Seiberg-Witten map Invariant Scatterings

We investigate Scattering amplitudes of the reversible $\theta$-exact Seiberg-Witten (SW) map based noncommutative (NC) quantum electrodynamics, and show explicitly the SW map invariance for all tree-level NCQED $2\to2$ proceses, including M\"oller, Bhabha, Compton, pair annihilation, pair production and light-by-light $(\gamma\gamma\to\gamma\gamma)$ scatterings. We apply our NCQED results to the $\gamma\gamma\to\gamma\gamma$ and $\gamma\gamma\to\ell^+\ell^-$ exclusive processes, convoluted to the ultraperipheral lead $^{208}$Pb ion-ion collisions, recently measured by the ATLAS collaboration at LHC. We demonstrate that $\gamma\gamma\to\gamma\gamma$ is the more appropriate channel to probe NC scale $\Lambda_{\rm NC}$ while both are less efficient than some other probes.


I. INTRODUCTION
Lorentz symmetry, as a cornerstone of modern physics, appears with the Einstein formulation of the special theory of relativity in 1905. A classical example maintaining that Lorentz symmetry is the Dirac-Born-Infeld model [1][2][3], which introduces an upper bound for the electric field near the electron eliminating electron self-energy divergences, while another is famous Euler-Heisenberg classical Maxwell Lagrangian density [4]. Since than, Lorentz symmetry is one of the symmetries whose violation has been challenged many times by many experimental attempts and bounds obtained.
In the modern string theory framework regarding the effect of spontaneous breaking of Lorentz symmetry, one of the most striking observations is that via tensor field vacuum expectation values the low-energy effective theory can be expressed by usual gauge fields or gauge fields deformed by space-time noncommutativity (NC). These two versions of effective theory can be subsequently connected by highly nonlocal expressions, called the Seiberg-Witten (SW) map [5].
In recent years tremendous progress has been achieved in the field of the on-shell scattering amplitudes of quantum gauge theories [126,127]. The NC (super) Yang-Mills (NC(S)YM) without the SW map have been found to share a good amount of recently discovered abstract scattering amplitude structures with their commutative counterparts at both tree and one-loop levels [128,129]. On the other hand, scattering amplitudes of SW mapped NC gauge theories are mainly studied on an explicit process-by-process basis. An expansion over the NC parameter θ was also used in various early studies which covered a fairly large number of processes. Furthermore, while it has been shown formally that SW map-based equivalence exists between on-shell background field effective actions [130][131][132], there is not yet a work that studies explicitly the scattering amplitudes in an NC gauge theory defined via a reversible SW map and compares with its counterpart without the SW map.
In this work, we fill this vacancy by demonstrating explicitly that all tree-level two-by-two scattering amplitudes in an NCQED model defined with a reversible θ-exact SW map are identical to their unmapped counterparts which is compatible with the formal equivalence proven before [130][131][132].
As applications of this newly found identity, we revisit the NCQED two-by-two processes and, in particular, lightby-light scatterings and lepton pair production processes in the context of ATLAS ultraperipheral lead 208 Pb ion-ion collision measurements [133][134][135][136] and show that LbyL scattering is better probe to the NC scale of these two. ATLAS results were already compared with the standard model (SM) predictions [137][138][139] and used in constraining various models beyond the SM [140][141][142][143][144][145][146].
The paper is structured as follows: In Sec. II. we describe the Moyal NC deformation and the SW map-induced actions by means of the θ-exact SW maps for a photon and charged fermion. In Sec. III. we prove the SW map invariance of scatterings amplitudes in the matter and the gauge sectors of NCQED. In Secs. IV-VI. we revisit and compute the cross section of exclusive processes in NCQED and SM: Møller, Bhabha, Compton (first computed by Klein and Nishina 1929 [147]), dilepton pair annihilations and productions, and γγ → γγ scatterings (first published by Karplus and Neuman in 1951 [148]) showing complementarity between NCQED and QCD [42][43][44][45][46][47], and present them in the form of 3D figures of ratios of cross sections σ NC /σ SM as functions of the incoming energy and the NC scale. Section VII is devoted to the SM and NCQED computations of recent ATLAS Collaboration measured PbPb(γγ) → Pb * Pb * γγ and PbPb(γγ) → Pb * Pb * + − collision cross sections. Section VIII contains discussions and conclusions, respectively. In Appendixes we use main conventions and notations, like equations of motions (EOM) and on shell conditions from Peskin and Schroeder [149], and give the NCQED Feynman rules (FRs), the NC phase factors including ϕ integrals, and details of vanishing of the SW map-induced amplitude for γγ → γγ scattering. For relevant integrals and other mathematical issues among Wolfram Mathematica [150], we are using mainly Gradshteyn and Ryzhik [151].

A. General SW map considerations
We consider the minimal NCQED model in terms of the minimal NC gauge and charge fermion fields, A µ and Ψ respectively, which lives in the adjoint representation of the NC gauge group U(N): where ( ) denote the Moyal-Weyl star-product. The relation between the NC and commutative quantities or fields is given by the θ-exact Seiberg-Witten map. In the above action (1) we do apply the generalized SW maps of the NC gauge parameter Λ, NC gauge and fermion fields A µ and Ψ, in terms of commutative gauge parameter λ and commutative-physical-gauge and fermion fields a µ and ψ, respectively. Extending results obtained in Refs. [76,77], by applying a method based on the following consistency condition transformations [13,14] we have been working out in detail the θ-exact SW map for the noncommutative gauge field strength F µν up to the third order of ordinary U(N) gauge field a µ [90]. So in terms of commutative U(1) gauge field strength f µν , we have found the following solution: F µν (e · a µ , θ µν ) = ef µν + F e 2 µν + F e 3 µν + O e 4 , f µν = ∂ µ a ν − ∂ ν a µ , where the term for the SW map up to the e 2 order F e 2 µν is fairly universal [90,91,104] F e 2 µν (x) = e 2 θ ij f µi 2 f νj − a i 2 ∂ j f µν .
Here, we use well-known Moyal-Weyl -and the 2 -product definitions: where, in fact, 2 -product is nonassociative; however, it is commutative. Now we give the e 3 power of the θ-exact field strength F e 3 µν [SW map (I)-induced solution in Eq. (65) from Ref. [90]], needed for all further computations: with new star product being defined in Refs. [90,92]. A general SW map expansion of the NC U(N) charged fermion field Ψ(x), in terms of commutative gauge and fermion fields (a, ψ) respectively, is given as where the right-hand side of Eq. (9) represents the leading-order charged fermion SW map, derived long ago [5]. The bullet product (•) in Eq. (9) is defined as Above the • product is not associative, but it satisfies the integral identity, which is easy to prove by expanding, integrating by part, and resumming the resulting series. At the end of this section, note that, to obtain the complete two-electron-two-photon vertex Γ µν (ēeγγ) (necessary for any quantum-loop computation) in momentum space, we also need a second order in gauge field θ-exact SW map for the electron field, Ψ (2) (x). Using the SW differential equation method [76] in the same way as we did in Ref. [90], we have also obtained the Ψ (2) (x) term of the SW map expansion (9) explicitly: B. Minimal θ-exact SW mapped NCQED action From minimal noncommutative U(N) action (1) we perform the SW map expansion in terms of commutative -physical-fields (a µ , ψ) and write the generalized manifestly gauge invariant actions with a minimal number of fields: The solution of the θ-exact SW map in terms of commutative gauge field strengths f µν was resolved from the SW differential equation [90] giving the following minimal action 2 : where additional star products could be found in Refs. [90,92]. Next we give θ-exact electron-photon action up to the e 1 order Sψ f ψ as There are also two-electron-two-photon action relevant terms Sψ f 2 ψ with additional parts which are proportional to free field equation for either ψ orψ. Therefore, since they vanish due to the EOM they are irrelevant to the tree-level scattering processes we are heading for. Bearing this in mind we collect now the relevant terms at the e 2 order which does not vanish on shell Sψ a 2 ψ relevant : Extracting the relevant matter part of the two-electron-two-photon coupling terms we integrate (17) by part in such way that all partial derivatives hit ψ, thus changing the form of Sψ a 2 ψ to From the above actions (14,15,16, and 18), we obtain the Feynman rules which are given in Appendix A, respectively.

Minimal gauge sector: Relevant parts of three-and four-photon couplings
Since the triple-photon interaction, based on a fairly universal second-order order SW map for the gauge field, is already given as Eq. (14), here we continue with higher-order contribution.
Expanding (1) up to the order a 4 µ by using (12) we get the following general form for the four-photon interaction: where the two distinct solutions for the e 3 order gauge field strength have been found and given explicitly in Ref. [90]. After using solution (I) [Eq. (65) [90]) from (6)− (8) and (19), we obtain In the second term of Eq. (19), we are not using the complete (8) for F e 3 µν , but only the relevant part, which then for the universal gauge field SW map A e 2 µ = − e 2 2 θ ij a i 2 (∂ j a µ + f jµ ) gives producing Altogether, from Eq. (19) we then have From the above action (23), we obtain the relevant FRs and give them in Appendix A.
C. Pametrizations of the NC θ matrix and unitarity of the NCQED theory We parametrize the 4 × 4 antisymmetric θ matrix as follows: where the c µν -matrix elements are identical in all reference frames resulting in Lorentz symmetry breaking and run within It is important is to stress that, for the case θ 0i = 0, ∀ i, the NC theory is not unitary, i.e. acausal behavior [78,79]. Only the condition θ 0i = 0 keeps the theory unitary and causal. However, to keep the possibility of nonzero space and time NC contributions to the novel particle physics effects, i.e., the physical noncommutative effects, we invoke a covariant generalization of θ 0i = 0 requirements through the so-called unitarity condition, known as the quantum or perturbative unitarity condition [80]. Namely, the theory with θ 0i = 0 can be converted into one with only θ ij = 0 by a suitable observer Lorentz transformation, since the presence of observer Lorentz invariance implies that there are no difficulties with perturbative unitarity provided by θ µν θ µν > 0. Analogous methods do apply for noncommutative theories with θ 0i = 0, which lead to the so-called lightlike noncommutativity defined in Ref. [79]. One may combine both spacelike and lightlike into the following set of requirements for unitarity: The lightlike noncommutativity takes place when the equality holds in the first part of Eq. (25) otherwise, the conditions allow a boost to B θ = E θ = 0, i.e. the spacelike noncommutativity.

A. Electron and positron processes
In accord with Figs.1-4 here we use the two-by-two kinematics 1(k 1 ) + 2(k 2 ) → 3(k 3 ) + 4(k 4 ), and momentum conservation k 1 + k 2 = k 3 + k 4 which defines Mandelstam variables for arbitrary 2 → 2 scattering:  Fig.1 by using Feynman rules (A1) and/or (A2) we read out the following S-matrix element: Here M M I+II represents sum of the non-SW(I) map-induced and the SW(II) map-induced contributions to the amplitude, respectively. Performing trivial integrations over the δ functions and splitting amplitudes into non-SW mapped NCQED (M It , M Iu ) and the SW mapped parts of NCQED (M IIt , M IIu ), respectively, we have the following Møller scattering amplitude M M I+II : with vertices Γ I and Γ II being given in Appendix A as FRs (A1( and (A2), while propagators we are taken from Ref. [149], respectively. So the above amplitude M It represents the t-channel contribution from the NCQED without SW map, while M IIt stands for the t-channel amplitude arising in the NCQED theory with the SW map. The same notations are going to be used for sand u-channel amplitudes, further on. Using detailed vertices from (A1) and (A2) and applying them in amplitudes (29) and (30), we find that all the SW map-induced NCQED parts of amplitudes and/or diagrams in the left part of Fig.1, including both the pure NC Γ IIµ × Γ µ II and the cross terms of the type Γ Iµ × Γ µ II respectively, due to the free field equations vanish on shell trivially, giving the SM result which is shifted only by the noncommutative phases: This property was first observed for fermions in the adjoint representation of the NC U(1) some years ago [88].
For the in-out momentum conservation δ (4) (k 1 + k 2 − k 3 − k 4 ) one can show the following relations, in general: Using the relation (1 .) above one can show that NCQED Møller scattering amplitude shares the same permutation symmetry as its QED counterpart and respects the fermion statistics accordingly.
2. Bhabha scattering: e − e + → e − e + Second, we examine the NC effects arising from the contributions of the two relevant diagrams to the Bhabha amplitude M B . From the sum of right diagrams in Fig.1 we read out the following S − 1 matrix element: Trivial integrations over the δ functions and splitting amplitudes into parts coming from non-SW mapped NCQED (B It , B Iu ) from the SW mapped parts of NCQED (B IIt , B IIs ), respectively, we have the following Bhabha scattering amplitude M B I+II : Using vertices from (A1) and (A2) and applying in the amplitudes (37) and (38) we show, due to the free field equations, the trivial on-shell vanishing of the SW map induced amplitudes and producing the SM result shifted by the noncommutative phases: B. Electron and photon processes 1. Compton scattering: γ e → γ e From the sum of diagrams in Fig.2 we read out the following Compton scattering (S − 1) ≡ S C matrix element: with the total amplitude M C I+II being split into the non-SW(I) and the SW(II) map-induced contributions to the electron and photon interactions and/or Compton scattering amplitude, respectively. Feynman rules for vertices FIG. 2: Feynman diagrams contributing to the Compton scatterings e − γ → e − γ, with free field conditions as usual from Peskin and Schroeder [149]. Momenta ki, i = 1, 2, 3, 4, placed in the counterclockwise way start in the left upper corner, with respecting Mandelstam variable or channels (s, t, u) being given in Eq. (26).
, while propagators are defined in Ref. [149], respectively. After performing trivial integrations over δ functions we obtain the following sum of amplitudes: where γ 0 Γ † µ γ 0 = Γ * µ , γ 0 Γ † µν γ 0 = Γ * µν , and γ 0 Γ † µνρ γ 0 = Γ * µνρ . Above, amplitude M C Is represents the s-channel contribution from the NCQED without SW map, while M C IIs stands for the s-channel amplitude induced into the NCQED by the SW(II) map. The same is valid for the uand t-channels, respectively. For instance, only the first term Γ µ I in (A1) represents the Feynman rule for the NCQED with non-SW(I) map-induced terms, while the additional two terms in Eq. (A1) arise due to the SW(II) map. Second, only the first line V µ1µ2µ3 I of Eq. (A6), represents the FR for the NCQED theory with non-SW(I) map-induced contributions. Other notations of the diagrams in Fig.2 and corresponding contributions given in Eqs. (43) are self-evident, and Ward identities in matter sector being satisfied.
Note that the four-field-vertex contact term in Fig.2 does not exists without the SW(I) map, i.e., only the nonvanishing part is induced by the SW(II) map. In Appendix A, this is anticipated by writing explicitly in Eq. (A4) vanishing of that contact term part on shell due to the free field equations: (Γ µν I ) αβ = 0. Thus, we called it as irrelevant. However, the SW(II) map-induced term (Γ µν II ) αβ = 0 in Eq. (A4) is necessary to cancel the SW(II) map-induced peaces in the other three diagrams in Fig.2. Next, we show that by explicit computations of s-, u-, and t-channel three-field-vertex diagrams from Fig.2 and Eqs. (43) for the on-shell case.
Finally from Appendix A using FR (A4), in the notation of the fourth diagram in Fig.2, we have where It can then be shown that Thus, the sum of the SW map-induced contributions to the on-shell Compton scattering amplitude vanishes.
2. Dilepton pair annihilation and production: From the sum of diagrams, for annihilation (left panel in Fig.3), and for production (right panel in Fig.3), we read out the following S − 1 matrix elements, i.e. the M A and M P amplitudes, respectively: Because of the same topological and Lorentz structures of contributing diagrams in left and right panels in Fig.3 as in the Compton scattering (Fig.2), the same conclusion of vanishing the SW(II) map-induced contributions is valid for the pair annihilation and pair production processes, respectively. So those, together with (31), (39), and (53), represent the extraordinary property of SW maps in the matter sector of NCQED. This same property we shall prove for the pure NCQED gauge sector.

C. Photon-photon process
In this section we present noncommutative effects arising from the contributions of the four relevant diagrams to the γγ → γγ scattering amplitude M γ from Fig.4. We shall further on consider two different basis to express our total amplitude after summing up diagrams. Those are, first the amplitudes in terms of the Lorentz decomposition basis. Second are the photon amplitudes expressed in the helicity basis, along the analysis of [126,128,129], i.e., on the line of the scattering amplitude ideas in recent trends of theoretical physics progress.
Lorentz decomposition is important for proof the SW(II) map invariance of scattering amplitudes on shell, respectively. Now note that after or when proving of vanishing of the SW(II) map-induced contributions, the Ward identities have to be shown only for the non-SW(I) map induced parts of the scattering amplitudes, which we shall actually perform first in the next subsections. (56) Here each vertex Γ in the diagrams is the sum of the non-SW(I) map, and SW(II) map-induced relevant parts, from FRs (A5)-(A17). By splitting (56) into non-SW (Γ I ) and SW (Γ II ) parts, we have the following amplitude for the sum of s−, t−, and u-channel three-photon diagrams, plus the four-photon one as:

Non-SW(I) map-induced γγ → γγ amplitudes and Ward identities
Total non-SW(I) map-induced contributions are: Taking into account FRs and the decompositions of vertices, the total on-shell non-SW map-induced amplitude (58) shall have the following form: where the coefficients a i , i = 1, 2, 3 shall be determined later by explicit computations. One could simplify iM γ I by employing the following Moyal-Weyl -product Jacobi identity in momentum space: One of the momenta, k 4 , has a fixed place, and the remaining three are cyclically permuted. The left side vanishes due to the momentum conservation and antisymmetry among two indices of θ µ1µ2 . Note that the structure of our Jacobi identity in momentum space (60), generated by the Moyal-Weyl product applied to the s-, t-, u-channel scattering amplitudes given in Fig.4, arises from color-kinematic correspondence between the U(1) and the U (1) gauge theories [126,128,129]. Taking into account the SW map, that is, after the U (1) SW −→ U(1), one has only usual U(1) gauge symmetry. The same is valid for the inverse SW map, thus, one could say that the structure (60) is invariant under the U(1) SW map. Finding concrete frameworks of the alternative derivation which provide a geometric interpretation of such structure, in color-kinematic correspondence or duality between gauge theory and perturbative gravity at the tree-level scattering amplitudes, was recently performed and published in [127].
Using Eqs. (58) and (60) and FRs from Appendix A, we get usual Lorentz decomposition of the on-shell non-SW map amplitude (M γ I Lorentz ) in the following form: From the above it is easy to show that Ward identities in the pure gauge sector are fulfilled too. Namely, with the replacement µi (k i ) → k µi i in Eq. (61), we have found that k µi i · M γ I Lorentz µi = 0, ∀i = 1, ..., 4; i.e., it always vanishes, as it should. Q.E.D.

SW(II) map-induced γγ → γγ amplitude
Because of our first result for Compton scattering in momentum space (53), i.e. that the θ-exact SW(II) mapinduced contributions from the sum of diagrams in Fig.2 vanish nontrivilay, it was clear that we have to prove that the same property holds in the pure gauge sector of NCQED, i.e. that from Fig.4 the sum of s-, t-, and u-channels (56), plus 4-photon term from (A7), vanish too. After extracting SW(II) map induced amplitude M γ II Lorentz from (57) we have to show that: The underlying idea for the SW map invariance of scattering amplitudes is that for each vertex term in Feynman rules generated by the invertible SW map there exists a collection of diagrams to cancel it on shell. The general proof for such cancellation is already sketched in our equivalence and duality papers [131,132]. Proof of Eq. (62) is given in Appendix C.

A. Møller scattering cross section
From the sum of matrix elements (32) and ( 33), with the square of total amplitude and after summing and averaging over spins for the massless unpolarized case, we find Phases in the pure tand u-channel terms of (63) cancel out, while total phases in the interference cross terms of Eq. (63) turn into the ∓ of the form (1 .) in Eq. (34), i.e., ∓2(k 3 θk 2 + k 4 θk 1 ), respectively. From the above, one obtains the differential cross section in the head-to-head massless Møller scattering. Using (B6) and (B7), and after rearranging Eq. (63) we obtain three terms representing QED contributions in agreement with Ref. [149], while the fourth one is coming from the NCQED phase factors, confirming the expression earlier given in Ref. [42].
To obtain the total cross section with the NC correction included we shall compute the phase space integral in a general -NCM-frame of head-to-head collision. In the NCM frame after integration over k 3 , the delta function over energy ω 3 then yields a differential cross section formula for an arbitrary unpolarised two-by-two process in the polar coordinate system: Since we shall work out all processes for the massless case the cross section (64) represents quite general form in the NCM frame. For relevant processes we just have to compute all terms in spins M 2 , and than integrate over the final state angles. We next compute the Møller cross section of head-to-head collision for the massless case in the NCM frame defined in Eqs. (B1). Using Eq. (B4), the relevant phase factor for Møller, Eqs (B6) and (B7), for the general case of space-time noncommutativity we obtain contributions to the Møller scattering, for all types of noncommutativity.
Taking into account two identical particles in the final state by integrating over dΩ we obtain the NC part of the Møller cross section in the NCM frame as a function of only incoming energies ω 1,2 and the NC parameters M and Λ NC , respectively: with M given in Eq. (B7). Expanding the above cross section in the regime of small energies and large Λ NC , one obtains which is regular at both x → ±1 limits. Therefore the full destructive cross section (65) is regular at these limits too.
In the CM frame ω 1 = ω 2 = ω 4 ; thus, M → 2M . For small argument expansion (65) simplifies into an asymptotic cross section at low energies, suppressed by the NC scale Λ NC to the fourth power: We notice thatM is proportional to the length of the projection of the vectorB θ = Λ 2 NC B θ onto the plane transverse to the incoming particle axis. So for the same length of B θ the noncommutative contribution would be maximized when it is on the transverse plane, as expected.

B. Bhabha scattering cross section
From the sum of matrix elements (40,41), and after summing and averaging over spins for the massless unpolarized case, we have which gives the differential cross section for the head-to-head massless unpolarized Bhabha collision. Again the phases in the pure tand s-channel terms of (68) cancel out, while total phases in the interference cross terms of (68) turn into the ∓ of the form (2 .) in Eq. (34), i.e. ∓2(k 3 θk 2 − k 4 θk 1 ), respectively. To determine phases in the NCM frame (B1) we are using relevant phase factor from (B4). The phase factors for Bhabha from (B6) and (B7) for NCM shows that the Bhabha scattering receives contributions from all NC types. The same procedure as for Møller, gives the noncommutative part of the Bhabha differential cross section in the NCM frame, and as a function of only incoming energies ω 1 and ω 2 and the NC parameters c 03 , A, and Λ NC , respectively: The small argument expansions of cosine and Bessel functions in (69) gives which is regular at x → ±1 limits. It is interesting that Bhabha cross section in the CM frame of head-to-head collision for the massless case, i.e. where ]. This means that in the CM frame and for the pure spacelike noncommutativity where E θ = 0, the NC contribution to Bhabha cross section vanishes. However the general condition (25) still allows the NC contribution to be nonzero. In this case, one can express E θ as projection along the z axis E θ as well as perpendicular to the z axis E ⊥ θ , so that A = 2Λ 2 NC |E ⊥ θ | and |c 03 | = Λ 2 NC |E θ |. The cross section (70) for energies below the scale Λ NC in the CM frame then simplifies to Thus for fixed |E θ | (which in the lightlike noncommutativity |B θ | = |E θ | ∝ Λ −2 NC case fixes the NC scale too), the NC contribution would maximize when E θ = E ⊥ θ . Finally, we illustrate oscillatory behavior of Møller and Bhabha scattering cross sections in 3D plots from Fig.5, which show almost no oscillation for Møller due to the absence of cosine, while there is wild oscillation for the Bhabha case.

V. CROSS SECTIONS FOR ELECTRON AND PHOTON PROCESSES
In this section, we analyze Comptons ± γ → ± γ, as well as dilepton pair + ell − → γγ annihilation, and pair γγ → + ell − productions ( = e, µ, τ ) and γγ → γγ, respectively. The above processes in NCQED are important by themselves, however, diphoton and dilepton productions together with Bhabha scatterings also serve as a source of the so-called, long-distance effect contributions to the exclusive γγ → γγ scatterings and shall be, in particular, considered later.

A. Compton scattering cross section
After rearranging phases (34), we obtain the sum of the remaining non-SW map-induced amplitudes as: where two terms (73) and (75) without NC phases arising from the first two diagrams in Fig.2, are the same as in Klein-Nishina [147] expressions, while (74) with extracted phases from (72) represents NC correction coming from the three-photon vertex contribution to the t-channel fully described by the non-SW map term (46). After average over initial spins and summations over final spins with contractions over photon polarizations, we compute the absolute square of the M C for the unpolarized case as: From the above equation, it is clear that M C s 2 + M C u 2 builds Klein-Nishina formula. Also in the massless average By extracting out the ordinary QED contributions, following Peskin and Schroeder [149], in terms of Mandelstam variables and for the massless unpolarized case, from Eq. (76), we have obtained the noncommutative correction to the absolute square of the full Compton amplitude M C : Jumping shortly to Eqs. (72)-(77), we found a differential cross section for massless Compton scattering confirming results obtained before in [45][46][47], where first two terms represents the Klein-Nishina formula for standard QED. However, the third term (77) containing the phase factor arises from the t-channel diagram in Fig.2 containing a triple-photon coupling vertex, existing in all Moyal versions of the NCQFT [47], and all of its interference with the sand u-channel diagrams and with the interference between the sand u-channel diagrams, as presented in Eq. (76).
Continuing with NCM frame, having definitions (B1), and (B3), and after variable change (cos ϑ → x) and integrating over ϕ, we obtain the following massless NC part of Compton differential cross section as function of incoming energies ω 1,2 , the NC parameters c 03 and C, and the NC scale Λ NC , [see Eq. (B7)], respectively: while in the CM frame, where ω 1 = ω 2 = ω 4 , of head-to-head collision and, from Eq. (78) we obtain the following NC correction to the cross section: which is valid for various types of noncommutativity.

Collinear singularity in NCQED Compton scattering
It is easy to notice the similarity between the NCQED amplitude square, and the quark-gluon scattering amplitude qg → qg in QCD, which has the t-channel term (77) and (78) added to the amplitude square besides the term that is identical to the Compton scattering in the commutative QED, which makes the complete (QED+NCQED) differential cross section for a massless electron divergent at both forward and backward limits respectively, as shown in Fig. 6. Next we can also notice that the NCQED γe → γe process possesses a t-channel collinear singularity similar to qg → qg. On the other hand, these two singularities are not exactly the same. The NCQED γe → γe amplitude square diverges by t −1 because of the NC factor as shown in Eqs. (77)-(79) 3 , while qg → qg diverges by t −2 [149].
Since to our knowledge the collinear singularity in the NCQED Compton process has not attracted many studies, we invest here some efforts to study its properties. One question one may ask is whether this collinear singularity can be suppressed by certain reasonable choices of NC parameter θ µν . Limiting to the head-to-head scattering scenario (78) and (79), we notice that the collinear singularity is suppressed if C = (c 01 + c 13 ) 2 + (c 02 − c 23 ) 2 = 0. Now a nonvanishing NC contribution to Compton cross section exists only if c 03 = 0 when C = 0. Once we inspect these constraints more carefully we notice that θ µν would have to be neither spacelike nor lightlike NC, to satisfy both conditions. We conclude that it is not possible to suppress the collinear singularity by an appropriate choice of θ µν .
Next, we attempt to introduce a regulator for the collinear singularity in the NC cross section integral by hand. We consider the following simplified scenario: CM frame, spacelike θ µν , and s 4Λ 2 NC . We then obtain the following approximative formula for the cross section We then introduce a small dimensionless shift δ to the 1 − x denominator [149] in Eq. (80), making it Evaluating this integral and keeping only terms till O(ln δ), we get Based on the divergence order one may then choose δ = s/Λ 2 NC to ensure that On the other hand this regulator δ, appropriate or not, would limit the validity of σ C NC to well below Λ NC . Thus finding a more natural and/or less limited way to regulate the t-channel singularity of NCQED Compton scattering would remain a relevant open problem for future research.

B. Dilepton pair annihilation cross section
Using the crossing symmetry in accord with the left plot in Fig.3 we replace momenta (k 1 → −k 4 ), (k 2 → k 1 ), (k 3 → −k 2 ), and (k 4 → k 3 ) in the Compton amplitude square (77) and obtain an unpolarized differential cross section in terms of Mandelstam variables with the first two terms out of three representing QED contributions in agreement with Ref. [149], while the third one is coming form the NCQED triple-photon coupling phase factors [42]. Next following the same procedure as in the Compton section by using the NCM frame defined in Eq. (B1), from Eq. (B7), we shall find contributions for the general case of the NC space-time. After variable change and integration over ϕ, we obtain destructive noncommutative contribution to the unpolarized pair annihilation differential cross section in the NCM frame as a function of incoming energies ω 1,2 , the NC parameters c 03 and A-given in (B7)-and Λ NC , respectively: describing the pair annihilation cross section showing not only explicit dependence of the pair annihilation cross section on the NC scale, but also in the head-to-head annihilation geometry revealing all types of space-time noncommutativity, just like for Compton scatterings. Taking into account two identical particles in the final state we should take 1/2 of the needed integral in forming a total cross section. From Eq. (85) we next give the pair annihilation cross section in the CM frame of head-to-head collision for the massless case, with a perfectly regular integral which carries pure timelike noncommutativity since A → 2 c 2 01 + c 2 02 : Interestingly, there is no collinear singularity whatsoever. In the case of pure spacelike noncommutativity, cross section (86) vanishes identically, as it should. However the more general conditions (25) again allow (86) to be nonzero. The leading noncommutative contribution at low energies can be expressed as follows: So for fixed |E θ |, the NC contribution to the cross section at low energies is maximized when E θ = E θ , which is opposite to the Bhabha scattering case.
Pair annihilation shows in the NCM frame all types of noncommutativity, while in the CM frame the NC correction becomes pure timelike which could be transferred into the lightlike one, and it is illustrated in the 3D left plot in Fig.7. Mild oscillation of the NCQED/QED cross section ratio gets stable by approaching unity for lightlike NC and when Λ NC and the energy pass ∼1 TeV values. In the case of CM frame at small energies the NC correction to the total cross section is destructive and finite (87).

C. Dilepton pair production cross section
Again using the crossing symmetry in accord with the right plot of Fig.3, we can replace momenta ( (77) and obtain a differential cross section for the head-to-head unpolarized massless pair production where the first two represent QED contributions [149], while the third one is coming form the triple-photon coupling phase factors [47]. The same procedure as in the Compton section in the NCM frame (B1) after variable change gives the noncommutative part of pair production total cross section, and as a function of only incoming energies ω 1,2 , the scale Λ NC , and only the NC parameter c 03 , respectively. So, first, after using the relevant phase factor for pair production from Eqs. (B4) and (B5), we have found the destructive NC contribution to the differential cross section again with no collinear singularity. Notice that the pair production cross section of head-to-head collision for the massless case is fully timelike and depends on E θ only, which is easy to understand, since this is the only possible nonvanishing term in the NC coupling (k 1 θk 2 ), the two incoming momenta which are collinear in the head-to-head scattering geometry. This coupling vanishes for a pure spacelike NC parameter yet remains allowed by the general unitary constraint (25). Next, from Eq. (89) we compute the pair production total cross section in the CM frame of head-to-head collision for the massless case, ω 1 = ω 2 = ω 4 . For energies smaller than scale, s/2Λ 2 NC < 1, the expansion of cosine function in Eq. (89) gives asymptotically Finally we illustrate numerical analyses of both full cross sections in the CM frame, i.e., the pair annihilation (86) and pair production (89) as functions of scale Λ NC , incoming energy and for lightlike NC parameters and present them as 3D left and right plots in Fig.7, respectively.
Inspecting the left plot in Figs.7 one realizes that in the CM frame the pair annihilation total cross section ratio of the NCQED (86) with one obtained in QED, as a function of energy, the lightlike noncommutativity, and the NC scale, having oscillatory behavior for small E and scale values, than rises and starts to stabilize around unity when both Λ NC scale and the incoming energy E approach ∼ 1 TeV scale. Similar oscillatory behavior one may see in Fig.18 in Ref. [42].
Lepton pair production shows in the NCM frame pure timelike type of noncommutativity, which could be transferred into the lightlike one. There is no collinear singularity whatsoever, and in the case of CM frame and at small incoming energies the NC correction to the cross section has a destructive finite value (90). It is illustrated in the right plot in Fig.7, which due to cosine shows wild oscillatory behavior at small NC scale and large energy values and up to 0.5 TeV agrees with Fig. 2 in Ref. [47]. However, the right plot in Fig.7 does show double oscillations, than dropping and stabilization of the NCQED/QED cross section ratio around unity for lightlike NC when Λ NC and energy pass ∼1 TeV values.

A. Helicity amplitudes of NCQED photon-photon scattering
Since the NCQED γγ → γγ is invariant under the SW map, the calculation of the scattering cross sections can follow the general results of tree-level photon scattering amplitudes in NCQED without the SW map. The basic fact is that there exists a one-to-one correspondence between the color ordering in QCD and a -product ordering in NCQED [128,129]: The tree-level -product ordered NCQED photon scattering amplitudes can then be shown to be identical to their QCD gluon scattering counterpart. The full helicity amplitudes of NCQED γγ → γγ scattering can be obtained by summing over all -product orders. The nonvanishing helicity amplitudes up to a total reflection of all photon helcities may be expressed as follows [42]: The amplitude square is then a sum over all nonvanishing helicity amplitude squares, which yields, after some arithmetic Here we notice two remarkable properties: First, the amplitude square (93) is closely connected to its gg → gg counterpart in QCD, as expected. A conversion to the latter (up to an overall normalization factor) can be achieved by replacing three NC factors with three identical QCD factors in accordance with Eq. (91): The sum over the Mandelstam variable fractions in Eq. (93) then boils down to which is exactly the same kinematic factor that occurs in the differential cross section of QCD exclusive free gluon (gg → gg) scattering [149].
On the other hand, unlike QCD, all the collinear singularities from the factions of Mandelstam variables in Eq. (93) are canceled by the corresponding collinear zeros in the NC factors. This property can also be shown at helicity amplitude level with some help from the star product Jacobi identity (60), which allows us to transform the amplitudes (92) into the following form The cancellation of collinear singularities is then straightforward.

B. Light-by-light cross section in NCQED
Using a helicity amplitude decomposition method or signature, we shall determine the NC contributions to the total cross section of the exclusive γγ → γγ process starting with the NC amplitudes (92) and than adding on the SM amplitudes [138,[152][153][154][155][156][157][158][159][160][161][162][163][164][165]. Averaging over initial and summing over final helicities in the NCM frame, from Eq. (64) we obtain a cross section containing three terms: the SM one σ γ SM 2 , the SM interference with NCQED σ γ SM×NC , and pure NCQED term σ γ NC 2 , respectively. Note that, since the SM contribution dependence on energy is rather tedious [157][158][159][160][161], there exist various approximations in the literature [152][153][154]. Namely, a very high-energy regime is relevant for further LC γγ experiments, while relatively low-energy limit is interesting because the ATLAS Collaboration actually measured a diphoton final state in the central detector with only less than 30 GeV diphoton invariant mass.

Pure NCQED contributions to the γγ → γγ cross section
We compute the pure NC contribution to the γγ → γγ cross section starting with the helicity amplitude square (93). 4 Generally, after taking into account two identical particles in the final state, we have obtained for arbitrary energies in the NCM frame the following pure NCQED exclusive cross section σ γ NC 2 : with above integrals over dϕ being given in Eqs. (B11). The above cross section (97) is valid for an arbitrary energy regime and for arbitrary scattering angles, which, together with the SM×NC interference term, give noncommutatative correction to the total cross section in the laboratory of NCM frame. The above integralsÎ 1324 , andÎ 1423 contain new NC constant G given in (B7).
Choosing the CM frame with ω 1 = ω 2 = ω 4 , and the pure spacelike noncommutativity where c i0 = 0, ∀i = 1, 2, 3, from Eq. (B7) we have A → 0, G = C = M/2, and M → 2M . The linear combinations of ϕ-integrated phase factors (B9) and (B11) satisfy the following relations: and we finally have expression for cross section which vanishes in the absence of spacelike noncommutativity [see (B7)], as it should. Performing expansions in Bessel functions (98) we have found that all terms proportional to the (1/Λ NC ) 4 cancel out, while the next terms of (1/Λ NC ) 8 order of expansion contribute. So the first nonvanishing contribution than reads and shows correctness of our computations of Eqs. (97) and (98). Note that, after using CM frame and NC phase integrals (B11) one can easily see that the above x-integrals contain no collinear singularities.
where the ϕ-integration of the NC helicity amplitudes (92) gives: We define the structures of I ± integrals in Eq. (102) in Eq. (B10) with respect to the NC type, and with respect to the NCM frame. Taking the above integrals from (B10) for the spacelike noncommutativity only, one see that the deference of integrals (I − − I + ) 1234 vanishes.
Since the interference contributions for spacelike NC, independently of frame, arises from I 1423 ± integrals only, we take the ϕ-integrated NC amplitudes from Eq. (102) and the real part of SM amplitudes from [152][153][154][155] and find the interference terms for the unpolarized cross section at high energies: In the NCM frame for spacelike noncommutativity, M = 0 (B7), the high-energy regime expression relevant for the further experiments is good enough for numerical integrations. Like for the pure NC 2 case we chose the CM frame and from (B10) and (103) obtain which for the small argument (sM /4Λ 2 NC < 1) expansion of the Bessel function eliminates (1 − x 2 ) in denominator: The top left plot in Fig.8-log scale-shows the angular distributions as functions of x in the CM frame. How "flat" the NCQED contribution is depends on the ratio √ s/Λ NC . This ratio varies roughly from one to three in the scenarios considered in the PbPb part. The plots are evaluated at √ s = Λ NC = 100 GeV. In this case the NC contributions are already significantly larger than the standard model. Note that the pure NCQED contribution and interference contributions are at similar scale.
In the SM, LbyL scattering [152,155] is tree-level forbidden and goes only via quantum loops, 5 while in the NCQED it exists at tree level. In the 3D plot of Fig.8, we illustrated the exclusive ratio of LbyL cross sections σ γ NC /σ γ SM of the NC contributions (99)+(104) with the SM one in the CM frame, as functions of incoming energy E and the NC scale Λ NC . Integration over x = cos ϑ was performed with cutoff regulator = 10 −7 . The absence of cosine and presence of Bessel functions only produce mild oscillatory behavior which starts to show up gradually. To see transparently such behavior, we take a wide range of energy and relatively low NC scale. Thus, in the upper 3D plot in Fig.8, right corner, there are two peaks when energy grows very high and for the small NC scales, where the first corresponds to the pure NC contributions (99)-large due to the small value of the NC scale-while the second peak corresponds to the interference (104), respectively. The 3D plot in Fig.8 shows behavior similar as demonstrated further by the lines of In this section we continue searching for the physical phenomena of noncommutativity of spac-etime coordinates via ATLAS 208 Pb-ion experiments producing γγ and + − final states, sketched diagrammatically in Fig.9. Particularly, in an attempt to estimate a bound to the scale of spacelike NCQED, we take the convoluted exclusive cross sections and, under the assumption σ f T heory σ f Experiment for dominant f = γγ, + − channels, compare with convoluted ATLAS measured cross section. Note that produced gg andqq exclusive but virtual states in lead experiments get realized as real jets and real meson pair final states captured in detectors, respectively.
In the PbPb(γγ) → Pb * Pb * γγ reaction the incoming 208 Pb ions have survived the electromagnetic interaction, with a possible electromagnetic excitation due to the energy loss via photon emissions, denoted by ( * ). Hence, the final state consists of two low-energy photons and no further activity in the detector-in particular, no reconstructed charged-particle tracks originating from the IP; see [135]. Although the cross section for exclusive γγ → γγ process is tiny, various techniques can be used to study it indirectly [139].

A. Kinematics of the ultraperipheral ion scatterings
In the ultraperipheral scattering one considers the electromagnetic fields of the incoming heavy ions to be a spectrum of real photons moving along the beam axis. This simplifies the incoming photon kinematics in the Laboratory (nCM) frame to be a head-to-head collision. Thus, we take the PbPb(γγ) → Pb * Pb * γγ cross section as a convolution of exclusive theoretical differential cross section (107) [ (97) and (103) plus the SM one from Refs. [152][153][154][155][156]] with the incoming photon flux factors, integrated over the range of observed solid angle and measured outgoing photon energies [133,135].
The incoming photon energy spectrum space (ω 1 , ω 2 ) is usually reparametrized by the diphoton invariant mass m γγ , rapidity Y , and the volume element transformation: To determine the rapidity Y of the produced outgoing diphoton mass which, for symmetric systems, is maximal at Y = 0 when ω max 1 = ω max 2 ≈ γ/b min with b min being the minimum separation between the two equal charged nuclei of radius R N , we use for the lead ion spectrum, the impact parameter-dependent expression integrated from b min to infinity, with the requirement b min = R N plus a correction equivalent to the geometrical condition | b 1 − b 2 | > R min = 2R N to ensure that all collisions occur without hadronic overlap and breakup of the colliding beams. Propagated uncertainties to the final cross sections are of the order of ±20% for lead-lead ion collisions, covering different formfactor parametrization and the convolution of the nuclear photon fluxes [137].

B. Criteria of the ATLAS events selection denoted as the ATLAS cuts
In the ATLAS experiment ultraperipheral PbPb(γγ) → Pb * Pb * γγ events [133][134][135] the two incoming photons emitted by the 208 Pb ions are almost collinear, i.e., almost head to head: thus, they are selected by the following rules: -small diphoton transverse momentum p T and acoplanarity.
-diphoton invariant mass m γγ larger than 6 GeV. -transverse energy E T of each photon larger than E T ≥ E 0 = 3 GeV.
The first condition allows us to assume that incoming photons are real and collinear. This assumption simplifies the constraint on transverse energy to be only one, since the transverse momentum of outgoing photons must equal each other. The above conditions shall be translated into constraints implied on the theoretical cross section calculation.
We continue by assuming k 1 and k 2 as incoming momenta and outgoing momenta being k 3 and k 4 = k 1 + k 2 − k 3 , respectively. Then pseudorapidities y 3,4 of outgoing depend only on the angle to the z axis: where Y from Eq. (108) is the rapidity of the total momentum (k 1 + k 2 ) = (k 3 + k 4 ). By this we can express the last constraint and we have first simpler constraint on Y 1 2 To include the transverse energy constraint E T ≥ E 0 requires a bit of care. Using diphoton invariant mass m γγ , we can express this constraint as For a typical cut which requires transverse energy E T ≥ E 0 and both outgoing particles to bear pseudorapidity y 3,4 ≤ y 0 , the full sophisticated integral domain is defined as follows: and that was used in all of our further numerical integrations in fiducial phase space.

C. Equivalent photon approximation and fiducial cross section
Photons originating from the high-energy 208 Pb-ion nuclei can be viewed as the photon beam in the equivalent photon approximation (EPA) [166][167][168]. Because of the coherent action of all the protons in the nucleus the electromagnetic field surrounding each fast-moving nucleus with the charge Ze is very strong and it is approximated by a distribution of (almost) real photons moving along the beam direction. The simplest version of this approximation is to introduce a photon number function n(ω), where ω is the monophoton energy. Under such a convention convolution of exclusive cross section (107) with the photon number function in the fiducial phase space, was expressed in Refs. [163][164][165] as with simplest photon number functions n(ω i ) from the monopole form factor. However this did not quite take into account the ultraperipheral nature of the process: Here, γ = √ s NN /(2m N ) is the Lorentz relativistic factor for the Nth ion nucleus, and r 2 is the mean squared radius of that Nth nucleus. An improved version of the monopole photon number function N (ω, | b|) is constructed, including the " b-impact parameter" which is a two-dimensional vector that marks the position of the nucleus from the position of impact on the plane perpendicular to the beam direction [156]. Photon number function N (ω, b), being expressed in terms of the second kind modified Bessel function K 1 , is determined by the nucleus elastic form factor: In the ATLAS experiment, from 208 Pb-ion nuclear data [169] one finds ξ = 0.088 GeV, γ is the 208 Pb-ion Lorenz factor, and m u = 0.931 GeV is the atomic mass unit. The differential cross section in the fiducial phase space is then expressed as the following sevenfold integral: with R min = 2R ( 208 Pb) = 14 fm 71 GeV −1 , where in practice the fourfold integration over impact parameters b 1 and b 2 can be performed by defining the improved luminosity function d 2 Lγγ dmγγ dY : Kinematic cuts Collaboration σ γ SM (1 − loop) monopole (nb) mγγ > 5 GeV, |yi| <7 ATLAS 382 mγγ > 5 GeV, pt > 2 GeV, |yi| <7 ATLAS 190 pt >0.9 GeV, |yi| <0.7 ALICE 120 pt >5.5 GeV, |yi| <2.5 CMS 12 Taking this into account the cross section of PbPb(γγ) → Pb * Pb * γγ is expressed in the fiducial phase space by convoluting (107) with luminosity function (118) in the following form: Using this photon number function from one-loop and monopole approximations we first compute numerically the fiducial total cross section of the PbPb(γγ) → Pb * Pb * γγ collision in the SM for the ATLAS cuts and found it to be 57 nb, which is comparable to the previously reported values [133,135]. Two other ATLAS kinematic cuts discussed in the literature were also listed in Table I. In the same Table I we have also presented additional two cross sections, that is the 120 and 12 nb coming from the ALICE and CMS experiments, respectively. In the case of monopole approximation our integrated cross section at the SM one loop given in Table I relatively well agree with numbers published in Table I in Ref. [138]. D. NCQED contribution to PbPb(γγ) → Pb * Pb * γγ reaction for ATLAS cuts Applying the protocol tested in the former subsection, we calculate the combined contribution from both the SM one-loop and NCQED tree-level γγ → γγ amplitudes in the ATLAS PbPb(γγ) → Pb * Pb * γγ experiment, with 208 Pbion center of mass energy √ s NN = 5.02 TeV [135], which is sensitive to most of the 4π solid angle, implying that the integral (117,119) ranges over the ATLAS cuts: ϑ cutoff with = 0.0102, with 0 ≤ ϕ ≤ 2π, m Pb =0.9315 GeV, and γ = 2693, giving the data span as a function of diphoton invariant mass m γγ , presented in Fig.10 and Table II. It seems that in the LbyL scattering measurements ATLAS has recorded only events with less than 30 GeV diphoton invariant mass [ Fig.3(a) in Ref. [134], or the same figure as Fig.2(b) in Ref. [135]]. Inspecting Fig. 4 in the Appendix in Ref. [134], which displays event 453765663 from run 366994 with diphoton invariant mass of m γγ = √ s = 29 GeV, we conclude that this LbyL scattering event and other recorded events certainly belong to the class of exclusive  relatively low-energy (with respect to m t ) processes, which is in accord with the conclusion in Ref. [138] that the cross section for elastic γγ → γγ scattering could be measured in the present 208 Pb-ion collisions only for subprocess energies smaller than 30 GeV. We first consider matching the reported experimental fiducial cross section of 78 nb [135] by combination (107)-(119), which yields an unimpressive scale Λ NC of about 72 GeV, a value which had been ruled out some time ago. Since the relevant NC scale is small in the ATLAS PbPb(γγ) → Pb * Pb * γγ scenario, we notice that pure NC amplitude M γ NC 2 gives the main contribution to the fiducial cross section diphoton invariant mass distribution peak value of dσ γ dmγγ (right panel in Fig.10), which occurs moderately above Λ NC as expected from the Bessel function J 0 dependence [see Eqs. (97) and (103)] on the scale Λ NC . So the high-energy peak value of dσ γ dmγγ in the left panel in Fig. 10 is a direct consequence of the pure NC amplitudes and determines the possibility of observing excessive events due to the NCQED.

E. Recent ATLAS lead experiments
Proposing that the difference between LHC-ATLAS measured PbPb(γγ) → Pb * Pb * γγ scattering cross section in ultraperipheral lead-lead collisions and the SM prediction is mainly due to the noncomutativity of space-time (106), by using Eqs. (92), (106) and (113)- (116) we obtain the total cross section in the fiducial phase space to be 59.8 nb, relatively close to the experimental mean value, however only at Λ NC = 100 GeV. Inspecting Table II one may clearly see that present ATLAS experiment basically gives only the NC scale smaller than 100 GeV. Namely, from the left panel in Fig. 10 and Table II we see that second peak values of ∼ (1, 0.1, 0.01) nb/GeV exist for Λ NC (53,72,100) GeV, respectively. In addition, after checking the distribution of cross section with respect to diphoton invariant mass m γγ , we conclude that such a scenario does not explain the observation of the ATLAS experiment, since the cross section induced by NCQED amplitude comes from a wide band at m γγ 100 GeV, as nicely presented by the ATLAS Collaboration in Fig.2(b) in Ref. [135]. Given the current integrated luminosity accumulation speed (∼ 1 nb −1 /yr) for current LHC, we doubt that any stronger bound of Λ NC 100 GeV can be achieved from the present-day ATLAS PbPb(γγ) → Pb * Pb * γγ experiment.
The lepton pair production in the framework of ATLAS lead experiments shall be presented and discussed further.
F. LbyL scatterings in the next-generation collider experiments

Convoluted cross section versus diphoton invariant mass distribution
Because of the importance of the possibility to discover noncommutativity of space-time, we shall now discuss Fig.11 in some more detail. The first left dashed and solid line large peaks, up to m γγ ∼ 100 GeV, correspond to the dσ γ SM dmγγ and the dσ γ SM+NC dmγγ diphoton fiducial cross section distributions, while the second solid lines peaks correspond to the sum of interference and the pure NCQED terms, respectively. This outstanding second peak reflects the evolution of NC factors with respect to energy scales: When energy scales become larger than Λ NC , the NC factors become oscillatory and bounded. Consequently, the NC amplitudes are very small at very low energies where the SM contribution dominates and then increase quickly and compete with the exponentially decreasing luminosity factor [156] to give the rising side of the NC second peak. Such new feature is a genuine peculiarity of our θ-exact NCQED.
Since the future higher-energy experiments could increase the experimental sensitivity to the noncommutative effects drastically, we also estimate potential improvements one would expect from the next generation of colliders,   in particular, upgraded CERN HL-LHC 6 up to √ s NN =14 TeV, the proposal for next-generation Chinese hadron collider SppC with energy √ s NN =70 TeV [170,171], and from the Future Circular Collider (FCC) proposal up to √ s NN =100 TeV [172][173][174][175][176][177][178]. Assuming that all the kinematic ATLAS cuts remains the same, while only the energy scale or Lorentz factor scales up 5, 7, or 20 times with respect to the current ATLAS value √ s NN =5.02 TeV, we estimate the noncommutative scale Λ NC corresponding to a high-energy dσ γ dmγγ maximum with ∼ 0.01 nb/GeV magnitude and present that in Table III and Fig.11. We find that Λ NC are about 2.5, 3.1, or 5.2 times ∼ 100 GeV, for the nowadays ATLAS energy scale 5.02 TeV, respectively. Also the left plot in Fig.11 and Table III Table III), shifted with respect to that in the left plot in Fig.10. Such improvements are considerable, yet still insufficient to make this kind of experiment(s) convenient for bounding Λ NC when comparing to other known bounds, unless the integrated luminosity can be further improved by multiple scales of the next-generation hadron colliders. Because of the SM W loops the left and right panels of Fig.11 at m γγ between 150 and 200 GeV show a small dent arising from the W-loop contributions to the pure SM and the SM×NC interference terms, respectively. Namely, W contribution starts to show up by inducing a sharp drop or turn between 100 and 200 GeV presented in Fig. 1 in [155]. That dent is unfortunately experimentally invisible.
The existence of a second peak, with maximum m γγ | max given in Table III, reflect the evolution of NC factors with respect to energy scales: When energy scales are much smaller than Λ NC , the NC factors are increasing as monomials with high power. Once the energy scales become larger than Λ NC the NC factors become oscillatory and bounded. Consequently, the NC amplitudes are very small at very low energies where the SM contribution dominates and then increase fast and compete with the exponentially decreasing luminosity factor [156] to give the rising side of the NC second peak. So when the energy scale goes beyond Λ NC the NC amplitudes start to deviate from monomial increase, so that dσ γ dmγγ falls down quickly. The subsequent oscillatory behaviors of the NC factors are fully suppressed by the luminosity function and cannot be seen in this process.
In the case of the θ-expanded NCQED model cross sections presented in the right panel in Fig.11 (dotted lines for θ-expanded model) show the behavior of dσ γ dmγγ with the same two peaks. Here we give example for two higher energies, red lines correspond to √ s NN = 25.10 TeV (Λ NC = 257 GeV), and the blue lines are for √ s NN = 35.14 TeV (Λ NC = 311 GeV), respectively. So the same NC peak shows up but a bit higher and shifted from about 700 to about 1050 GeV, that is shifted up for about 30% in diphoton energy. The same shall work for other energy cases, like in Fig.11, showing this way explicitly that the important NC peak shows up in both the expanded and unexpanded theory, showing that it is an genuine feature arising from the Moyal-Weyl manifold of the NCQED. Our estimate show that like in the original ATLAS case in Fig. 10, the NCQED contribution in the next-generation hadron collider ultraperipheral heavy ion scattering scenarios also manifests as a second peak of dσ γ dmγγ at a diphoton mass range moderately higher than Λ NC (see Fig. 11). Also, both panels in Fig.11 transparently show that the point at which NCQED starts decoupling 7 from the SM, and the NCQED contribution second peaks, as functions of ( √ s NN , Λ NC ) follows more or less a similar sort of pattern.
It is also important to note that the SM PbPb(γγ) → Pb * Pb * γγ background in the NC second peak region is a few orders of magnitude lower than the NC second peak. This simply states that at each of the NC second peaks, i.e. at each m γγ | max from Tables II and III, in the left plot in Fig.11, the corresponding 11 could be a bit misleading because of the log-log scales. It is a quite wide band if the diphoton invariant mass scale is linear. The hardest problem is still the absolute value of the peak which will require having totally 1000 nb −1 integrated luminosity. To be absolutely clear to have a signal would require about ten events at the peak. According to the CERN ATLAS and CMS Collaborations, further planes of upgrading the LHC to HL-LHC, and the Chinese proposal for SppC [171], as well as within the FCC [174][175][176][177][178][179][180] proposal, to increase luminosity by at least a factor of 10 beyond today's LHC is quite possible, and ten events at the peak are, in fact, accessible.

The ϑ and ϕ angular distributions of convoluted LbyL scattering
The broad NC peak discussed in the last subsection raises one additional question on how to distinguish it from other (currently unknown) physical processes within the same regime. It is long known that NCQED breaks Lorentzian symmetry and induces nontrivial dependence of the differential cross section to the transverse angle ϕ. We have also learned from the left plot in Fig.8 that the γγ → γγ differential cross section also depends on the longitudinal angle ϑ in a unique way. So we investigated the PbPb(γγ) → Pb * Pb * γγ differential cross section with respect to x = cos ϑ and ϕ for ATLAS and its FCC analogy scenarios. The results are summarised in Figs. 12 and 13. As we can see from the plots, the x dependence of the PbPb(γγ) → Pb * Pb * γγ differential cross section follows the same trend as the γγ → γγ process in the left plot in Fig.8. The NC contributions are flatter than the SM contribution yet the difference is not significant within the angular range considered.
The ϕ dependence of the different cross section is, on the other hand, considerably more significant. The NC contributions vanish at the orientations where the transverse momentum is parallel to the transverse projection of B θ while reaching the maximum at the orientations perpendicular to the transverse projection of B θ . This can be easily explained by observing the fact that in a head-to-head collision geometry, the NC factor k 1|2 θk 3|4 is proportional to | k 1,2 · (B ⊥ θ × k ⊥ )|. We therefore conclude that the ϕ variation of the differential cross section is likely an appropriate 14 TeV, ΛNC = 311 GeV, lower plots). In each scenario, the left 3D plot shows a total differential cross section with respect to x and mγγ. The right plot compares the behaviors of the SM (red line), interference (green line), and NCQED (blue line) contributions as well as the total differential cross section (black dashed line) with respect to x when mγγ is at the NC peak position listed in Table III. signature for the NCQED contributions, given that enough events could be available for determining the anisotropy.

G. Noncommutative background processes in the LbyL scatterings
In the ATLAS Letter [135], a large portion of material is, as it should be, devoted to the very important analysis of background processes. Namely the dominant background processes, i.e., the central exclusive productions (CEPs) gg → γγ and γγ → e + e − as well as other fake-photon background contributions, 8 are in the ATLAS experiment estimated from data, and the statistical significance against the background-only hypothesis is found to be 8.2 standard deviations [135]. From the perspective of our NCQED contributions to the exclusive γγ → γγ process, we discuss sources to the noncommutative background contributions in the LbyL collisions which are represented by the following exclusive three-level processes.
i) The first two are dilepton pair annihilation and productions in the CM frame. Our Fig. 7 shows oscillatory behavior of both processes, while from Eqs. (85) and (89) we have σ A;P QED > 0 (see [149]), and σ A;P NC < 0, i.e. both upper plots) and FCC ( √ sNN=35.14 TeV, ΛNC = 311 GeV, lower plots). In each scenario, the left 3D plot shows the total differential cross section with respect to ϕ and mγγ. The right plot compares the behaviors of the SM (red line), interference (green line) and NCQED (blue line) contributions as well as the total differential cross section (black dashed line) with respect to ϕ when mγγ is at the NC peak position listed in Table III. The transverse angle ϕ is chosen to be zero at the direction of the projection of B θ onto the transverse plane.
iii) Both, the 2-cascade and the 3-cascade processes shall be understood as a kind of long-distance effect, contributing to the γγ final state. However, due to the sign switch in the NCQED contributions they should partially cancel general CEPs as dominant background contributions to the fake γγ final state, see the discussion in Ref. [135]. iv) Exclusive Compton and Møller scatterings cannot produce a fake γγ final state at all. v) All the above is much welcome since that way the noncommutative backgrounds become even more suppressed. Therefore, we conclude that NCQED background contributions to the LbyL scatterings should be in the heavy ion collision scenario(s) experimentally invisible, i.e., the NCQED framework contribution to the γγ final state, as a true single final state, becomes more exposed and dominated; see Figs. 10 and 11.

H. Lepton pair production
One may also consider NCQED contributions to the PbPb(γγ) → Pb * Pb * + − reaction via tree-level exclusive pair production processes γγ → + − . The full spin averaged exclusive differential cross section in NCQED is given in Eq. (89). We notice that the NC factor is sensitive to θ 03 component only in the geometry of the exclusive  Table III. γγ → + − processes on an ATLAS PbPb(γγ) → Pb * Pb * + − setting. Here we use lightlike noncommutativity and set |θ 03 | = | − θ 13 | = Λ −2 NC and by convolution estimate the NCQED correction to the PbPb(γγ) → Pb * Pb * + − process. We assume exactly the same ATLAS kinematic cuts and energy scales to calculate convoluted lepton pair production from (89) in analogy to the left panel in Fig. 11. Resulting diphoton fiducial cross section distributions dσ P QED dmγγ and the dσ P QED+NC dmγγ are displayed in Fig.14, which shows that NCQED plots give a negative correction to the PbPb(γγ) → Pb * Pb * + − process. The relative magnitudes tend to be large at high energies and not much different from the QED background. At the same time, the absolute magnitudes in the same range are inherently very small. Also, it is interesting that theoretical decouplings of NCQED from ordinary QED at the ATLAS experiment energy of 5.02 TeV (black lines in Fig.14) start to show up gradually at maximal excess of diphoton invariant mass ∼ 100 GeV, 3 times higher than in the case of LbyL scattering. Because of the cosine dependence of energy in dilepton cross section (89) oscillatory behavior with respect to the LbyL scattering starts to show up at a bit higher energy, i.e., when approaching a diphoton invariant mass of ∼200 GeV. We therefore conclude that PbPb(γγ) → Pb * Pb * γγ collision producing exclusively two monophotons is a much better probe to the NC scale than PbPb(γγ) → Pb * Pb * + − process.

VIII. CONCLUSIONS
In this work we present an explicit proof that all tree-level two-by-two scattering amplitudes in NCQED are invariant with respect to an invertible SW map. This surprisingly simple result is in accordance with our prior formal analysis that U(1) NC(S)YM with and without the SW map are equivalent to each other on shell [131,132]. Our result, like its more formal precedent, is nontrivial because the reversible θ-exact SW maps from noncommutative to commutative fields are highly nonlocal 9 . For this reason the results on the redefinition of local fields do not automatically apply to NCQED with the SW map. Motivated by this newly found explicit invariance, we revisit the differential and total cross sections of NCQED two-by-two processes.
It is, however, second to note that the collinear singularities of QCD processes do not match their NCQED counterparts. The NCQED Compton scattering exhibits the same t-channel collinear singularity as QCD quark-gluon scattering but at a lower power. And, while NCQED LbyL scattering shares the same color-or star-product-ordered amplitudes with QCD free gluon-gluon scattering amplitude, the extra momentum-dependent noncommutative factors completely cancel the collinear divergences from the fractions of Mandelstam variables and make the NCQED γγ → γγ process collinear divergence-free.
Comparing with the upper bounds derived from tree-level NC processes, we consider the collinear singularity in the NCQED Compton scattering as more worthy for further investigations in the near future. If we adopt an analogy to QCD here then this singularity appears to suggest an unknown and nontrivial soft physics of NCQED, which would be instrumental in canceling the IR singularities in the hard process(es). The difference between collinear singularities in NCQED and QCD, in particular the lack of it in the NCQED γγ → γγ process, seems to indicate that the conjectured soft physics may be very nontrivial by itself. One would further wish that such completion could result in a theory compatible with the long-known negative one-loop β function in U(1) NCYM [85] and/or the (in)famous UV-IR mixing [93,95,98,105]. On the other hand, the scaling is QCD. A full investigation along this direction lies beyond the scope of this work. The authors would, nevertheless, be truly grateful if some progress can be achieved soon.
As an application of the theoretical progress made in this work, we investigate the possibility of detecting NCQED signals in the ultraperipheral PbPb(γγ) → Pb * Pb * γγ and PbPb(γγ) → Pb * Pb * + − future scattering experiments at HL-LHC and FCC, presented in Figs. 11 and 14, respectively. We follow the equivalence photon approximation and employ a monopole form factor to calculate the total cross section of both processes. Our results indicate that the PbPb(γγ) → Pb * Pb * γγ channel probes NC scales better than PbPb(γγ) → Pb * Pb * + − . Yet neither channel could probe really large scales (Λ NC 0.5 TeV) even after we extrapolate the ion energy to beyond the next-generation hadron collider proposal(s).
The first term Γ µ I in Eq. (A2) represents FRs for the NCQED without the SW map and is in agreement with Refs. [42,45,47], while additional terms denoted as Γ µ II , are arising due to the θ-exact SW map. Note that the Lorentz structure of the second term in Eq. (A2) is the same as the NC neutrino-photon coupling, in the case of massive neutrinos [86,88].
From the two-photon-two-electron diagram in Fig.15 the regular recipe yields: which is symmetric under simultaneous pair exchanges (ν, k 4 ) ↔ (µ, k 1 ). Here, Γ µν I represents a term irrelevant for the on-shell tree-level computations; thus, we indicate that fact by equating it with zero. This FR is coming from the action (17) and/or (18) induced by the θ-exact SW maps for gauge field strength (5) and fermions (9), respectively. Considering the gauge sector by employing a straightforward reading-out procedure from S f 3 (14) for the third diagram in Fig. 15 we have the Feynman rule Γ µ1µ2µ3 for the triple-photon vertex in momentum space [92]: with vertex function V µ1µ2µ3 I+II (k 1 , k 2 , k 3 ) and F 2 (k 2 , k 3 ) being taken from Refs. [92,104]: Here again, only the first line of Eq. (A6), with function 2 sin k2θk3 2 instead of F 2 (k 2 , k 3 ), from SW NCQED triplephoton FR (A5) and (A6), represents the FRs for the NCQED theory without the SW map, in agreement with Refs. [42,55], respectively.

b. Four-photon coupling: General considerations and FRs
Finally, the e 3 order SW map for the gauge field strength produces the four-photon interaction diagram in Fig. 15 arising from the action S f 4 (15). The Feynman rule Γ µνρτ for fourth diagram in Fig.15 is much more complicated, even with additional gauge freedom κ's fixed to unity. Those FRs are generally given in detail in Eqs. (16), (B.1), (B.2), (B.4)-(B.6) in Ref. [92]. After the inspection of tadpole contributions to the photon polarization tensor for two different SW maps (I) and (II), represented with gauge freedom parameters (κ, κ i )'s and (κ, κ i )'s, respectively, in Ref. [92] we have found that for κ = κ 1 = κ 2 = κ 3 = κ 4 = κ 1 = κ 2 = κ 3 = κ 4 = 1 the sum of bauble and tadpole contributions to polarization tensor (I) given by Eq. (37) and contributions to polarization tensor (II) (38) are equal. Thus, we shall use the FR from Ref. [92] for the SW maps (I) and (II) with all κ's=1 and with momenta k i in Fig.15 being the incoming ones. Because of number of typos in FRs from Appendix B in Ref. [92], next we repeat all, but corrected, FR equations: where S 4 denotes permutations over all ({k i , µ i }, ∀i = 1, 2, 3, 4), pairs simultaneously. Capital indices I + II, as before from Eqs. (A1)-(A6), indicate a splitting of total contribution into the one I arising from the action without the SW map and the second II, as a contribution induced by the SW mapping. First we split Γ A into five and Γ B into three pieces, respectively: where clearly only the first term in Eq. (A9), the V A1 , belongs to the class of non-SW map-induced terms, since that term was induced due to the -commutator term of the NCYM gauge Lagrangian ; see details in [42]. What follows are additional, but irrelevant terms: Tensor structures of remaining Γ i , i = 1, 2, 3, 4, 5, terms are listed next where the above products induced functions in momentum space: F 2 , F 3 and F (I) are defined in Refs. [90,92], respectively. Tensor structures of V i 's are given next: For any additional details, we refer to Ref. [92]. However, the irrelevant terms are proportional to the free field equation for a µ (x) , therefore, since they vanish due to the EOM and gauge condition, they are irrelevant to the tree-level scattering processes we are heading for. Thus, we collect now the relevant terms at the e 2 order which does not vanish on shell. As we mentioned at the beginning of the paper in Sec. II, we shall work out only the FRs from relevant gauge action given in Sec. II.B. Thus, the irrelevant FRs (A10)-(A12) we are giving just for the sake of completeness, but we do not need them further on. However, one should not forget that those irrelevant terms are indeed very much relevant for arbitrary loop and/or non-Abelian computations.
c. Four-photon coupling: Relevant terms FRs Starting with action (12) and obtaining relevant part (23), from where we already have FRs arising from the very first term in Eq. (23) presented in Ref. [92] and denoted as Γ A and/or V A in (A7) and (A9), respectively. Thus FRs from second, third, and fourth terms in Eq. (23) we determine next. Note that only 2 is present twice, thus producing at the very end terms proportional only to F 2 (k 1 , k 2 )F 2 (k 3 , k 4 ), etc. The relevant four-photon vertex in accord with fourth diagram in Fig.15 is where V A is exactly the same as terms given in Eqs. (A7)-(A9). The W 1 is coming from the second term in the first line, while W 2 , and W 3 are coming from first and second terms in second line of Eq. (23) respectively. After we split the above W 1 into three terms, we have: Taking the case of photons there is s + t + u = 0 from where, together with energy conservation, we express a pair of incoming energies ω 1,2 in terms of pair of outgoing energies ω 3,4 , and vice versa: Now, by using noncommutative matrix θ and kinematics, from Eqs. (24) and (B1), respectively, we write down analytically all phase factors generated by the NCQED introduced in the previous section, and in the NCM frame of head-to-head 2 → 2 collision necessary further to compute and analyze NC contributions to all electron and/or photon scattering and/or annihilation processes: c 03 (cos ϑ − 1) + sin ϑ (c 01 − c 13 ) cos ϕ + (c 02 − c 23 ) sin ϕ , c 03 (cos ϑ + 1) + sin ϑ (c 01 + c 13 ) cos ϕ + (c 02 + c 23 ) sin ϕ , c 03 ω 1 (cos ϑ − 1) + ω 2 (cos ϑ + 1) (B4) 2. The ϕ integrations over the noncommutative phase factors Using Eq. (B4), after decomposing sinus functions in Eqs. (92) and (93), and for other processes, with the help of [151] we may neglect terms which in the integration over dϕ give zeros 10 and obtain all contributing phase factors needed: 10 Note that while computing the ϕ distribution of certain processes we have used non-ϕ integrated, i.e. full NC phase factors given in Eq. (B4).
cos(k 1 θk 2 ± k 3 θk 4 ) = cos cos(k 1 θk 3 + k 2 θk 4 ) = cos with the following definitions of the noncommutative coefficients Now, since the SM amplitudes M hi SM , ∀i = 1, ..., 5-see [152][153][154][155][156]-do not depend on ϕ, the form which contains only the linear type of phase factor products 2 sin α sin β = cos(α − β) − cos(α + β) shall be applied for computations of interference terms in Eq. (107). Second, from above general form we have to integrate over dϕ a square phase factor coming from Eq. (93) as 4 sin 2 α 2 sin 2 β 2 = 1 − cos α − cos β + 1 2 (cos(α + β) + cos(α − β)). Combining both types of phase, the 2 sin α sin β and the 4 sin 2 α 2 sin 2 β 2 , we found two types of integral I, andÎ, i.e. I ± , andÎ 0,± respectively: amplitudes, after making simple replacement Λ 2 NC → 2Λ 2 NC in (B6) phases, and using (B8) structure, we obtain the integrals we need: and we note that for the spacelike case c 03 = 0 the difference (I 1234 Now we find dϕ integrals over the (B9) structure of phases as: (B11) Explicit cancellation of the SW(II) map-induced contributions to γγ → γγ process from Fig.4 in the particular case of Moyal U(1) NCQED is very demanding, because some quite nonstandard tensorial analysis within a sum of relevant Feynman diagrams is needed. That is the reason why lengthy tensorial contractions in a sum of s−, t− and u-channel 3γ-vertex diagrams plus 4γ-vertex diagram from Fig.4 executed by hand and computer is given in detail next, showing explicit vanishing of M γ II , i.e. Eq. (62). We start with details of 3γand 4γ-vertices and relevant amplitudes, in the (s, t, u) channels, generated by both the non-SW(I)|SW(II) map-induced terms, respectively. a. 3γ-vertices, (s, t, u) channels: Non-SW(I)|SW(II) map-induced terms Using FRs (A5) and (A6), and from Fig.4, and after splitting the non-SW parts from the SW parts, applying momentum conservations and free field equations, we obtain the following twice six non-SW(I)|SW(II) map-induced 3γ-vertex contributions : needed to show the Ward identity for Eq. (61) and vanishing of scattering amplitude (62).
b. 3γ-diagrams, (s, t, u) channels: SW(II) map-induced amplitudes The sum of 3γ-vertex diagrams defined in Fig.4 gives the following amplitude: which produces all nine terms, from s, t and u channels in Eq. (C3) separately, as: and with self-evident shorthand notations: The above 3γ-vertex diagram s-, t-, u-channel amplitudes (C4), (C5), and (C6), respectively, should be cancelled with the corresponding s-, t-, u-channel 4γ-vertex diagram amplitudes we give next.
e. Cancellation 2 The remaining first two terms in the ninth lines from (C4), (C5), and (C6) are 1 st two terms in 9 th lines of (C4) + (C5) + (C6) and we compare them with the sum of the corresponding contributions to the four-photon diagram in Fig.4 denoted as a sum of all S 4 permutations of Γ A3 (C10) and W 11 (C13), respectively: It is easy to see that Eq. (C21) exactly cancels the sum of the remaining first two terms arising from the ninth lines in Eqs. (C4), (C5), and (C6) and given as Eq. (C20), respectively. Q.E.D.
The meaning of the above cancellations (C18)−(C25) producing nontrivial zero for LbyL (62), together with nontrivial zero in Compton case (53) and with trivial zeros for Møller (31) and Bhabha (39) cases, is that the SW map-induced contributions to all scattering amplitudes vanish, proving thus explicitly that in both the matter and the gauge sector of the on-shell U(1) NCQED scattering amplitudes with and without SW maps are equal or equivalent one to each other, as indicated generally in Refs. [131,132].