Relativistic Coulomb $\displaystyle\bm{S}$-factor of Two Spinor Particles with Arbitrary Masses

It is well known from quantum electrodynamics (QED) that the so-called Coulomb resummation factor plays an important role in describing the system of two charged particles near the threshold Schwinger70 . The resummation performed on the basis of the nonrelativistic Schrödinger equation with the Coulomb potential, $\displaystyle V(r)=-\alpha/r$, where $\displaystyle\alpha$ is the fine structure constant, leads to the Sommerfeld–Gamov–Sakharov factor Sommerfeld ; Gamov:28 ; Sakharov which, for two particles of equal masses, reads as:

where $\displaystyle v_{\rm nr}$ denotes the velocity of each particle in the center-of-mass frame. At small values of $\displaystyle\;{\alpha/v_{\rm nr}}$, the $\displaystyle S$-factor (1) behaves as $\displaystyle S_{\rm nr}\,\sim 1+\pi\alpha/(2v_{\rm nr})$, which implies that at low velocities the corresponding cross sections increase. As originally mentioned by Sommerfeld Sommerfeld , this enhancement is directly related to the wave function of the relative motion in the Coulomb field evaluated at the origin, $\displaystyle|\psi(0)|^{2}$, i.e. to the probability to find two interacting particles close to each other. Such an increase of the $\displaystyle S$-factor is often referred to as the Sommerfeld effect. A similar phenomenon was also predicted and then confirmed by Gammov Gamov:28 in nuclear reactions when interacting particles overcome the Coulomb barrier and by Sakharov Sakharov in electron-positron pair production. Intensive studies of the Sommerfeld-effect demonstrate the crucial role it plays in understanding a large variety of inelastic processes with heavy particles moving slowly in a thermal environment, see, e.g., Refs. Hisano:2003ec ; Cassel:2009wt ; Kim:2019qix ; Mahbubani:2019shd ; Dremin:2020vdb ; ArkaniHamed:2008qn ; Binder:2019erp . These processes are encountered in many physical situations, and nowadays there is renewed interest in detailed investigations of the Sommerfeld effect. A classic example is given by astrophysical nuclear reactions taking place within the electromagnetic plasma of stars. In particle physics, one may mention heavy Dark Matter particles produced by pair annihilation in the early Universe. If Dark Matter in the Universe contains interactions with the Standard Model particles, the annihilation may give visible imprints in the cosmic rays Hisano:2003ec . Within quantum chromodynamics (QCD), an increase in the $\displaystyle S$-factor may occur in heavy quark and anti-quark pair annihilation into light quarks and gluons within a quark-gluon plasma environment generated in heavy ion collision experiments. Bound states contribute to the annihilation process, and enhancement factors of up to $\displaystyle\sim 100$ can be encountered, see, e.g., Ref. Kim:2019qix . Other examples where the Sommerfeld effect is crucial, include threshold production of heavy states at colliders and partial decay rates when the products have large phase space suppression. The Sommerfeld effect in boosting the dark matter signal with Coulomb resonances has been considered, to some extent, in Ref. Mahbubani:2019shd ; Cassel:2009wt , where the $\displaystyle S$-factor was computed within a nonrelativistic approach for attractive and repulsive Coulomb and Yukawa (between neutral particles) potentials. Details of the theoretical consideration of the Sommerfeld enhancement in Dark Matter can be found in, e.g., Refs. ArkaniHamed:2008qn ; Binder:2019erp .

Another manifestation of the enhancement of the Sommerfeld-Gamow-Sakharov factor can be clearly seen in abundant creations of di-lepton pairs with low masses in heavy-ion collisions Dremin:2020vdb , which is directly related to the physics at the NICA and LHC colliders. It should be noted that most of the theoretical studies of the $\displaystyle S$-factor have been performed within nonrelativistic approaches originating from the Schrödinger equation. Coming back to Eq. (1) one shall stress that, in spite of the expansion of the $\displaystyle S$-factor in a power series $\displaystyle(\alpha/v)^{n}$ with respect to the coupling constant $\displaystyle\alpha$ reproduces the threshold singularities of the Feynman diagrams, in the threshold-near region, $\displaystyle v\to 0$, the parameter $\displaystyle\alpha/v$ can no longer be adequate to cut off the perturbative series. Consequently, the $\displaystyle S$-factor should be taken into account in its entirety.

A similar situation also arises also in QCD because in describing a quark-antiquark system near the threshold, $\displaystyle s_{th}=(m_{q}+m_{\bar{q}})^{2}$, the expansion parameter $\displaystyle\alpha_{s}/v$ becomes singular when $\displaystyle v\to 0$ App-Politzer75 ; PQW . Here $\displaystyle\alpha_{s}$ denotes the strong coupling constant, and the quark velocity $\displaystyle v$ for the case of equal masses, $\displaystyle m_{q}=m_{\bar{q}}\equiv m$, reads as

where $\displaystyle\sqrt{s}$ is the total energy of the considered two particles in their c.m. system. Therefore, in order to obtain a meaningful result, threshold singularities of the form $\displaystyle(\alpha_{s}/v)^{n}$ should be summed, which can be achieved by solving the corresponding relativistic problem. It is important to note that the problem of accounting for the relativistic effects in a few body system is a more general and longstanding task. Up to now there are no reliable relativistic equations derived from the first principles to describe few body bound and/or continuum states near the threshold. For this reason, one is forced to use phenomenological or quasi-phenomenological formalisms to relativistically describe these systems. One can mention several approaches defining the relativistic wave functions of few-body systems. One of the approaches is based on the fully covariant and Lorentz-invariant Bethe-Salpeter (BS) formalism BS . As is well known, in Minkowski space the BS equation is singular, hence in order to avoid uncertainties in numerical calculations, one usually performs the Wick rotation of the equation to Euclidean space and solves the BS equation along the imaginary energy axis. Such a representation is extremely useful for covariant calculations of diagonal matrix elements which are the same in both Euclidean and Minkowski spaces. For instance, within the BS formalism one can relativistically describe the main properties of few-body bound states such as glueballs LPK ; Fischer , quark-antiquark systems (mesons) Drkin1 ; Dorkin2 ; ALkofer , two-nucleon bound state (deuteron) dorkin3 , etc. However, not all physical observables can be obtained from calculations in Euclidean space by the inverse Wick rotation back to Minkowski space. Consequently, one needs to solve the BS equation directly in Minkowski space by finding new mathematical methods allowing one to overcome the difficulties resulting from the singularities of the BS equation, c.f. Ydrefors ; Karmanov . Also, within the BS formalism one encounters difficulties related to the problem of probabilistic interpretation of the BS amplitudes and BS vertex functions. This problem can be partially solved by considering different approximations to the BS equation like the Gross Covariant Spectator Equation CSE .

Another strategy of relativization of calculations consists in elaborating relativistic analogues of the usual three-dimensional Schrödinger equation. These approaches are mainly based on the 1949 seminal paper Dirac , where P.A.M. Dirac proposed several peculiar representations of the Poincaré group, in strict relation with the choice of possible space-time hyper surfaces without a time-like direction. Among them the most popular is the approach based on the Light Front dynamics which allows for a specific description of the dynamics of relativistic interacting systems LF ; Pace .

Eventually, a tempting and rather popular approach is the one based on the quasipotential (QP) equations in Lobachevsky space LogunovTav ; Kadyshevsky68 ; KadMS:72 ; Faustov73 . The QP equations are differential ones with the structure very similar to the Schrödinger equation; however, the interaction quasipotential is now energy-dependent. Note that at vanishing curvature the solution of the QP equation reduces to the known nonrelativistic approaches. Albeit the QP wave functions, being three-dimensional, admit a probabilistic interpretation, they at the same time have the main advantages (renormalizability, analyticity, etc.) of the completely covariant field theory, see e.g. Ref. KadMS:72 .

For the first time, the relativization of the $\displaystyle S$-factor (1) in the case of two particles with equal masses was obtained in Refs. Fadin88 ; Fadin95 , and the result is reduced to the replacement of the nonrelativistic velocity $\displaystyle v_{\rm nr}$ in Eq. (1) by the relativistic one, $\displaystyle v_{\rm nr}\rightarrow\,v$. Exactly the same form of the $\displaystyle S$-factor but with the change $\displaystyle v_{\rm{nr}}\rightarrow\,v/(1+v^{2})$ was later on suggested in Ref. Hoang97 . Note here that the relativistic generalization of the $\displaystyle S$-factor is obviously not unique, since there are numerous ways of expressing the nonrelativistic velocity in terms of the invariant energy $\displaystyle\sqrt{s}$. Another form of the relativistic generalization of the $\displaystyle S$-factor was obtained in Ref. YoonWong:0005 . The relativistic $\displaystyle S$-factor for two particles of arbitrary masses was for the first time presented in Ref. Arbuzov94 (see also Refs. Arbuzov2011 ; Arbuzov2019 ). It was considered within the framework of a version of the relativistic quantum mechanics, on the basis of the Schrödinger equation for the wave function $\displaystyle\Psi(t,{\bm{x_{1}}},{\bm{x_{2}}})$ in a specific frame of reference, $\displaystyle{\bm{p}}_{2}=-{\bm{p}}_{1}m_{2}/m_{1}$, and treating $\displaystyle{\bm{p}}_{1,2}$ as differential operators.

In the present paper, we employ the relativistic quasipotential (RQP) approach Kadyshevsky68 ; KadMS:68 ; KadMM:70 ; KadMS:72 , which, in our opinion, is the most suitable one for the goal of investigations of the effects of relativisation of the $\displaystyle S$-factor for systems consisting of two relativistic, spin 1/2, particles with arbitrary masses $\displaystyle m_{1}$ and $\displaystyle m_{2}$. The pseudoscalar, vector, and pseudovector systems are considered, and the behavior of the corresponding $\displaystyle S$-factors is analyzed in the nonrelativistic, relativistic and ultrarelativistic limits.

Our paper is organized as follows. In the next section, we present the $\displaystyle S$-factor calculated within the RQP approach for two spinless relativistic particles. Explicit, analytical expressions for the $\displaystyle S$-factor of two spinor particles with equal and arbitrary masses are presented in Secs. III and IV, respectively. The results of numerical calculations for pseudo-scalar, vector and pseudo-vector systems are presented in Secion V. The conclusions and summary are collected in Secs. VI and VII.

In this section we consider the $\displaystyle S$-factor corresponding to systems of two relativistic spinless particles with equal masses $\displaystyle m_{1}=m_{2}=m$. It is important to note that the use of the RQP approach for our task is justified by that the BS amplitude $\displaystyle\Phi_{\rm BS}(x),x=(x_{0},\bf{x})$, that parameterizes, e.g., the Drell ratio $\displaystyle R(s)$ in QCD (see also Ref. Adel-Yndurain:95 ), is evaluated at $\displaystyle x=0$ BarbieriCR73 . Consequently, the BS amplitude can be related to the RQP wave function in momentum space, $\displaystyle\Psi_{\chi}({\bf p})$, and in the configuration representation, $\displaystyle\psi_{\chi}({\bf r})$, as

where $\displaystyle\chi$ is the rapidity related to the total c.m. energy of particles $\displaystyle\sqrt{s}$ as $\displaystyle\sqrt{s}=2m\cosh\chi$, $\displaystyle\lambda=1/m$ is the Compton wavelength of the particle of mass $\displaystyle m$, $\displaystyle d\Omega_{\bf p}=(md{\mathbf{p}})/p_{0}$ is the invariant space volume in the Lobachevsky space realized on the hyperboloid $\displaystyle p_{0}^{2}-{\mathbf{p}}^{2}=q_{0}^{2}-{\mathbf{q}}^{2}=m^{2}$. According to Eq. (3), the $\displaystyle S$-factor within the RQP approach is defined in terms of the corresponding wave function in the continuum, $\displaystyle\psi_{\chi}({\bf r})$, as follows Milton-Solovtsov_ModPL:01

The nonrelativistic replacement of the amplitude $\displaystyle\chi_{\rm BS}(x)$ by the wave function, which obeys the Schrödinger equation with the Coulomb potential, leads to formula (1).

Based on the RQP approach, Milton and Solovtsov obtained the following expression Milton-Solovtsov_ModPL:01 :

where $\displaystyle{\tilde{\alpha}\;}$ denotes the coupling constant in the Coulomb-like potential $\displaystyle V(r)=-\tilde{\alpha}/r$. The use of Eq. (5) in QCD requires the replacement of $\displaystyle{\tilde{\alpha}\;}\rightarrow 4\alpha_{s}/3$, where $\displaystyle{4}/{3}$ is due to the SU(3) color factor. Here, it is worth emphasizing that in the above mentioned method Milton-Solovtsov_ModPL:01 , the possibility of transformation of the RQP equation from the momentum space to the relativistic configurational representation in the case of two particles of equal masses KadMS:68 has been widely employed. Note also that the Coulomb quasipotential formally has the same form as the nonrealistic Coulomb potential since its behavior in the momentum Lobachevsky space corresponds to the static quark-antiquark potential $\displaystyle V_{q{\bar{q}}}\sim{\bar{\alpha}_{s}}(Q^{2})/Q^{2}$ for which $\displaystyle{\bar{\alpha}_{s}}(Q^{2})\sim 1/\ln Q^{2}$ plays a role of effective charge SavrinS80 . Thereby, one accumulates a dominant effects induced by the running QCD coupling.

The function $\displaystyle X(\chi)$ in Eq. (5) can be expressed in terms of the velocity $\displaystyle v$ defined by Eq. (2) as $\displaystyle X(v)=\pi\tilde{\alpha}\sqrt{1-v^{2}}/v$. It becomes obvious that the shape of the $\displaystyle S$-factor in the relativistic case remains the same as in Eq. (1), albeit with a replacement

The presence of the square root $\displaystyle{\sqrt{1-v^{2}}}\,$ in the denominator is essential since it provides the correct expected relativistic limit ($\displaystyle v\rightarrow 1$) of the $\displaystyle S$-factor in Eq. (5) Milton-Solovtsov_ModPL:01 ; SOP-Ch10 . Notice that the relativistic limits of the $\displaystyle S$-factors from Refs. Hoang97 ; YoonWong:0005 differ substantially from the relativistic limit ($\displaystyle v\to 1$) of the $\displaystyle S$-factor corresponding to Eq. (5). A relativistic Coulomb-like resummation factor for arbitrary masses and orbital moment $\displaystyle\ell\geq 1$, called the $\displaystyle L$-factor, was investigated in Ref. SOP-ChTMF-11 .

Applications of the relativistic $\displaystyle S$-factor (5) in describing some hadronic processes can be found in Refs. MSS_Adler:01 ; SS_NonLin:02 ; MSS:06 . Also, the factor (5) was applied in Ref. Milton:08 to reanalyze the mass limits obtained for magnetic monopoles, which might have been produced at the Fermilab Tevatron.

Generalization of the relativistic $\displaystyle S$-factor (5) to the case of two relativistic particles of unequal masses $\displaystyle m_{1}$ and $\displaystyle m_{2}$ can be written in terms of the velocity $\displaystyle u$ as ( for details see Refs. SOP-Ch10 ; SOP-ChTMF-11 )

where the subscript “uneq” indicates the quantities related to the case of unequal masses, and the velocity $\displaystyle u$ is determined by the expression

Here $\displaystyle m^{\prime}=\sqrt{m_{1}m_{2}}$ is the mass of an effective particle with 3-momentum $\displaystyle{\bf\Delta}_{k^{\prime},m^{\prime}\lambda_{\mathcal{Q}}}$ and energy $\displaystyle\Delta_{k^{\prime},m^{\prime}\lambda_{\mathcal{Q}}}^{0}$ proportional to the total c.m. energy of particles, $\displaystyle\sqrt{s}$, see Refs. KadMM:70 ; KadMS:72 :

where $\displaystyle\mu=m_{1}m_{2}/(m_{1}+m_{2})$ is the reduced mass of a composite particle with 4-momentum $\displaystyle\mathcal{Q}=q_{1}+q_{2}$ and total mass $\displaystyle M_{\mathcal{Q}}=\sqrt{\mathcal{Q}^{2}}$; $\displaystyle\Delta_{k^{\prime},m^{\prime}\lambda_{\mathcal{Q}}}^{0}$ and $\displaystyle{\bf\Delta}_{k^{\prime},m^{\prime}\lambda_{\mathcal{Q}}}$ are the time and spatial components of the 4-vector $\displaystyle\Lambda^{-1}_{\lambda_{\mathcal{Q}}}k^{\prime}=\Delta_{k^{\prime},m^{\prime}\lambda_{\mathcal{Q}}}$ of the effective relativistic particle in Lobachevsky space corresponding to the Lorentz boost $\displaystyle L=\Lambda^{-1}_{\lambda_{\mathcal{Q}}}$, where $\displaystyle\lambda_{\mathcal{Q}}$ is the 4-velocity vector of the composite system, $\displaystyle\lambda_{\mathcal{Q}}=(\lambda_{\mathcal{Q}}^{0};\bm{\lambda}_{\mathcal{Q}})=\mathcal{Q}/\sqrt{\mathcal{Q}^{2}}$ (see Refs. KadMM:70 ; KadMS:72 ; ChernYAF2014 for details):

In the case when $\displaystyle m_{1}=m_{2}\equiv m$, the velocity (7) of the effective particle reduces to the velocity $\displaystyle v$, Eq. (2). The 3-momentum $\displaystyle{\bf\,k^{\prime}}$ of the effective particle in Lobachevsky space is related to the relative relativistic velocity $\displaystyle{\bf\,v}$ of the composite system by the following expression KadMS:72

Recall that within the RQP approach KadMM:70 ; KadMS:72 the modulus of the relative relativistic velocity $\displaystyle|{\bf{v}}|$ of two particles can be expressed in terms of their total c.m. energy $\displaystyle\sqrt{s}$ (see Ref. SOP-Ch10 ) as

that is exactly the same as in, e.g., Ref. Arbuzov94 . From Eqs. (7) and (11) one can infer that

It is obvious that the relativistic limit ($\displaystyle u\to 1$) of the $\displaystyle S$-factor considered in Ref. Arbuzov94 ¹¹1In Ref. Arbuzov94 the notation $\displaystyle T(|\bf v|)$ is used instead of $\displaystyle S(|\bf v|)$ adopted in the present paper.

differs from the $\displaystyle S$-factor defined by Eq. (6) for which the relativistic limit is $\displaystyle S_{\rm uneq}(u)\overset{{u\to 1}}{\,\,\xrightarrow{\hskip 11.38092pt}1}$. However, for small values of $\displaystyle u$ the $\displaystyle S$-factors defined by Eqs. (6) and Eq. (13) provide the same result.

For the case of two spin-1/2 particles interacting via a Coulomb-like potential in $\displaystyle s$-state ($\displaystyle\ell=0$), the wave function in the configuration representation and the corresponding $\displaystyle S$-factor were thoroughly investigated within the RQP approach in Refs. skachkov ; ChYAF2019 ; CKS-2020 . Furthermore, a closed analytical expression for the $\displaystyle S$-factor has been found

where $\displaystyle\Gamma(2+i\tilde{\rho})$ is the familiar Euler gamma function, $\displaystyle{{}_{2}}F_{1}(1+iB,-i\tilde{\rho};2;1-e^{-2\chi})$ is the hypergeometric function and the subscript “s” refers to spinors. In Eq. (14) the quantities $\displaystyle X_{{\rm{eq,s}}}(\chi)\;$ and $\displaystyle\tilde{\rho},\,B$ are defined as

where the parameters $\displaystyle a$ and $\displaystyle b$ for different total spin of the system are

It can be shown that the relativistic limit, $\displaystyle v\to 1$ ($\displaystyle\chi\gg 1$), of the $\displaystyle S$-factor is

which is valid for all values of the spin parameters $\displaystyle a$ and $\displaystyle b$.