Production of true para-muonium in linearly polarized photon fusions

True muonium (TM) -- the bound state of $\mu^+\mu^-$ -- has not been discovered yet. It was demonstrated that searching for TM via $\gamma\gamma$ fusions in heavy ion collisions is feasible due to the enhancement of the atom number. We study the production of the true para-muonium (pTM) in the collisions of linearly polarized photons in the experiments of heavy-ion collisions, calculate the production rate as well as the transverse spectrum of pTM, and explore the discovery potential in nuclear experiments. Our results show that there is a significant correlation between the linearly polarized photon distribution and the transverse momentum distribution of pTM. The optimal kinematic region of the generated pTM is identified, which can provide a theoretical guide to the detection of pTM in experiments.

The µ + µ − bound state, referred to as true muonium (TM), is a short-lived, exotic atomlike particle composed of a muon and its antiparticle, the antimuon.It is a purely quantum electrodynamics (QED) bound state.The lightest QED atom, the positronium (e + e − bound state), was observed more than 75 years ago [1] and studied extensively.Even the so-called muonium (µe bound state) [2], πµ atom [3], and the dipositronium [4] [(e + e − )(e + e − ) molecule] have been discovered and studied more, while the TM remains unobserved, even though the possibility of its existence was discussed [5] soon after the clarification of the leptonic nature of the muon [6][7][8] in the mid-20th century.The investigation of TM can contribute to our broader understanding of the Standard Model of particle physics and may offer clues to the existence of physics beyond this well-established framework [9][10][11][12].
Muons are heavy, unstable relatives of electrons, with a mass approximately 207 times that of an electron.Due to the short lifetime of muons and their antimatter counterparts, TM exists only fleetingly [13][14][15][16], making it challenging to study and observe directly.The muon's relatively short lifetime, on the order of microseconds [17], restricts the timescale for the formation and stability of TM.Many sophisticated experimental techniques have been employed to detect and study these ephemeral particles, often relying on high-energy accelerators and advanced detection methods.It was proposed to detect the TM in the physical processes at high-energy experimental apparatus, such as π − p → (µ + µ − )n [13], γZ → (µ + µ − )Z [13,18], eZ → e(µ + µ − )Z [19][20][21], [18,[22][23][24][25][26] (where Z indicates a heavy nucleus), µ + µ − collisions [14], η → (µ + µ − )γ [27,28], e + e − collisions [29][30][31], K L → (µ + µ − )γ [32], etc.Among these processes, the e + e − → (µ + µ − ) and γ(µ + µ − ) is of particular interest because there is no hadron involved and thus is a pure QED process.It was demonstrated in Ref. [30] that the TM could be discovered at electron-positron colliders, if the µ + µ − resonances above the threshold are also taken into account, because the states with highprincipal quantum numbers cannot be distinguished from the resonances just above the threshold due to the beam energy spread, which production rates are enhanced by the Sommerfeld-Schwinger-Sakharov factor [33].Based on this, it was proposed to search for TM with fool's intersection storage rings [e + e − → (µ + µ − )] discussed by Bjorken [34] in which the electron and positron beams are arranged to merge at a small angle in order to give rise to a strong boost for the produced TM, and to search for the true para-muonium (pTM) and true orthomuonium (oTM) using the initial and final radiation technology [e + e − → γ(µ + µ − )] on BESIII and Belle-II experiments [35,36].
Another promising process is the heavy-ion collisions.If the heavy ions collide with impact parameter larger than the twice of the radius of the nuclei, i.e., the ultraperipheral collisions (UPC), the strong interaction is suppressed, ensuring γγ fusion in UPC a QED dominant process [37].The electromagnetic field strength is scaled with the atom number Z. Hence comparing with electron-positron collisions, the production rate is enhanced by a factor Z 4 in γγ fusions in UPC.For example, in lead-lead collisions the rate will be enhanced by 82 4 ∼ 4.5 × 10 7 .This makes the study of rare processes possible in heavy-ion collisions.As a consequence, the light-by-light scattering, predicted by quantum theory but forbidden by classical electrodynamics, was observed in Pb+Pb(γγ) → Pb * +Pb * +γγ process by ATLAS [38] and CMS [39], based on the approach proposed in [40].
On the theory side, the calculation can be carried out with the equivalent photon approximation where the virtual photons are regarded as quasi-real photons, which is also known as the Weizsäcker-Williams method [41][42][43].The quasi-real photons, radiated from the relativistic nuclei, can be either unpolarized and linearly polarized.The linearly polarized photon distribution is correlated with the transverse momentum and the azimuthal distribution of the produced particles.It was proposed that the linearly polarized photon distribution can be extracted from the azimuthal-asymmetries of the lepton pair production in UPCs, see, e.g., Ref [44].The cos 4φ asymmetry has been measured by STAR collaboration [45].
The transverse momentum distribution provides useful information for experimental research.Although the total production rate of pTM in heavy-ion collisions has been studied by previous works, however, the transverse momentum distribution of pTM has never been extensively explored, especially in the small transverse momentum region.Moreover, the effect of the linearly polarized photon is never explored.In this work, We will calculate the transverse momentum dependent differential cross-sections and study the effect of linearly polarized photons on the transverse momentum distribution of the pTM.
The pTM-the 1 S 0 bound state of µ − µ + , can be produced by γγ fusion, which is shown in Fig. 1.When the virtuality of photon is small, one can utilize the equivalent photon approximation to calculate the differential cross section.In this work, we take the transverse momentum distribution of pTM into account.If the transverse momentum of pTM in the final state is small, the differential cross section takes the form of transversemomentum-dependent (TMD) factorization [46][47][48], dσ = 1 2s where √ s is the collision energy per nucleon pair in the center-of-mass frame, p a and p b are the momenta of the photons from nucleus A and B, respectively, and q is the momentum of pTM.H is the partonic scattering amplitude of γγ → pTM multiplied by its complex conjugate, and Γ A,B is the photon correlation matrix of the nuclei A and B, respectively, defined as the Fourier transform of the correlator of electromagnetic tensor as where g µν T = g µν − n µ nν − n ν nµ is the transverse metric and the light cone vectors are n = 1 √ 2 (1, 0, 0, −1) and n = 1 √ 2 (1, 0, 0, 1).The transverse part of a vector a is then expressed as a µ T = g µν T a ν .F µν = ∂ µ A ν −∂ ν A µ is the electromagnetic tensor, and z = z − n + z T ; distinguished from the gluon case, there is no Wilson line.f γ 1 is the unpolarized photon distribution function, while h ⊥γ 1 is the distribution of linearly polarized photons.The effect of h ⊥γ 1 has not be considered in the previous research of TM production.In general case, h ⊥γ However, when the photons carry very small longitudinal momenta k + , nearly all the photons are linearly polarized, in this case one has T ) [44].This relation will be utilized in the numerical discussions.
The function H can be expressed as where M µρ is the amplitude of γγ → pTM.It can be calculated as where m is the mass of muon, R(0) is radial wave function of pTM at the origin.The mass of pTM is approximated by 2m.Simplify the expression, the amplitude becomes where we have made the approximation p a ∼ x a P + A n and p b ∼ x b P − B n in the partonic amplitude, P A and P B are the momenta of nucleus A and B (per nucleon), respectively.The collision energy squared s (per nucleon pair) can be expressed as s = (P The transverse Levi-Civita tensor is ǫ µρ T = ǫ µρνσ n ν nσ .Then the differential cross section becomes, where α ≈ 1/137 is the fine structure constant, x a,b = 2m √ s e ±y , and y ≡ 1 2 ln q + q − is the rapidity of pTM, with q + and q − denoting the "+" and "−" components of the pTM momentum q in the light cone coordinate system, respectively.A very similar formula for 1 S 0 quarkonium production has been derived before, which is expressed in terms of the gluon distributions [48,49].When deriving the above results we have adopted R(0) 2 = α 3 m 3 2n 3 with n being the principal quantum number; summing over n leads to a factor ζ(3).It is interesting to note that the formula doesn't depend on the muon mass m at first glance because all the m dependence is through the variables x a,b = 2m √ s e ±y .Integrating over q T , one can find that the term associated with h ⊥γ 1 h ⊥γ 1 disappears and get so the linearly polarized photon distribution will not modify the q T integrated cross section, as well as the total cross section.However, the linearly polarized photon will modify the transverse momentum distribution of the pTM, as we shall demonstrate below.The total cross section can be derived by integrating out y, where iT ) are the integrated photon distributions.
According to the Weizsäcker-Williams method, the unpolarized photon distribution for a nucleus is where M p is the proton mass, Z is the atom number and F is the electric form factor of the nucleus.The form factor is often parametrized using the Woods-Saxon distribution [50] F where ρ 0 is the normalization factor, R WS is the radius, and d is the skin depth.The exact analytical expression of the form factor F can be derived by doing the Fourier integral.The result reads where Φ(z, s, a) ≡ ∞ n=0 z n (n+a) s is the Hurwitz-Lerch transcendent.The last line in the above expression is extremely tiny because of the exponential, thus one can neglect it and get an approximation of the Woods-Saxon distribution, and the normalization factor ρ 0 is evaluated as Numerical result shows that Eq. ( 12) is a very good approximation of Eq. ( 10), and we will adopt it for further convenience in most of the numerical calculations.We note that the same expression has been derived in Ref. [51].Another commonly used form factor in literature is the one from the STARlight Monte Carlo (MC) generator [52,53], where R A = 1.1A 1/3 fm, A is the mass number of nuclei and a = 0.7 fm.The normalization factor A is the nuclear density.It is also a good approximation of the Woods-Saxon distribution.In Fig. 2, we plot the Woods-Saxon distribution, the form factor in Eq. ( 12), and the STARlight form factor of Pb, where R WS = 6.62 fm, and d = 0.546 fm.The comparison of the total cross FIG. 2. The Woods-Saxon form factor, the form factor adopted by this work, and the form factor in STARlight MC generator of Pb, where RWS = 6.62 fm, and d = 0.546 fm.Too see the differences between the form factors more clearly, we "zoom in" by plotting in a particular region.
sections computed from the two form factors will also be presented later.Based on the above results, we present the numerical analysis herein, for the Au-Au collisions at the RHIC and Pb-Pb collisions at the LHC.The nucleus radii are R WS = 6.38 fm for 197 Au and R WS = 6.62 fm for 207 Pb, while the corresponding skin depths are d = 0.535 fm for 197 Au and d = 0.546 fm for 207 Pb [54].As discussed earlier, one has h ⊥γ 1 (x, k 2 T ) ≃ f 1 (x, k 2 T ), i.e., all photons are linearly polarized.The case that photons are partially polarized will be discussed later.
To delve into the differential cross section dependence on √ s, we fix y = 0.The resulting plot illustrates the differential cross section versus transverse momentum q T of pTM.We consider three distinct center-of-mass energies per nucleon pair: √ s = 100 GeV, 200 GeV, and 300 GeV of Au-Au collisions, which are around the RHIC collision energy √ s = 200 GeV per nucleon pair.The graphical representation is depicted in Fig. 3. Integrating over the rapidity y one can have the y-integrated differential cross section dσ/dq 2 T , which is shown in Fig. 4. One can find that the differential cross section increases with the rising of √ s, which is because that the photon density increases when x goes smaller.To see how the produced pTM is distributed with its rapidity, we integrate over q T and get the q T -integrated differential cross section dσ/dy as a function of y-the rapidity of pTM-which is presented in Fig. 5 for Au-Au collisions and Fig. 6 for the Pb-Pb collisions.Once more, we let √ s = 100 GeV, 200 GeV, 300 GeV for Au-Au mode, and take √ s = 2.76 TeV, 5.02 TeV for Pb-Pb mode which corresponds to the center-of-mass energy per nucleon pair at the LHC.One can find that the rapidity of pTM at the RHIC is around |y| < ∼ 5, while |y| < ∼ 8 at the LHC.FIG. 6.The qT integrated differential cross section dσ/dy as a function of y in Pb-Pb collisions, where y denotes the rapidity of pTM.We take the collision energy √ s = 2.76 TeV and 5.02 TeV per nucleon pair in the center-of-mass frame.
To see how the polarization of photon affects the production of pTM, we introduce the degree of polarization r, which is defined as the ratio of the linearly polarized and unpolarized photon distributions, i.e., r ≡ h r is generally a function, but for the sake of simplicity, we assume r is a number and present the differential cross sections with r = 0, 1/2, 1 in Fig. 7.When r = 0, there is no linearly polarized photon, while for r = 1 all of the photons are linearly polarized.One can find that the linearly polarized photons strongly effect the transverse momentum dependence in the small-q T region, while for larger q T the differential cross section is not affected.It is interesting to figure out that, the differential cross section for fully linearly polarized photons decreases to zero when q T → 0, and the differential cross section approaches the maximum value around several MeV.Integrating out y and q T leads to the total cross section.We plot the total cross sections as the functions of √ s for both Au-Au and Pb-Pb collisions, which is shown in Fig. 8. Again, one can conclude that the total cross section increases when √ s increases.We list the total cross sections in Table I.The cross sections of pTM production reported in [22] are 0.15 µb for goldgold mode at RHIC, and 1.35 µb for lead-lead collisions at LHC.Our results in Table I are larger, but in agreement with their results in magnitude.As a coproduct of this analysis, the cross sections for the production of para-positronium (pPM) and true para-tauonium (pTT) are also presented in Table I.We note that the total cross section for true tauonium production via two photon fusions in UPCs was calculated in Refs.[55][56][57]; our result is larger because the UPC events are not singled out in the traditional TMD formalism.In Fig. 9, we explore the effects of different choices of nuclei form factors on the total cross sections.It indicates that the result calculated with the form factor in Eq. ( 12) matches well with the result calculated with the form factor adopted by STARlight, especially when √ s is not too large; when √ s is large, there may be visible differences between the two cross sections.
It is also intriguing to explore the average transverse momentum of pTM.We define the average transverse momentum as With the differential cross sections calculated above, one can compute the numerical results of q T , which is plotted in Fig. 10.We take the Au-Au collisions as an example.At the RHIC energy √ s = 200 GeV, q T ≃ 1.4 FIG. 9.The total cross section of pTM production in Pb-Pb collisions, computed with the form factor in Eq. 12 and the one from STARlight MC generator.
MeV.One can find that the average transverse momentum decreases when √ s increases.We plot q T for both r = 0 and r = 1 cases, and find that the two curves coincides.Therefore, although the linearly polarized photons modify the shape of q T distribution, the average value q T is not affected by the polarization.
To summarize, we study the true para-muonium production with photon-photon fusions in heavy ion collisions.The transverse distribution of true para-muonium is taken into explored, and to achieve this purpose after taking the linearly polarized photon distribution into account.Our results indicate that the differential cross section of true para-muonium production in the small q T region is affected by the linearly polarized photons in nuclei significantly, and exhibits a maximum when q T is around several to 10 MeV at the RHIC, while the maximum of differential cross section is located at lower q T at the LHC.Detecting pTMs with such low momenta presents a challenge in current experiments at the LHC, as the minimum transverse momentum measured is approximately 1 GeV [58][59][60][61].However, it is crucial to acknowledge that transverse momentum distributions have the potential to serve as valuable guides for future ex-  I.The total cross sections of pTM production in two gamma fusion in Au-Au and Pb-Pb collisions.In the second row we list the cross sections of para-positronium production; while in the third row we list the cross sections for true para-tauonium.perimental analyses.The total cross section is also computed, indicating that significantly considerable amounts of true para-muonium should be produced.Our work can also be extend to the true ortho-muonium production.

2 [mb/GeV 2 ]FIG. 3 . 2 [mb/GeV 2 ]FIG. 4 .
FIG.3.The differential cross section as a function of the pTM transverse momentum qT in Au Au collisions.We take the collision energy per nucleon pair as √ s = 100 GeV, 200 GeV, 300 GeV that are near the RHIC energy.The differential cross section exhibits an increase with the rising values of √ s.

FIG. 5 .
FIG.5.The qT integrated differential cross section dσ/dy as a function of y in Au-Au collisions, where y denotes the rapidity of pTM, and the collision energy √ s = 100 GeV, 200 GeV, 300GeV per nucleon pair in the center-of-mass frame.

2 [mb/GeV 2 ]FIG. 7 .
FIG.7.The differential cross section computed with r = 0, r = 1/2 and r = 1, in Au-Au collisions, where r is the degree of photon polarization, and y is the rapidity of pTM.The collision energy per nucleon pair is taken as √ s = 200 GeV.

FIG. 8 .
FIG.8.The total cross sections of pTM production in Au-Au and Pb-Pb collisions.

FIG. 10 .
FIG.10.The average transverse momentum qT as the function of √ s in Au-Au collisions, computed in the case of r = 1 and r = 0, where r is the degree of polarization.