Higgs and vector unparticle production via collision in the randall – sundrum model

VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 4 (2019) 93-98 93 Original Article  Higgs and Vector Unparticle Production via   Collision in the Randall – Sundrum Model Nguyen Thi Hau1,*, Dao Thi Le Thuy2 1Hanoi University of Mining and Geology, 18 Pho Vien, Dong Ngac, Hanoi, Vietnam 2Faculty of Physics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam Received 14 September 2019 Revised 08 November 2019; Accepted 11 November 2019 Abstract: This paper studies the production of Higgs boson and vector U  unparticle, which has been proposed as an option for    collision by s, t, u-channels in the Randall-Sundrum model. The cross-section is presented and numerical evaluation is detailed. The study results reveal that the cross-section increases as fast as 1.8 2Ud  . The advantageous directions to collect Higgs boson and U  are either the same or opposite to the initial muon beams by s-channel. The U  exchange contribution is much larger than muon exchange contribution. Keywords: Randall-Sundrum model, cross-section, Higgs, vector unparticle, muon. 1. Introduction The discovery of Higgs boson in 2012 at the LHC [1, 2] verify the correctness of the standard model, but it still has many unanswered issues [3]. In order to solve this remaining problems, the extended models are proposed. In this paper, we are interested in two extended models, namely the Randall- Sundrum model and unparticle physics. The Randall-Sundrum model [4] is one of the extended models that brings many new physical consequences. This model extends 4-dimensional space-time with x coordinates to 5-dimensional space-time with coordinates ( , )x  . The fifth dimension is a single 1 2/S Z orbifold of radius r. The 5- ________ Corresponding author. Email address: nguyenthihau@humg.edu.vn https//doi.org/ 10.25073/2588-1124/vnumap.4375 N.T. Hau, D.T.L. Thuy / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 4 (2019) 93-98 94 dimensional space-time has two 3-branes placed at two fixed points, the Planck brane (UV brane) at 0  and the TeV brane (IR brane) at   . Unparticle physics proposed by Georgi [5] in 2007, which includes the standard model fields and the Banks-Zaks fields [6]. The two fields interact through the interchange of particles with a large mass scale UM . In unparticle physics, there are scalar U , vector U  and spinor sU unparticles. Their interactions with standard model particles are presented in Ref. [7]. In the previous paper we have studied the effect of vector unparticle on some of the high energy processes in the Randall-Sundrum model [8-10]. In this article, we discuss the U  production in the process hU     in the Randall-Sundrum model. The paper is organized as follows. The Feynman rules for the vector unparticle interactions with leptons and Higgs boson; the Higgs boson interactions with leptons and photons are given in section 2. The calculation results of the cross-section of    collisions are discussed in section 3. Finally, in section 4 we give a brief summary and discussions. 2. Formalism As already mentioned, in this work we only consider the vector unparticle in the unparticle physics and the Randall-Sundrum model. The interaction of vector unparticle with leptons according to the Feynman rules is shown in Fig. 1[11]. 51 1 (1 ) Ud U i        Fig. 1. Feynman rules for the interaction of vector unparticle with leptons. In Ref. [12] shows Feynman rules for the interactions of Higgs boson with photons and leptons in the Randall-Sundrum model (Fig.2). Based on the efficiency theory, we proposed the Feynman rule for the interaction of Higgs boson with vector unparticles in this model (Fig. 2a) following: 1 2 1 2. ; ,XiC k k k k X U         W ( ) 2 mig d b m    a) b) Fig. 2. Feynman rules for the interaction of Higgs boson with photons (vector unparticles) (a) and leptons (b). N.T. Hau, D.T.L. Thuy / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 4 (2019) 93-98 95 where 2 2( ) ( ) 2 i fV i c i i Y r i C g e N F b b g              [12] and UC is the coefficient that we included based on the efficiency theory and we evaluated the cross-section according to UC C . 3. The process hU     in the Randall-Sundrum model In the rest of the paper, we concentrate on the possibility of Higgs boson and vector unparticle production in the    collisions according to s, t, u-channels in the Randall-Sundrum model. The Feynman diagrams of the above processes are shown in Fig. 3. Fig. 3. Feynman diagram for Higgs boson and vector unparticle productions at    collision The matrix elements of the process hU     through by s, t, u-channels in Fig. 3a, b, c, respectively are given by the expression:  2 2 51 2 1 2 2 21 ( ) ( ) (1 ) ( ) . ( ).2sin( ) dudu s s U s sdu u i A M q C v p u p q k g q k k du                      (1) * 51 2 2 11 2 2 w ( ) ˆ( )( ) ( ) (1 ) ( ) 2 ( )u t td tu gm d b M v p q m k u p m q m                (2) 5 *1 2 2 11 2 2 w ( ) ˆ( ) (1 ) ( )( ) ( ) 2 ( )u u ud uu gm d b M v p k q m u p m q m                (3) where 1 2 1 2 1 1 2 2 1 2 1 2; ;s t uq p p k k q p k k p q k p p k            , 2 s s s q q g q             . The matrix elements squared for the different channel are given by: 2 2 2 2 2 21 2 1 2 2 1 1 21 2 1 2 ( ) {( ) [ 2( ) ( 2( )( ) ( ) )] 2sin( ) dudu s s U s s s sdu u s i A M q C q k p p p q p q p p q du q                 2 2 2 2 2 2 2 1 2 1 2 2 2 1 2 1 2 24 1 [2( )( ) ( ) (2( )( )( ) ( ) ( ) )s s s s s s s q p k p k p p k p q p q q k p p q q k q     2 2 2 1 2 1 2 2 2 1 2 22 1 (2( )( )( ) 2( )( )( ) 2( )( ) )]}s s s s s s p k p q q k p k p q q k p p q k q    , (4) N.T. Hau, D.T.L. Thuy / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 4 (2019) 93-98 96 2 2 2 2 21 2 1 2 1 2 1 11 2 2 w ( ) 4 {2( )( ) ( ) ( ) 2 ( )} 2 ( )u t t t t td tu gm d b M p q p q p p q m p p m p q m q m                , (5) 2 2 2 2 21 2 1 2 1 2 1 21 2 2 w ( ) 4 {2( )( ) ( ) ( ) 2 ( )} 2 ( )u u u u u ud uu gm d b M p q p q p p q m p p m p q m q m                . (6) The differential cross-section for hU     at a center-of-mass energy s is given by: 1 2 1 1 , cos 64 kd M d s p     (7) where 2 1 2( )s p p  ,  is the angle between 1p and 1k . The cross-section is plotted taking 1 1  , 1U TeV  [11], UC C , 500s GeV and 1 2Ud  [13], in Fig. 4. Fig 4. The cross-section of hU     as a function of Ud Here, the cross-section increases fastly as 1.8 2Ud  . Therefore, we evaluated it at 1.9Ud  . In Fig. 5 we charted the differential cross-section of the Higgs and vector unparticle production as a function of cos at 1.9Ud  . The center-of-mass energy is chosen as 500s GeV . The figure shows that the value of the differential cross-section by s-channel is much larger than t, u-channels. It reaches maximum values when cos 1   . For that reason, the advantageous directions to collect Higgs boson and vector unparticle are the same or opposite direction to the initial ,   beams. Finally, Figure 6 shows the range of the cross-section of hU     as a function of s at 1.9Ud  . It increases by s through s-channel and decreases with higher s through t, u-channels. For the vector unparticle exchange contribution, the higher the center-of-mass energy increases, the bigger the cross-section gets. For the muon exchange contribution, the higher the center-of-mass energy increases, the smaller the cross-section gets. Moreover, the value of the cross-section of s-channel is much larger than t, u-channels. N.T. Hau, D.T.L. Thuy / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 4 (2019) 93-98 97 a) b) c) Fig. 5. The differential cross-section of hU     as a function of cos Fig 6. The cross-section of hU     as a function of s N.T. Hau, D.T.L. Thuy / VNU Journal of Science: Mathematics – Physics, Vol. 35, No. 4 (2019) 93-98 98 Conclusions In summary, we have calculated the cross-section of process hU     by s, t, u-channels. The result shows that the cross-section increases fastly as 1.8 2Ud  . According to the s-channel, the advantageous directions to collect Higgs boson and vector unparticle are the same or opposite direction to the initial ,   beams. The vector unparticle exchange contribution is much larger than muon exchange contribution. Acknowledgments The authors would like to thank the sponsors of the Hanoi University of Mining and Geology for the basic science project in 2019, code T19-06. References [1] ATLAS Collaboration, Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1-29. https://doi.org/10.1016/j.physletb.2012.08.020. [2] CMS Collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30-61. https://doi.org/10.1016/j.physletb.2012.08.021. [3] Particle Data Group Collaboration, Review of particle physics, Chin. Phys. 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