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Dark matter (DM) is a special matter kind. The total amount of dark matter should be around five times bigger than that of ordinary matter. Currently, there are many dark matter models, such as the Cold Dark Matter model with a cosmological constant (ΛCDM) [1] and the vector-fermion dark matter model.
HNUE JOURNAL OF SCIENCE Natural Sciences 2019, Volume 64, Issue 10, pp 70-76 This paper is available online at http://stdb.hnue.edu.vn DOI: 10.18173/2354-1059.2019-0074 DARK MATTER FERMION PRODUCTION AT LEPTON COLLIDERS VIA PHOTON - DARK PHOTON - PHOTON EXCHANGE Le Nhu Thuc1 and Dao Thi Le Thuy2 Hanoi National University of Education Faculty of Physics, Hanoi National University of Education Abstract The dark matter fermion production has studied at lepton colliders via photon-dark photon-photon exchange, the results show that, the production cross-section at e e collider is the same as at collider when the initial beams are unpolarized or both the initial beams are left- or right-polarized In case of mix between both the initial beams are left-polarized with both the initial beams are right-polarized, the cross-section is very much bigger than the e e cross-section, and the cross-section strongly depends on the polarization of initial beams Keywords: Dark matter fermion, dark photon, unpolarized, polarized Introduction Dark matter (DM) is a special matter kind The total amount of dark matter should be around five times bigger than that of ordinary matter Currently, there are many dark matter models, such as the Cold Dark Matter model with a cosmological constant (ΛCDM) [1] and the vector-fermion dark matter model [2] The dark photon is a hypothetical hidden sector particle, proposed as a force carrier potentially connected to DM [3] This new force can be introduced by extending the gauge group of the SM with a new abelian U(1) gauge symmetry In this paper, we study the process l l via the exchange of photon-dark photon-photon when beams l , l are unpolarized and polarized, where l l are e e and ; and is dark matter fermion Specifically, we evaluate the contribution of dark photon on the cross-sections when the initial beams are polarized and unpolarized Received August 8, 2019 Revised October 18, 2019 Accepted October 25, 2019 Contact Le Nhu Thuc, e-mail address: thucln@hnue.edu.vn 70 Dark matter fermion production at lepton colliders via photon - dark photon - photon exchange Content 2.1 Interaction Lagrangian The effective interaction Lagrangian of photon ( ) and dark matter fermion ( ) was given by [4]: i Lint ( d ) F , (1) where F A A , and , d correspond to the magnetic dipole moment, and the electric dipole moment of the DMF The effective interaction Lagrangian for the dark photon ( V ) and photon ( ) with respective field strengths V and F is [5]: m2 1 Lint F2 V2 F V V VV eJ em A , 4 2 The corresponding Feynman rules are (2) i( 5d )(qˆ q ) q q i g 2 2 q mV mV Figure Feynman rules for the photon couplings with DMF and V propagator 2.1 The cross-section of the l l collision The corresponding Feynman diagrams for the pair production of dark matter fermion in l , l collision via V exchange are shown in Figure Figure The Feynman diagrams for the process l l via V exchange 71 Le Nhu Thuc and Dao Thi Le Thuy For unpolarized l , l beams, the square of matrix element is given by: e 4(qp2 )q ( p1k2 ) M 2 4( p k )( p q ) 4( p q )( p k ) 2 2 mV2 q mV 4(qp2 )( p1q)(k2 q) 4(qp1 )q ( p2 k2 ) 4(qp1 )(k2 q)( p2 q) mV2 mV2 mV2 2(qp1 )(qp2 ) q2 4 [(p p ) m ] l mV4 mV mV 2 q (k2 q) 4[(p1p ) ml ](k2 q) [8(2 d 2 )(qk1 ) 4( 2 d 2 )(k1q)] 4(qp2 )(qp1 ) 4(qp2 )(qp1 ) 4(qp1 )(q p2 ) 4(qp1 )(qp2 ) 4( p1 p2 ) 4( p1 p2 ) mV2 mV2 mV2 mV2 2(qp1 )(qp2 ) q2 2 4 [(p p ) m ] q 4[(p p ) m ]4 l l 4 mV mV mV [ 8(2 d 2 )(qk1 )(k2 q)+[4(2 d 2 )q (k2k1 ) 4( 2 d 2 )m2 q ] 4(qp2 )( p1k2 )(qk1 ) 2 4( p1k2 )( p2 k1 ) 4( p1k1 )( p2 k2 ) m2' 4(qp2 )( p1k1 )(qk2 ) 4(qp1 )( p2 k2 )(qk1 ) 4(qp1 )( p2 k1 )(qk2 ) mV2 mV2 mV2 2(qp1 )(qp2 ) q2 4 [(p p ) m ] l mV4 mV mV 2 2 (qk2 )(qk1 ) 4[(p1p ) ml ](k1k2 ) [ 4( d )q ] 4(qp2 )(qk1 )( p1q) 2 4( p2 k1 )( p1q) 4( p1k1 )( p2 q) mV2 4(qp2 )( p1k1 )q 4(qp1 )(qk1 )( p2 q) 4(qp1 )q ( p2 k1 ) mV2 mV2 mV2 2(qp1 )(qp2 ) q2 2 2 4 [(p p ) m ] q ( qk ) 4[(p p ) m ]( qk ) l 1 l [4( d )( k q)] 4 m m m V V V 8(qp2 )q ( p1q) 8(qp1 )q ( p2 q) 8( p1q)( p2 q) mV2 mV2 72 Dark matter fermion production at lepton colliders via photon - dark photon - photon exchange 2(qp1 )(qp2 ) q2 2 4 [(p p ) m ] q 4[(p p ) m ] q l l mV4 mV mV [ 4(2 d 2 )(k2 k1 )+4( 2 d 2 )m2 4(2 d 2 )(k2 k1 ) 4(2 d 2 )m2 4(2 d 2 )(k2k1 ) 4(2 d 2 )m2 ] (3) In case of both the l and l beams are polarized, we have M LL M RR e 4(qp2 )q ( p1k2 ) 4( p k )( p q ) 4( p q )( p k ) 2 2 mV2 q mV 4(qp2 )( p1q)(k2 q) 4(qp1 )q ( p2 k2 ) 4(qp1 )(k2 q)( p2 q) mV2 mV2 mV2 2(qp1 )(qp2 ) q2 4 (p1p2 ) 4 mV mV mV 2 2 q (k2 q) 4(p1p )(k2 q) [8( d )(qk1 ) 4( d )(k1q)] 4(qp2 )(qp1 ) 4(qp2 )(qp1 ) 4(qp1 )(q p2 ) 4(qp1 )(qp2 ) 4( p1 p2 ) 4( p1 p2 ) mV2 mV2 mV2 mV2 2(qp1 )(qp2 ) q2 4 (p p ) q 4(p p )4 2 mV4 mV mV [ 8(2 d 2 )(qk1 )(k2 q)+[4(2 d 2 )q (k2k1 ) 4( 2 d 2 )m2 q ] 4(qp2 )( p1k2 )(qk1 ) 2 4( p1k2 )( p2 k1 ) 4( p1k1 )( p2 k2 ) mV2 4(qp2 )( p1k1 )(qk2 ) 4(qp1 )( p2 k2 )(qk1 ) 4(qp1 )( p2 k1 )(qk2 ) mV2 mV2 mV2 2(qp1 )(qp2 ) q2 2 4 (p p ) ( qk )( qk ) 4(p p )( k k ) 1 2 [ 4( d ) q ] 4 m m m V V V 4(qp2 )(qk1 )( p1q) 2 4( p2 k1 )( p1q) 4( p1k1 )( p2 q) mV2 4(qp2 )( p1k1 )q 4(qp1 )(qk1 )( p2 q) 4(qp1 )q ( p2 k1 ) mV2 mV2 mV2 2(qp1 )(qp2 ) q2 4 (p p ) mV4 mV mV 2 q (qk1 ) 4(p1p )qk1 ) [4( d )(k2 q)] 8(qp2 )q ( p1q) 8(qp1 )q ( p2 q) 8( p1q)( p2 q) mV2 mV2 73 Le Nhu Thuc and Dao Thi Le Thuy 2(qp1 )(qp2 ) q2 4 (p p ) q 4(p p ) q 2 mV4 mV mV [ 4(2 d 2 )(k2 k1 )+4( 2 d 2 )m2 4(2 d 2 )(k2 k1 ) 4(2 d 2 )m2 4(2 d 2 )(k2k1 ) 4(2 d 2 )m2 ] , 2 e q2 m q (k2 q) ml2 (k2 q) l q mV mV mV RR M M LL [8(2 d 2 )(qk1 ) 4( 2 d 2 )(k1q)] q2 4 ml2 q ml2 [ 8( 2 d 2 )(qk1 )(k2 q)+[4( 2 d 2 )q (k2 k1 ) 4( 2 d 2 )m 2 q ] m ' m ' q2 2 ml2 (qk2 )(qk1 ) ml2 (k1k2 ) [ 4( 2 d 2 )q ] m m ' ' q2 2 ml2 q (qk1 ) ml2 (qk1 ) [4( 2 d 2 )(k2 q)] m m ' ' q2 4 ml2 ] q ml2 q [ 4( 2 d 2 )(k2 k1 )+4( 2 d 2 )m2 m m ' ' 4(2 d 2 )(k2 k1 ) 4(2 d 2 )m2 4(2 d 2 )(k2k1 ) 4(2 d 2 )m2 ] (4) M LR M RL M LR M RL and From the square of matrix elements above, we evaluate the differential cross section (DCS) as a function of cos by the expression: d k1 M d cos 64 s p1 (5) The results are shown in Figure for e e me , G ;5e 1V m 0,1068MeV ; Here we choose mV 103 GeV ; 10 12 , 2,3.1010 GeV ; d d , m 30MeV ; d 2, 26.1010 GeV [4]; s 3000GeV (CLIC) We see that the DCS is unchanged when cos changes from 1 to in the cases of the e , e beams are unpolarized, and as well as left-polarized or right-polarized Therefore, the direction to collect , is the same for direction to the e , e beams 74 [5], Dark matter fermion production at lepton colliders via photon - dark photon - photon exchange a) b) c) d) Figure The DCS as a function of cos Next, we proceed to evaluate the total cross section of these colliders as function of mass center energy s , it is shown in Figure The cross section increases while s increases from 200GeV to 3000GeV for the e , e beams unpolarized, and as well as left- or right-polarized While, the mixing cross section decreases, it can see from figure 3.3d In addition, when the e , e beams are left - polarized or right-polarized, the cross-section is twice times as large as the cross-section when the e , e beams are unpolarized This is shown in figure 3.3a and figure 3.3b For the cross section, we obtained the results the same for the e e , the cross section when the initial beams are unpolarized or both the initial beams are left - polarized or right - polarized In case of mix between both the initial beams are left-polarized with both the initial beams are right-polarized, from figure 3.2 c, d and figure 3.3 c, d, we can see that the cross-section is very much bigger than the e e cross-section 75 Le Nhu Thuc and Dao Thi Le Thuy a) b) c) d) Figure The cross – section as a function of s Conclusions The cross sections of the process l l depend strongly on the polarization of initial beams The direction to collect ( , ) not depend on the direction of the l , l beams The cross section increases when s increases for left or right-polarized and unpolarized of l , l beams The cross-section is very small, however, we maybe search for DM particles from l , l collisions if interactive energy is large enough REFERENCES [1] R Primack, 1997 Dark Matter and Structure Formation in the Universe Cite as: arXiv: astra ph/9707285 [2] T Ahmed, et al., 2018 Muti-componert dark matter: the vector and fermion case Eur Phys J C, 78: 905 [3] R Essig, J A Jaros, W Wester, P Hansson Adrian, S Andreas, T Averett, O Baker, B Batell, M Battaglieri, et al., 2013 Dark Sectors and New, Light, Weakly-Coupled Particles arXiv:1311.0029 [hep-ph] [4] Atanu Guha, Selvaganapathy J and Prasanta Kumar Das, 2017 Q-deformed statistics and the role of a light dark matter fermion in the supernova SN1987A cooling Phys Rev D 95, 015001 [5] Haipeng An, Maxim Pospelov and Josef Pradler, 2013 Dark Matter Derectors as Dark Photon Helioscopes Phys Rev Lett 111, 041302 76 .. .Dark matter fermion production at lepton colliders via photon - dark photon - photon exchange Content 2.1 Interaction Lagrangian The effective interaction Lagrangian of photon ( ) and dark. .. beams 74 [5], Dark matter fermion production at lepton colliders via photon - dark photon - photon exchange a) b) c) d) Figure The DCS as a function of cos Next, we proceed to evaluate the total... 8(qp1 )q ( p2 q) 8( p1q)( p2 q) mV2 mV2 72 Dark matter fermion production at lepton colliders via photon - dark photon - photon exchange 2(qp1 )(qp2 ) q2 2 4 [(p p )