadrons is created, and no photons can be
radiated, unless it has happened at the initial state.

##  6 Why Feynman rules cannot be applied to the graph in Fig. 2

Feynman propagator employed for the virtual hadrons before or after
interaction in Figs. 1 and 2, displayed in Eq. (6), is in fact the coherence
length of radiation, Eq. (2). Indeed, according to the definition (2), we have

$\frac{1}{p^{2}-m_{h}^{2}}=\frac{x_{1}(1-x_{1})}{k_{T}^{2}+x_{1}m_{h}^{2}}=\frac{l^{\gamma}_{c}}{2E_{1}}.$
(7)

This expression remains correct in the infra-red regime of vanishing photon
energy $\omega\to 0$, considered by Low, because $x_{1}=\omega/E_{1}$ vanishes
as well. In this limit $k_{T}=\omega$, so (7) is $\propto 1/\omega$.

Important is that the radiation length (2), i.e. the divergent Feynman
propagator (7), becomes infinitely long in the infra-red limit. This
demonstrates a blunder of the BM, presented in Fig 2, which assumes that the
outgoing particles appear momentarily from the interaction blob of finite
size. This picture contradicts the general principle of Landau-Pomeranchuk and
the observation by Low [1] about long coherence time of production of light
particles. Within the contemporary phenomenology, like Fock-state expansion,
parton model, multi-gluon radiation, etc., hadrons are composite particles,
whose fluctuations have long life-time at high energy. So the constituent
particles (partons) appear as fluctuations long before the interaction, which
brings them to mass shell. Correspondingly, photons can be radiated either by
the incoming, or outgoing charged parton, as is illustrated in Fig. 4.

Figure 4: Space-time pattern of particle production at high energies

An approach, based on the quark-gluon string model, supplemented with the
color-dipole model for photon radiation, was developed in [5] and led to a
good description of available data.

It is also worth commenting on the choice of reference frame. In the inelastic
process depicted in Fig. 2 there is only one natural axis, the momentum
direction of the colliding hadrons. We consider only longitudinal Lorentz
boosts along this direction, so $k_{T}$ and $x_{1}$ are invariant under such
boosts. Apparently, small $k_{T}$ of the photons relative to this axis does
not mean smallness of $k_{T}$ relative the momentum of the radiating hadron.
According to Eq. (2), the infra-red behavior $1/\omega$ of the propagator
means vanishing $x_{1}$ as well. However, experimentally detected photons may
have a small $k_{T}$, but usually they are radiated with rather large $x_{1}$.

##  7 Conclusions

  * •

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

  * •

The paper by Low considered a large rapidity gap process of diffractive
excitation of a hadron, $h_{1}\to h_{1}+\gamma$, which has little to do with
multiple hadron production spanning all over the rapidity interval between
colliding hadrons.

  * •

Unitarity between the elastic amplitude and multi-particle production cross
section is independent of the stage of multi-particle formation. Photon
radiation by the final state hadrons violates unitarity, because makes the
relation dependent on the stage where the cut is made.

Acknowledgements: This work was supported in part by grant ANID PIA/APOYO
AFB220004.

## References

  * [1] F. E. Low, “Bremsstrahlung of very low-energy quanta in elementary particle collisions,” Phys. Rev. 110, 974-977 (1958) doi:10.1103/PhysRev.110.974
  * [2] L. D. Landau and I. Ya. Pomeranchuk, “Limits of applicability of the theory of bremsstrahlung electrons and pair production at high-energies,” Dokl. Akad. Nauk Ser. Fiz. 92, 535 (1953).
  * [3] A.T. Goshaw et al., Phys. Rev. Lett. 43, 1065 (1079).
  * [4] K. Reygers, ”Soft photons: an experimental overview”, ALICE workshop, Oc. 13-15, 2020, Heidelberg.
  * [5] B. Z. Kopeliovich, M. Krelina, I. K. Potashnikova, K. Reygers, ”Modeling photon radiation in soft hadronic collisions”, Diffraction and Low-x 2022, Sept. 24-30, 2022, Corigliano Calabro, Italy.