Probing Space-Time Noncommutativity in the Bhabha Scattering
PProbing Space-Time Noncommutativity in the BhabhaScattering
Linda Ghegal
Mentouri Constantine1 University, Route Ain El Bey 25017, Constantine, Algeria.We investigate Bhabha scattering with the Seiberg-Witten expendednoncommutative standard model scenario to first order of the noncom-mutativity parameter Θ µν . This study is based on the definition of thenoncommutativity parameter that we have assumed. We explore the non-commutative scale Λ NC ≥ . . .
1. Introduction and motivation
The noncommutative standard model based on the noncommutativityof the space and time variables. The noncommutative space-time is defor-mation of the ordinary one that can be realized by representing ordinaryspace-time coordinates x µ by Hermitian operators ˆ x ν [ˆ x µ , ˆ x ν ] = i Θ µν = i c µν Λ NC (1)Where c µν are dimensionless parameters and Λ NC is the energy scale wherethe noncommutative effects of the space-time will be relevant. In the presentwork, we are especially interested in the spacetime noncommutative stan-dard model based on the Moyal-Weyl (WM) product [1] and Seiberg-Witten(SW) Maps [2]. In the WM product formalism, to define a field theory onnoncommutative spacetime, we replace the ordinary ordinary product bythe WM (cid:63) -product( f (cid:63) g )( x ) = exp (cid:18)
12 Θ µν ∂ x µ ∂ y ν (cid:19) f ( x ) g ( y ) | y = x (2)and the SW maps, express noncommutative fields and parameters as localfunctions of the commutative fields and parameters. (1) a r X i v : . [ h e p - ph ] O c t Linda˙GHEGAL printed on October 19, 2020
The existence of the Seiberg-Witten map to all orders, leads to expansionsfor matter field ψ , gauge fields V α and gauge parameters λ α as followingˆ ψ = ˆ ψ [ ψ, V ] = ψ + 12 Θ µν V µ ∂ ν ψ + i µν [ V µ , V ν ] ψ + O (cid:16) Θ (cid:17) (3)ˆ V α = ˆ V α [ V ] = V α + 14 Θ µν { ∂ µ V α + F µα , V ν } + O (cid:16) Θ (cid:17) (4)ˆ λ α = ˆ λ α [ λ, V ] = λ α + 14 Θ µν { ∂ µ λ α , V ν } + O (cid:16) Θ (cid:17) (5)The full description of the minimal noncommutative standard model(mNCSM), built on the U (1) Y ⊗ SU (2) L ⊗ SU (3) C group, and the completeFeynman rules derived from NCSM, including new interactions between thegauge bosons in the non-minimal NCSM version, are presented in [3]. Theextension of SM to noncommutative space-time with motivations comingfrom string theory and quantum gravity provides interesting phenomeno-logical implications since scale of noncommutativity could be as low as afew TeV, which can be explored at the present or future colliders. Due tothe breaking of Lorentz invariance for fixed Θ µν background, noncommuta-tivity of space time leads to dependence of cross section on azimuthal anglewhich is absent in Standard Model (SM) as well as in other models beyondthe SM. Thus, the azimuthal dependence of the cross section is a typicalsignature of noncommutativity and can be used in order to discriminate itagainst other new physics effects. We have found this dependence to bebest suited for deriving the sensitivity bounds on the noncommutative scaleΛ NC , and this is the main purpose of this work, is to derive the bounds onthe noncommutative scale Λ NC in the Bhabha scattering in the context ofmNCSM, by using the framework introduced in [3].The present manuscript will be organised as follows. In the next sec-tion we will present some bounds for the noncommutativity parameter Θ µν provided by the most recent papers. We will then conclude with the resultsthat we have abtained of noncommutative scale Λ NC in the Bhabha scatter-ing, in the framework of the mNCSM, using the SW maps to the first orderof the NC parameter Θ µν .
2. Overview of bounds on noncommutative scale
We will dedicate this section to the bounds which can presently be stud-ied in several processes and systems, including estimates for some exper-iments [4-16]. Most of them used noncommutative antisymetric constantmatrix Θ µν analogous to the eletromagnetic field strength tensor; denotethe time-like components c oi by (cid:126)E and the space-like components c ij by (cid:126)B . inda˙GHEGAL printed on October 19, 2020 However, they ofen use two different parameterizations for the noncommu-tative parameter Θ µν . The first one, is considered as elementary constantin nature, its direction is fixed in the specific celestial Cartesian reference,so one should take into account these rotation effects on Θ µν in this frame,before moving towards the phenomenological investigations. In the secondparameterizations, the values of the tow vectors are fixed as (cid:126)E = √ ( (cid:126)i + (cid:126)j + (cid:126)k )and (cid:126)B = √ ( (cid:126)i + (cid:126)j + (cid:126)k ). The phenomenological consequences of the non-commutative space-time have been explored in several studies for givingbounds on nonvomutative parameter. From the e + e − → qq subprocess andby comparing with the ALEPH and OPAL data of LEP in [4] found twoconstraints on the noncommutative space-time scale, 215 GeV ≤ Λ NC ≤ ≤ Λ NC ≤
310 GeV respectively. In [5], the authors in-vestigate the TeV scale signatures of NC space-time in the e + e − → γγ , e − γ → e − γ and γγ → e + e − using polarized beams in the mNCSM. Theprocess e + e − → γγ at the International Linear Collider (ILC), in the frame-work of non-minimal NCSM, with anomalous triple gauge boson couplings isstudied in [6]; knowing that, all these bounds deal with the first parameter-ization. And for the constant parameterization case, the bound Λ NC ≥ pp → γ, Z → l + l − at the LargeHadron Collider (LHC) in the framework of the non-minimal NCSM [7].The Higgs boson pair production e + e − → HH in the linear collider (LC)gives the range Λ NC = 0 . − . NC ∼ TeV. The Higgs boson productionprocess e + e − → ZH and SM forbidden process e + e − → HH were investi-gated in the framework of the mNCSM [10], with Feynman rules involvingall orders of the noncommutative parameter which are derived using reclu-sive formation of SW map. And by applying the same Feynman rules, theauthors in [11] have studied the Higgs plus Z-boson production at a futureelectron-positron collider to explore the sensitivity of future accelerator ex-periments to noncommutativity, and obtained as lower limit of Λ NC = 1 . NC ≥ . NC andprojected sensitivities for the future collider with neutrino-electron Scat-tering was summarized in [13]. The authors in [14] have also explored anexperiment to probe the effects of noncommutative structure in quantumoptical.Since the models considered by the various authors are different, andsince also the experimental setups probe different energy and length scales,therefore, one must be careful in the comparison of the various results ofthe noncommutative scale Λ NC . Linda˙GHEGAL printed on October 19, 2020
3. Results and conclusion
In this section, we investigate the effect of space-time noncommutativityon Bhabha scattering, with the ansatz for noncommutative parameter Θ µν that we have assumed and we provide the numerical results of our investi-gation. We are restricting ourselves only to the first order in Θ; thus theinterference between SM and NC term can provide required corrections tocross section. In order to check the sensitivity of Bhabha scattering in thecontext of noncommutative space-time and specially the phenomenologicaleffects resulting, we have defined the noncommutative parameter with thehelp of the gamma matrices, which the noncommutative structure is de-termined by some spinor background on which the gamma-dependent Θ µν acts. c µν = 12 (cid:16) σ µν + ( σ µν ) + (cid:17) (6)knowing that σ µν = i γ µ γ ν − γ ν γ µ ) (7)where γ µ are Dirac matrices.We study now, how the space-time noncommutativity affects the e − ( p ) e + ( p ) → e − ( p ) e + ( p ) scattering process, through the exchange of γ and Z bosonsat tree level (Via the s and t channel). The Feynman rules to the first orderin Θ are given in [3].Feynman rule for e ( p in ) − e ( p out ) − γ ( k ) vertex= ieQ f (cid:20) γ µ − i k υ (Θ µνρ p ρin − Θ µν m f ) (cid:21) (8) e ( p in ) − e ( p out ) − Z ( k ) vertex= ie sin 2 θ W { ( γ µ − i k ν Θ µνρ p ρin ) (cid:16) C fV − C fA γ (cid:17) − i µν m f [ p νin (cid:16) C fV − C fA γ (cid:17) − p νout (cid:16) C fV + C fA γ (cid:17) ] } (9)where Θ µυρ = Θ µν γ ρ +Θ νρ γ µ +Θ ρµ γ υ , C fV = T f − Q f sin θ W and C fA = T f with θ W is the Weinberg angle and Q f = ∓ e ∓ .also p out Θ p in = p µout Θ µν p νin = − p in Θ p out . The momentum conservationreads as p in + k = p out .The corresponding Feynman diagrams are shown in Fig. 1 inda˙GHEGAL printed on October 19, 2020 e − e + → ( γ, Z ) → e − e + in the NCSM The scattering amplitude to the first order in Θ can be written asFor the γ mediated diagram A γ = A γs + A γt (10)where A γs = e (cid:104) i ( p Θ p + p Θ p ) (cid:105) × (cid:20) v ( p , s ) γ µ u ( p , s ) u ( p , s ) γ µ v ( p , s ) (cid:18) is (cid:19)(cid:21) (11)and A γt = − e (cid:104) − i ( p Θ p + p Θ p ) (cid:105) × (cid:20) u ( p , s ) γ µ u ( p , s ) v ( p , s ) γ µ v ( p , s ) (cid:18) it (cid:19)(cid:21) (12)For the Z mediated diagram A Z = A Zs + A Zt (13)where A Zs = e sin θ W (cid:104) i ( p Θ p + p Θ p ) (cid:105) × (cid:34) v ( p , s ) γ µ Γ − A u ( p , s ) u ( p , s ) γ µ Γ − A v ( p , s ) (cid:32) is − m Z (cid:33)(cid:35) (14)and A Zt = − e sin θ W (cid:104) − i ( p Θ p + p Θ p ) (cid:105) × (cid:34) u ( p , s ) γ µ Γ − A u ( p , s ) v ( p , s ) γ µ Γ − A v ( p , s ) (cid:32) it − m Z (cid:33)(cid:35) (15) Linda˙GHEGAL printed on October 19, 2020 where Γ ± A = ( C eV ± C eA γ ),with s = ( p + p ) , t = ( p − p ) and u = ( p − p ) The spin averaged squared-amplitude is given by (See the Appendix Afor more details). (cid:12)(cid:12) ¯ A (cid:12)(cid:12) = 14 (cid:88) spins (cid:12)(cid:12)(cid:12) A γ + A Z (cid:12)(cid:12)(cid:12) (16)The differential cross section can be written as dσd Ω = 164 π s (cid:12)(cid:12) ¯ A (cid:12)(cid:12) (17)where θ and φ are polar and azimuthal angles respectively, with dσd Ω = d cos θdφ .We can obtain the cross section σ = σ ( √ s, Λ NC , θ, φ ) as σ = (cid:90) − d (cos θ ) (cid:90) π dφ dσd Ω (18)Our results are based on Feynman rules for NCSM given in [3]. We ana-lyze the total cross section in the presence of space-time noncommutativity.The results are shown in Fig. 2 (GeV)s500 600 700 800 900 1000 1100 1200 1300 1400 1500 ( pb ) s = 0.8 TeV NC L = 1 TeV NC L = 1.2 TeV NC L SM Fig. 2. The total cross section for the process e − e + → ( γ, Z ) → e − e + [pb] as afunction of center of mass energy E com = √ s [GeV] The ordinary SM is presented by black curve and the NCSM with differentcurves; green, blue and red, with the corresponding Λ NC = 0 . , . . inda˙GHEGAL printed on October 19, 2020 TeV, respectively. As can be seen, the noncommutative correction increaseson increasing the center of mass energy of the collisions, and we found thatthe total cross section departs significantly from the standard model valueas the machine energy starts getting larger than 1 . NC and anti symmetric matrix c µν , which is defined with thehelp of the gamma matrices, which the noncommutative structure is de-termined by some spinor background on which the gamma-dependent Θ µν acts, and we have shown how the positron-electron scattering process at thetree level is affected by space-time noncommutativity. The NC effects arefound to be significant for Λ NC = 0 . , . . E com = 1 . NC = 0 . NC = 1 . Acknowledgements
Linda GHEGAL would like to express her gratitude to Prof. FedeleLIZZI for his useful discussions, during the peroid of her stay at the Uni-versity of Naples Federico II, Italy.
Appendix A
The spin squared-amplitude can be written as [16] |A| = |A γs | + |A γt | + |A Zs | + |A Zt | − Re ( A γs A γ ∗ t ) − Re ( A Zs A Z ∗ t )+2 Re ( A γs A Z ∗ s ) − Re ( A γt A Z ∗ s ) + 2 Re ( A γt A Z ∗ t ) (A.1)with the several terms |A γs | = e s (1 + C ) × T r [ /p γ µ /p γ ν ] T r [ /p γ µ /p γ ν ] , (A.2) |A γt | = e t (1 + D ) × T r [ /p γ ν /p γ µ ] T r [ /p γ ν /p γ µ ] , (A.3) Linda˙GHEGAL printed on October 19, 2020 |A Zs | = e θ W ) ( s − m Z ) (1 + C ) × T r [ /p γ ν Γ − A /p γ µ Γ − A ] T r [ /p γ ν Γ − A /p γ µ Γ − A ] , (A.4) |A Zt | = e θ W ) ( t − m Z ) (1 + D ) × T r [ /p γ ν Γ − A /p γ µ Γ − A ] T r [ /p γ ν Γ − A /p γ µ Γ − A ] , (A.5) − Re ( A γs A γ ∗ t ) = − e st Re [(1 + i C )(1 + i D ) × T r [ /p γ ν /p γ µ /p γ ν /p γ µ ] , (A.6)+2 Re ( A γs A Z ∗ s ) = e θ W ) s ( t − m Z ) × Re (cid:20) (1 + 14 C ) T r [ /p γ ν / Γ − A /p γ µ ] T r [ /p γ ν Γ − A /p γ µ ] (cid:21) , (A.7) − Re ( A γs A Z ∗ t ) = − e θ W ) s ( t − m Z ) × Re (cid:20) (1 + i C )(1 + i D ) T r [ /p γ ν Γ − A /p γ µ /p γ ν Γ − A /p γ µ (cid:21) , (A.8) − Re ( A γt A Z ∗ s ) = − e θ W ) t ( s − m Z ) × Re (cid:20) (1 − i C )(1 − i D ) T r [ /p γ ν Γ − A /p γ µ /p γ ν Γ − A /p γ µ (cid:21) , (A.9)+2 Re ( A γt A Z ∗ t ) = e θ W ) u ( u − m Z ) × Re (cid:20) (1 + 14 D ) T r [ /p γ ν / Γ − A /p γ µ ] T r [ /p γ ν Γ − A /p γ µ ] (cid:21) , (A.10) − Re ( A Zs A Z ∗ t ) = − e θ W ) ( u − m Z )( t − m Z ) × Re (cid:20) (1 + i C )(1 + i D ) T r [ /p γ ν Γ − A /p γ µ Γ − A /p γ ν Γ − A /p γ µ Γ − A (cid:21) . (A.11)The factors C and D are given by C = p Θ p + p Θ p D = p Θ p + p Θ p The differential cross section is calculated using the center-of-mass framefor the Bhabha e − ( p ) e + ( p ) → ( γ, Z ) → e − ( p ) e + ( p ) scattering process,in which the four-momenta of the incoming and outgoing particles are givenby inda˙GHEGAL printed on October 19, 2020 p = p e − = √ s (1 , , ,
1) = ( E , (cid:126)P ) ,p = p e + = √ s (1 , , , −
1) = ( E , (cid:126)P ) ,p = p e − = √ s (1 , sin θ cos φ, sin θ sin φ, cos θ ) = ( E , (cid:126)P ) ,p = p e + = √ s , − sin θ cos φ, − sin θ sin φ, − cos θ ) = ( E , (cid:126)P ) , (A.12)In evaluating the matrix element square, we have ignored the mass of ingoingparticles ( m e (cid:39) (cid:126)P + (cid:126)P = 0 = (cid:126)P + (cid:126)P . In the relativist limit s (cid:29) m , we get s = ( E + E ) = 4 E (with E = E = E ), u = − s (1 + cos θ )and t = − s (1 − cos θ ).In our analysis, we have assumed an ansatz for Θ µ,ν (See Eq. (6)).The NC antisymmetric tensor Θ µ,ν is defined with the help of the gammamatrices. Using this definition we may write C and D as follows C = p Θ p + p Θ p = √ s NC [1 + sin θ (cos φ + sin φ )] , (A.13) D = p Θ p + p Θ p = √ s NC [sin θ (cos φ + sin φ )] . (A.14)REFERENCES [1] J. E. Moyal, Proc. Camb. Phil. Soc. , 99 (1949).[2] N. Seiberg and E. Witten, J. High Energy Phys . , 032 (1999).[3] X.Calmet et al., Eur. Phys. J.C , 363 (2002); B. Melic et al., Eur. Phys. J.C , 483 (2005); B. Melic et al., Eur. Phys. J. C , 499 (2005).[4] M. Haghighat, et al. Phys. Rev. D , 016007 (2010).[5] S.K. Garg, et al., J. High Energy Phys. , 024 (2011).[6] N.G. Deshpande, et al. Phys. Lett. B , 150 (2012).[7] J. Selvaganapathy et al., Phys. Rev. D , 116003 (2016).[8] P.K. Das, et all. Phys. Rev. D , 056002 (2011).[9] S. Aghababaei, et all. Phys. Rev. D , 047703 (2013).[10] W. Wang, et all. Phys. Rev. D , 045012 (2011).[11] M. Ghasemkhani, et all. Prog. Theor. Exp. Phys. (2014).[12] J. Selvaganapathy, et all. Int. J. Mod. Phys. A , 1550159 (2015).[13] S. Bilmi¸s et al., Phys. Rev. D , 073011 (2012).[14] S. Dey et al., Nucl. Phys. B , 578 (2017).[15] L. Ghegal and A. Benslama, Int. J. Mod. Phys. A , 1450199 (2014).[16] P. K. Das, et all. Phys. Rev. D77