The influence of CP violation on the mass of some particles in the NMSSM

TẠP CHÍ KHOA HỌC SỐ 2/2016 25 
 THE INFLUENCE OF CP VIOLATION ON THE MASS 
 OF SOME PARTICLES IN THE NMSSM 
 Nguyen Chinh Cuong, Pham Xuan Hung, Le Hong Thang1 
 1 Ha Noi National University of Education 
 Abstract: The Next to Minimal Supersymmetric Standard Model (NMSSM) is established 
 from the Minimal Supersymmetric Standard Model (MSSM) when a gauge chiral single 
 superfield Sˆ is added. The mixing of additional states leads to the appearance of the new 
 states which make the Higgs sector of the NMSSM changed in comparision with the 
 MSSM. In the NMSSM, there are seven Higgs bosons (while there are five ones in the 
 MSSM) with three scalar CP-even Higgs S1,2,3 (ms1< ms2< ms3), two pseudoscalar CP-
 odd Higgs P1,2 (mP1 < mP2) and a pair of charged Higgs h . The mixing of Gauginos and 
 Higgsions also lead to the new states in the Neutralino and the Chargino sectors. The CP 
 violation in the NMSSM has the influence on the mass of the three types of particle 
 mentioned above. The purpose of this study is to evaluate the influence of CP violation on 
 the mass of the Higgs bosons, the Neutralinos and the Charginos in the NMSSM. 
 Keywords: Higgs boson, Neutralino, Chargino, CP violation, NMSSM. 
1. INTRODUCTION 
 The extension of the Standard Model (SM) towards the supersymmetry [1,5] has 
brought a new hope to solve the hierarchy problem as well as the remaining shortcomings 
of the SM. The minimal extension requires the addition of the two SU(2) doublets Higgs. 
The vacuum expectation values (vevs) of Hu and Hd create the mass for quarks and leptons. 
The MSSM can be considered as the minimal extended model of the Higgs sector. The 
Lagrangian of the MSSM must contain the  parameter which is the supersymmetric 
(SUSY) mass parameter [6]. The question is why the  parameter has the same order with 
the soft SUSY breaking terms  MSUSY. This problem is called the -problem of the 
MSSM [7]. 
 A simple way to solve this problem is to create a  parameter in the same way as 
creating the mass for quarks and leptons in the SM. The  parameter is created by the 
1 Nhận bài ngày 10.01.2016, gửi phản biện và duyệt đăng ngày 25.01.2016. 
 Liên hệ tác giả: Nguyễn Chính Cương; Email: nccuong@hnue.edu.vn. 
26 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI 
Yukawa interaction of Hu and Hd with the scalar field. To ensure the order of the  
parameter, there must be the vevs of scalar field which is derived from the soft SUSY 
breaking condition. Then, the  parameter does not have the quantum number of the group 
SU(3)CLY SU(2) U(1) [7,8]. The added scalar field is the S singlet which is the 
complex scalar components of the chiral scalar superfield S$ . This is called the Next to 
Minimal Supersymmetric Standard Model (NMSSM) which is also denoted as the 
(M+1)SSM. 
 The search for the Higgs boson as well as new particles in the NMSSM has 
encountered many difficulties due to this has to be done in high-energy processes and 
mainly based on their decay products. In 2012, the Higgs boson with the approximate mass 
of 125GeV was confirmed by experiments and this has brought us new knowledges and 
chances. The mass of Higgs bosons, Neutralinos and Charginos is defined through the 
mass matrices which contain the CP violating phases [7]. To demonstrate this, we have 
studied the inluence of CP violation on the mass of the three types of particle mentioned 
above. 
2. CONTENT 
2.1. The Higgs bosons mass 
 The Higgs potential was constructed in the NMSSM in the following form: 
 22 * * T 2 2
 VVV FDsoft V  S(HHHH)(H 1122 1 H) 2  S
 112
 gHH2 * (g 2 g)(HH 2 * HH) * 2
 282 1 2 1 2 1 1 2 2
 2 1
 m2 H * H m 2 H * H m 2 S  A (H T H )S  A S 3 c.c (1)
 h111 h222 S 1 2 3
 In which , A, k, Ak, g1, g2 are the interaction coefficients 
 The Higgs doublets H1 and H2 can be developed in the form: 
 v11 S iAsin  H .cos
 H1 * , H2 , S = (x + X + iY) (2) 
 H .sin 
 v22 S iAcos 
 We obtained the mass-squared matrix of the neutral Higgs boson (S1, S2, X) 
and (A, Y): 
TẠP CHÍ KHOA HỌC SỐ 2/2016 27 
 2
 2 2kx 2 2mZ 2
 mcZ2 (A  )xt  (v  )s   x(A  kx) 2vxc    vs(A   2kx) 
 22 
 m2 kx x 
 2 2Z 2 2 2 (3) 
 MS  (v )s 2  x(A  kx) ms Z  (A  ) 2vxs    vc(A   2kx) 
 2 2 t 
 v2 A s
 2 2 2 2 
 2 vxc  vs (A 2kx) 2  vxs  vc (A 2kx) (2kx) kxA 
      2x  
 2 x(A kx)
   v(A 2kx)
 s 
 2 . (4) 
 MP 
 
 v(A 2kx) 3kxA 2  kv22 s v A s
 k 2 2x  2 
 Here, the parameters were derived from the tadpole minimum conditions [9], 
 22
v 12  175(GeV) . 
 The mass of charged Higgs pair was easily obtained as: 
 2
 2 (2R R  ) 
 mm22   ss (5) 
 h W 2 sin(2 )
2.2. The mass of Neutralinos and Charginos 
 Breaking the symmetry led to the mixing of Gauginos and Higginos and producing 
Neutralinos and Charginos from which the mass matrix of Neutralinos and Charginos was 
obtained as follows: 
 MsMc(MM)cs22 0 0 0 
 w w w w
 22
 (MM)cs w w MsMc w w m z 0 0
 M 0 0 m  x sin 2   x cos 2  0 (6) 
 % z
 0 0  x cos 2   x sin 2   v
 0 0 0 v 2kx
 M 2s mw
 M (7) 
 % 
 2c mw  x
2.3. CP violation in the Higgs sector of the NMSSM 
 We studied the CP violation (charged symmetry and parity symmetry) by using the 
effective potential method for the neutral Higgs sector in the NMSSM. We can assume that 
all parameters are real so that the Higgs potential can conserve the CP symmetry. 
However, the CP violation at the 1-loop level in the Higgs potential of the NMSSM may 
occur with some complex phases[7]. The CP violation of the Higgs potential could still be 
28 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI 
clearly shown through the radiative correction. The complex phases which appeared 
through the scalar Higgs bosons and pseudoscalar Higgs bosons mixings have presented 
the CP violation[7]. 
 The CP violation could be done in the NMSSM right at the tree-level in the Higgs 
potential, which was different in the MSSM. A non-trivial complex phase appeared after 
the three Higgs fields were redefined. These complex phases can be developed in the 1-
loop effective potential from the tree-level Higgs potential if the top squarks mass is 
degenerated. 
 22 2 2 2 2 1 3
 VS = mHh 1 mH h 2 mS S  AHHS 1 2 ASH.c (7) 
 12 3
 The soft SUSY breaking VS sector in the Higgs potential has two additional 
parameters A and A (both with mass dimension), three soft masses m ,m and m . In 
 λ K HH12 S
general, all λ, x, k and Ak could be complex. Among these, A and kAk could be adjusted 
to become real and positive by redefining the phases of H1, H2 and S. Thus, the tree-level 
Higgs potential could have at most a physical phase. We could choose the phase ϕ in x = 
x.ei[7]. This phase appeared when the scalar and pseudoscalar were mixed in the tree-level 
Higgs pontential of the NMSSM. It existed when one of the two couplings A and 
1
  A in the equation (7) was three times greater than the other coefficient [10]. 
3 
2.4. The inluence of CP violation on the mass of the particles in the NMSSM 
 To calculate the mass of the Higgs boson, Neutrinos and Charginos, we used the 
numerical methods to diagonalize the mass matrices (3-7). The eigenvalues of these mass 
matrices corresponded to the mass of the following particles: two neutral pseudoscalar 
Higgs bosons Pi (i = 1, 2), three neutral scalar Higgs bosons Si (i = 1, 2, 3), a pair of 
 0 
charged Higgs h , five Neutralinos %j (j = 1,.., 5) and two pairs of Charginos %j (j = 1, 2). 
Next, we chose the parameters of the NMSSM according to reference documents [9, 10]: x 
 i
= 178.e ; λ = 0.8; k = 0.1; tanβ = 3; sin α = - 0.58; Ak = 6 and Aλ = 486 to study the 
influence of CP violation on the mass of the particles. The results obtained are as follows: 
TẠP CHÍ KHOA HỌC SỐ 2/2016 29 
 Figure 1. The influence of CP violation on Figure 2. The influence of CP violation on 
 the mass of P1. the mass of P2. 
 The results in Figure 1 and Figure 2 show that the mass of the pseudoscalar Higgs P1 
is around 79GeV and it is in the range from 502.5 to 506GeV for the preudoscalar Higgs 
P2. The influence of CP violation on the mass of these two preudoscalar Higgs is small. 
When  increases from 0 to 0.3Rad, the mass of these Higgs only changes from 0.4% to 
0.6%. 
 The results in Figure 3 and Figure 4 show that the mass of the scalar Higgs S1 is in the 
range from 74 to 100GeV and it is in the range from 114 to 128GeV for the scalar Higgs 
S2. S2 can be considered as similar to the Higgs boson that was found by experiments in the 
year 2012. The influence of CP violation on the mass of these two scalar Higgs is quite 
large. When  is in the range from 0 to 0.3 Rad, the mass of Higgs S1 may increase around 
25% and it is 12% for Higgs S2. 
 Figure 3. The influence of CP violation on Figure 4. The influence of CP violation on 
 the mass of S1. the mass of S2. 
30 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI 
 Figure 5. The influence of CP violation on Figure 6. The influence of CP violation on 
 the mass of S3. the mass of h . 
 The results in Figure 5 and Figure 6 show that the mass of the scalar Higgs S3 is in the 
range from 497 to 500GeV and it is around 470GeV for the charged Higgs h . The 
influence of CP violation on the mass of these two scalar Higgs is quite small. When  is in 
the range from 0 to 0.3Rad, the mass of these Higgs changes from 0.1% to 0.5%. 
 Figure 7. The influence of CP violation on Figure 8. The influence of CP violation on 
 the mass of %1 . the mass of %2 . 
 The results in Figure 7 and Figure 8 show that the mass of Charginos is in the 
range from 61 to 63.5GeV and it is from 193.5 to 199GeV for Charginos . The 
influence of CP violation on the mass of these Charginos is quite small. When  is in the 
range from 0 to 0.3Rad, the mass of may increase around 4% and decrease around 3% 
for . 
TẠP CHÍ KHOA HỌC SỐ 2/2016 31 
 Figure 9. The influence of CP violation on Figure 10. The influence of CP violation 
 0 0
 the mass of %1 . on the mass of %2 . 
 0
 The results in Figure 9 and Figure 10 show that the mass of the Neutralino %1 is in the 
 0
range from 11.4 to 15GeV and it is around 36.6GeV for the Neutralino %2 . The influence 
of CP violation on the mass of the Neutralino is quite small while it is quite large for 
 . When  is in the range from 0 to 0.3Rad, the mass of may increase around 30%. 
 Figure 11. The influence Figure 12. The influence Figure 13. The influence 
of CP violation on the mass of of CP violation on the mass of of CP violation on the mass of 
 0 0 0
 %3 . %4 . %5 . 
 0
 The results in Figure 11, 12 and 13 show that the mass of the Neutralino %3 is around 
 0
96.3GeV, it is around 151GeV for %4 and in the range from 189 to 193GeV for the 
 0
Neutralino %5 . The influence of CP violation on the mass of Neutralinos and is 
 0
insignificant. When  is in the range from 0 to 0.3Rad, the mass of %5 may decrease 
around 2%. 
3. CONCLUSION 
32 TRƯỜNG ĐẠI HỌC THỦ ĐÔ HÀ NỘI 
 This paper studied the influence of CP violation on the mass of some new particles in 
the NMSSM, the following results have been obtained: 
 - The CP violation existed in the NMSSM at the tree-level of the Higgs potential. A 
non - trivial complex phase appeared after the three Higgs fields were redefined. 
 - The mass of Higgs bosons, Neutralinos and Charginos in the NMSSM was defined. 
 - The influence of CP violation on the above mentioned particles was evaluated and 
some interesting results have been obtained: The influence of CP violation on the mass of 
 0
S1, S2 and %1 is quite large (from 12% to 30%); The influence of CP violation on the mass 
 0
of %1 , %2 and %5 is quite small (from 2% to 4%); For the other particles, the influence of 
CP violation on their masses is insignificant (below 1%). These results should be 
considered in the studies and the searches for new particles in the NMSSM. 
 This paper has not only given us a deep understanding on the NMSSM but also 
contributed to the searches for Higgs as well as confirming the detection of Higgs in the 
experiment. 
 REFERENCES 
1. Edward Written (1981), “Dynamical Breaking of Supersymmetry”, Nuclear Phys. B188 513. 
2. Edward Witten (1981), “Mass hierarchies in supersymmetric theories”, Phys. Let. B105 267. 
3. Harrison B. Prosper, Mikhail Danilov (2002), “Techniques and Concepts of High-Energy 
 Physics XII”, Kluwer Academic Publishers. 
4. R. K. Kaul and P. Majumdar (1982), “Renormalisation of the Fayet-Illiopoulos term in 
 supersymmetric spontaneously broken U (1) gauge theories”, Nucl. Phys. B199 36. 
5. T. K. Hemmick et al (1990), “Search for low-Z nuclei containing massive stable particles”, 
 Phys. Rev. D41 2074. 
6. H. Pagels and J. R. Primack (1982), “Supersymmetry, Cosmology, and New Physics at 
 Teraelectronvolt Energies”, Phys. Rev. Lett. 48 223. 
7. Ana M. Teixieira, G.C Baramco, F. Kruger and J.C. Romao (2001), “Sponta neous CP 
 Violtion in the Next-to-Minimal supersymmetric Standard Model” hep-ph/0110350v2. 
8. U. Ellwanger, C. Hugonie and A. M. Teixeira (2010), “The Next-to-Minimal Supersymmetric 
 Standard Model”, Phys. Rept. 496, 1. 
9. Radovan Dermisek (2010), "Light charged Higgs in the NMSSM", Physics Department, 
 Indiana University, Bloomington, IN 47405, USA, 1012.3487vl. 
10. Ulrich Ellwanger (2011), "Higgs Bosons in the Next-to-Minimal Supersymmetric Standard 
 Model at the LHC", hep-ph/1108.0157. 
TẠP CHÍ KHOA HỌC SỐ 2/2016 33 
 ẢNH HƯỞNG CỦA VI PHẠM CP TỚI KHỐI LƯỢNG 
 CỦA MỘT SỐ HẠT TRONG NMSSM 
 Tóm tắt: Mô hình chuẩn siêu đối xứng gần tối thiểu (NMSSM) là mô hình được xây dựng 
 dựa trên mô hình chuẩn siêu đối xứng tối thiểu (MSSM) khi bổ sung một siêu trường đơn 
 gauge chiral sˆ . Sự pha trộn của các trạng thái thêm vào dẫn tới xuất hiện các trạng thái 
 mới làm cho phần Higgs của NMSSM có nhiều thay đổi so với MSSM. Trong NMSSM sẽ 
 có 7 boson Higgs (còn trong MSSM có 5 boson Higgs), với ba Higgs vô hướng - CP chẵn 
 S1,2,3 (ms1< ms2< ms3) cùng hai Higgs giả vô hướng - CP lẻ P1,2 (mP1 < mP2) và một cặp 
 Higgs mang điện H . Sự trộn lẫn của các Gaugino và các Higgsino cũng làm xuất hiện 
 các trạng thái mới trong phần Neutralino và Chargino. Vi phạm CP trong mô hình 
 NMSSM sẽ ảnh hưởng tới khối lượng của ba loại hạt nói trên. Nghiên cứu này sẽ đánh 
 giá mức độ ảnh hưởng của vi phạm đối xứng CP đối với khối lượng của các boson Higgs, 
 Neutralino và Chargino trong NMSSM. 
 Từ khóa: Higgs boson, Neutralino, Chargino, vi phạm CP, NMSSM