( TẠP CHÍ KHOA HỌC Trường ĐHSP TPHCM ) ( Vo Quoc Phong et al ) ISSN TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH TẠP CHÍ KHOA HỌC KHOA HỌC TỰ NHIÊN VÀ CÔNG NGHỆ HO CHI MINH CITY UNIVERSITY OF EDUCATION JOURN[.]
TRƯỜNG ĐẠI HỌC SƯ PHẠM TP HỒ CHÍ MINH HO CHI MINH CITY UNIVERSITY OF EDUCATION TẠP CHÍ KHOA HỌC JOURNAL OF SCIENCE KHOA HỌC TỰ NHIÊN VÀ CÔNG NGHỆ ISSN: NATURAL SCIENCES AND TECHNOLOGY 1859-3100 Tập 16, Số (2019): 144-151 Vol 16, No (2019): 144-151 Email: tapchikhoahoc@hcmue.edu.vn; Website: http://tckh.hcmue.edu.vn NEW WEAK INTERACTION SIGNAL IN THE 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)� MODEL Vo Quoc Phong, Nguyen Thi Trang Department of Theoretical Physics – VNUHCM-University of Science, HCMC, Vietnam Corresponding author: Vo Quoc Phong – Email: vqphong@hcmus.edu.vn Received: 29/10/2018; Revised: 24/12/2018; Accepted: 25/3/2019 ABSTRACT ± According to the framework of 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)F Model (2-2-1 model), W ± (be like W in the standard model), and W±, �′ decays will be discussed The W± decay width1is equal to 2.1 GeV, consistently to SM and experimental data The W± decay width is very large, in which the main contribution to this decay is the channel containing exotic quarks Furthermore, it is found that the lepton rate decay of �′ accounts for the bulk Keywords: weak decays, Extensions of SM, SM Introduction The standard model (SM) has many successes in explaining physical phenomena at 100 GeV However, this model still has many shortcomings, such as the inability to explain the material-antimatter asymmetry phenomenon or the matter of dark matter Therefore, extending this model is a necessity Weak interactions are known as the swaps via �0 and W± These two bosons are fully covered in SM However, at the energy scale larger than 200 GeV, weak interactions may occur throughout new bosons which can be described in the extended SM The 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)F Model (2-2-1 model) is one extension of SM, which has the simplest group structure However, there are three coupling constants, three VEVs; two exotic quarks which are in a doublet of 𝑆𝑈(2)2 group; one new charged and one new neutral gauge bosons which are larger than 1.7 TeV (Chuan-Hung Chen and Takaaki Nomura, 2017) This model has two new gauge bosons which can play an essential role in the early universe The non-SM particles, such as Z’ is searched by LHC, whose estimating mass is about a few TeV The decay channel of this new particle is also an interesting concern and calculated However, in different models, the decay channels are different, because of their interactions with SM particles Researchers work with the 2-2-1 model, find the vertex coefficients in the possible decay channels of the two new propagators (Z’, W 2) and then calculate their decay width These decay channels are the signals of the weak interaction in the TeV scale, larger than the energy scale in SM, 200GeV After calculating these decay channels, we can know Vo Quoc Phong et al TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM which is dominant and give experiment the range to look for signals of new particles, or when calculating the higher loop of interaction, we can choose which new particles contribute This article is organized as follows In Sect.2, a short review of the 2-2-1 model In Sect.3 and 4, We show and calculate the channels of W and �′ which are the new signals of weak interaction at the 1-TeV scale Finally, in Sect.5, we summarize and discuss these decay Review on 2-2-1 model In this model, the SM gauge symmetry is extended to the 2-2-1 model, including the particles of the SM and some new particles The SM particles belong to the representations of 𝑆𝑈(2)1 ⊗ 𝑈(1)F and are singlets of 𝑆𝑈(2)2 Some new particles include Higgs doublets of 𝑆𝑈(2)2, Higgs singlet S’ and vector-like quarks (VLQ) doublets of 𝑆𝑈(2)2, �′� = (𝑈′, �′) The electric charge operator � = �(1) + �(2) + �, with �(1,2) = �3 and � is Pauli matrix 3 To clarify, the explicit representations of the particle generations in this model which include the particles in the standard model and the new particles are recorded as follows 𝑢� 𝑐� �� i �� = ( ) ; (� ) ; ( ) �� ߥ𝜇�� ߥ𝑐 �� ߥ𝑒� i ( � = 𝑒 ) ; (𝜇 ) ; (� ) � � � 𝑈� = 𝑢�; 𝑐�; �� �� = ��; ��; �� �� = 𝑒�; 𝜇�; �� The particles in the new model include VLQ, Higgs �2 doublet, and S’ singlet Unlike the standard model, VLQ doublet in this model includes both the left and right polarizations 𝑈′ ′ � �(�) = ( ′) � �(�) � derivative is as The covariant �𝜇 = ∂𝜇 − i�i�(i)�� − i�F��𝜇, (1) � i𝜇 � where �i and � (� = 1,3 ) are the gauge coupling and gauge field of 𝑆𝑈(2)i �F and �𝜇 i 𝜇 gauge field of 𝑈(1) �(i)= �� and � are coupling and � are the Pauli matrices Y is the F � hypercharge of a particle When3 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)F group breaks down to 𝑈(1)F, the gauge fields � , � and �𝜇 of 𝑈(1)F will be mixed so that we have two 1𝜇 2𝜇 massive neutralfield gauge bosons � and �′ and one massless photon we obtain charged gauge W± and W± which are defined by W± = (�1Moreover, ∓ �2)/√2 i i i Tập 16, Số (2019): 144151 TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM The new Yukawa interaction is written following as: � = [−�F ��F � ′ � 𝑆 ′ − �� ��F �2 �� − �� ��F �˜2 �� − �ƒ ��F � ′ � + ℎ 𝑐], (2) only quarks t and b in the standard model are coupled the VLQs in Yukawa interaction The Higgs potential has two doublets; �1 is the SM Higgs doublet and �2 is heavy Higgs doublet of 𝑆𝑈2(2), 𝑉(�1, �2, 𝑆′) = ∑ [𝜇2�†�i + �i(�†�i)2] + 𝜇2𝑆′2 + �𝑆𝑆′4 i i 𝑆 i i=1,2 +𝜇𝑆𝑆′3 + 𝑆′(𝜇1𝑆�†�1 + 𝜇2𝑆�†�2) +�12�†�1�†�2 + �1𝑆𝑆′2�†�1 + �2𝑆𝑆′2�†�2, (3) † �i +i�0) �i = (�i+ℎ i i , √2 𝑆′ = (4) �𝑆+𝑆 √2 , where �±,0 are unphysical Nambu-Goldstone bosons and ℎ1,2, 𝑆 are the physical i scalar bosons Table Masses of bosons and fermions in the 2-2-1 model �2(�, �2, �𝑆) Particles �2 ± �2�2 �1 �2 ± 2� 22 �2 �2 ~� � �2 2 �1 � �2 � (� )+ �′ �′4�2 + �4�2 �2 ~�2F 2 �2 = �2 �22 − �′ 2�1�2 �2 = �2 2�2�2 ℎ � ��𝑆 � ℎ1 = ℎ2 𝑆 2�𝑆�2 + � � ��2 �2~�2 F = �2 � O 𝜇2𝑆�2 − 2√2 2√2�𝑆 ƒ2� �2 �F (�ƒ + 𝑈 𝜇1𝑆�2 + 3𝜇𝑆�𝑆 √2 �𝑆 ) Tập 16, Số (2019): 144- TẠP CHÍ KHOA HỌC - Trường ĐHSP �2 �2 ~�2F = TPHCM � (�ƒ + � �F 151 O √2 �𝑆) Vo Quoc Phong et al TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM W± �𝑛� W± Decay the mixing between VLQ and SM-quarks will be not considered In order to calculate the scattering vertex factor of fermions with W− and W−, we base on the Lagrangian, � 𝜇 �𝜇 ��,� + ∑ i�f � 𝜇 �𝜇 �f �f𝑒��i�𝑛� = ∑ i�, ′ � � f f − Wi W1− → 𝑒 + −i ߥ𝑒1 W− → 𝜇 2√2 Vertex �1 ( �𝜇 − �𝜇�5) −i �1 (�𝜇 − �𝜇�5 ) 2√2 𝜇 �1 −i (� − �𝜇�5 ) + ߥ W− → � + ߥ ߥ 𝑐 2√2 − W1 → � + 𝑢ߥ �1 −i (�𝜇 − �𝜇�5) 𝑢� 2√2 W1− → � + 𝑢ߥ �1 −i 2√2 W1− → � + 𝑢ߥ (�𝜇 − � 𝜇 � ) 𝑢� �1 −i (�𝜇 − �𝜇�5) 𝑢� − W2√2→ � + 𝑐ߥ −i �1 𝑉𝑐�(�𝜇 − �𝜇�5) 2√2 �1 − W → � + 𝑐ߥ 𝑉1 (�𝜇 − � 𝜇 � ) −i 𝑐� 2√2 W− → � + 𝑐ߥ 𝑉 −i �1 (�𝜇 − �𝜇�5) 2√2 𝑐 � � −i �𝜇 W2− → � ′ + 𝑈̅ ′ߥ √2 According to the Golden rule for 2-body decays in the CM frame (see detail in C Patrignani et al., 2016; D.Bardin and G.Passarino, 1999), the decay width is Γ�→� = +� 𝑘�C |��→� 𝑐 8𝜋ℏ�W +� |2 , (5) where 𝑘 is the momentum of �1 and �2, and c=1 (in ‘Godiven’ unit) Besides, we consider in the CM frame and obtain as, 2 ( − (�1 � − �2 − �2)], | �→𝑙+ߥ� 𝑙 [ −� )22 32 � −�2)2 −(� |� F F ̅F|2 =2� [�2 − ( − �2 − �2) + 3�2�1], (7) � →� +𝑈 2 �2 (6) TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM − with � = �� , � is the factor that depends on �1 and �2 Vo Quoc Phong et al (a=1 for leptons, � = 𝑉ij for quarks), 𝑉ij is obtained from the experimental value (C Patrignani et al., 2016), 0.97434 0.2250 0.0035 (8) |𝑉ij| = 0.9735 0.0411 ) 0.22492 In Eq ((5), the decay widths are calculated in tree order so we need to add QCD corrections Finally, the formula for decay widths is obtained, Tập 16, Số (2019): 144151 TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM ⎧ = (Γ� −→𝑒+�ߥ Γ� + Γ� −→𝜇 +�ߥ 𝑒 + Γ� −→𝑐+�ߥ ) (1 + 1 �� ) 𝜋 � 𝜇 +(Γ� −→� +𝑢ߥ + Γ� −→�+𝑢ߥ + Γ� −→�+𝑢ߥ + Γ� −→�+𝑐 ߥ + Γ� − →�+𝑐 ߥ + Γ� − →� +𝑐 ߥ) (1 + 2𝛼s) �𝑐 ⎪ 1 1 ⎨ ⎪ � +Γ + ̅F F 2�� (1 + �2 →� +𝑈 ) 3𝜋 where, ��=��(MW)=0.1255 (C Patrignani et al., 2016) and �𝑐 = 1;2𝑙𝑒p��𝑛 2 { ��(� ) 1.405� (� ) 12.77�3(� ) (9) 𝜋 ) �𝑐, 3𝜋 = 2.108 �𝑒𝑉, Γ�2 = (Γ� −→𝑒+ߥ�𝑒 + Γ� −→𝜇 +�ߥ𝜇 + Γ� −→𝑐+�ߥ� )��(1 + �𝑐 = (1 + W + 𝜋 � 𝜋2 W � + W (10) ) ; 𝑞𝑢��𝑘 𝜋3 In case ��2 = 1.7 �𝑒𝑉 , �𝑈F = ��F = 750 �𝑒𝑉 and �2 = 2, we obtain Γ�2 = 15.07243 �𝑒𝑉 �′ Decay The branch decay width of �′ → X1X2 is given rule (S M Boucenna, A Celis, J Fuentes-Martin, A Vicente and J Virto, 2016a; 2016b), Γ= 2 √�(�2 �F ,� ,� ) |�ߥ|2 , 16𝜋�3 �F � (11) with |�ߥ|2 is the average value of square amplitude respectively, �1,2 is mass of two particles at final state, and �(�2 F , �2, �2) = �4F + �4 + �4 − 2(�2�2F + �2�2F + �2�2) (12) � We set �1,2 � = �1,�2 2 � � and obtain ��F √�(�2F , �2, �2) = √�4F [1 + �4 + �4 − 2(�2 + �2 + �2�2)] = �2F √�(1, �2 �2) (13) � � 1 2 � The Eq (11) is written as follows: � (1,� � ) 22 |�ߥ| , Γ= (14) 16𝜋��F |�ߥ|2 F = |� F |2�2 F {12���� �1�2 + (�2 + �2)[2 − � − − (� − � 2)2]},(15) (� ff) � � � where, ƒ = ߥ𝑒, 𝑒, 𝑞 are fermions in SM �1 = �2 = �f ′ � �ƒ = � F with fermions are slighter � than quarks top, the decay width (� → ) is calculated in SM with �1 = �2 = 0, ��F ƒƒ = � ′f ′f ′f ′f and |� | + |� | Γ= |� | + |� | , 𝑐W � � 𝑉 TẠP CHÍ KHOA HỌC - Trường ĐHSP ′2f �𝑐�2��F � TPHCM (�′ = → ƒƒ) � Γ2��F | W 24𝜋𝑐 𝑉 (|� � ′2 ′ f ), +| |� � ′� Tập 16, Số (2019): 144151 (16) 2) ( (�′ − (17) = → ��) (|� + | W ) |� � | �� 8𝜋𝑐 𝑉 �� Quark top has �1 = �2 = , where �� = 173.21�𝑒𝑉 Within limit �� = 0, 𝑐� = ��F We have the interaction coupling in table with �� = W and ��𝑐 �,� � � � � � = 0, � = �� = 0.3 ��,� � Vo Quoc Phong et al TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM Table � ′ and � in limit 𝑐� = 1, �� = 0, ��,� = ′ 𝑉 � � � f �𝑙 = �𝑒,𝜇,� ′ 1� 𝑉 ���� 𝑙 = 𝑒, 𝜇 � ���� −5 𝑢, 𝑐 �, � � �� �� ���� - �− ′ 1� � ���� 2 − ���� 12 ���� 18 �W |�′𝑉|12 + |�′ |2 �2 �2 � � � 17 � � 2 � � � � � �� � �18 ��� � �2) (21 (� 31� )� � (� � )( − ��𝑐 � � � (− (�� ) )+ �2 �2 �� � W � ) 2 W2 � 2� � � (� ) (− �2 )�2 + � �2 2 � ��𝑐� 2 � W 2 � � Finally, the decay width for the different decay modes are: Γ( ′ 2 ) � → νߥν Γ( ′ �2��F = 24�𝑐 W �2�� F � → 𝑙𝑙) = × 24�𝑐 W × �W��, 2 (18) �W��, (19) �2 � Γ(� ′ → 𝑢ߥ 𝑢, 17 2� � , �F × 𝑐ߥ𝑐 )2 = 8�𝑐W 18 W � ′ ߥ Γ( � �� F (20) � → ��, �ߥ�) = W × � � , 8�𝑐 18 W � Γ �′ → �2� 2 � ��) =F ( � − − ��2 ) × (� ) � 2( 8�𝑐2 �) �2 � � �1 ( ߥΓ �′ → 𝑐 ,( − ) − � W �2� F ( � � ��) =8�𝑐� W × (��)4( (21) (22) (23) Vo Quoc Phong et al TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM + �� Besides (� → �ℎ), an important decay, ∗µ � ��h = 𝑐�F�h�µ�ε�FF ε�F , ′ ∗ ∗ 2 �� � 𝑐 we2 have � )2 W µ ∗� ��h = 𝑐�F�h�µF�F ε�F ε� , |�ߥ �h |2 2 �2 � (1 + �2 )2], |𝑐 |− [2 = �2+F =|��h| F with �ℎ = �ℎ,� F ,z vector and � (p)� (p) = −� ��W� �� 2 Γ(�′ → �ℎ) = So, the decay width �′ → �ℎ is � � � h � = Where �𝜇 is orthogonal polarization � �ℎ p𝜇p� ∗ z 4�� and vertex factor 𝑐 ��F 𝜇 � �h (24) 𝜇� � � F W � (�W−𝑐)2 12 2 × [2��2 (1, + �,�) ℎ � 192𝜋 z z (1 + �2 − �2)2] z ℎ The last is the decay (�′ → W + W − ) with the average value of decay square amplitude 𝜇 ∗� � − = + � � �∗�− , 𝑐 Γ − + �� � ∗ �FW W ∗ ���+�− = 𝑐 � 𝜇� � FW+W− � F � + � � �F ∗𝜇F �F Γ𝜇F�F�F ��F ��+ ��− , 10 (25) Tập 16, Số (2019): 144151 TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM |� �� |2 = │ where �� = and 1 |2 = │ 𝑐 + − |2�2 F � (1, � , � )(1 + 20�2 + 12� ), (26) �W � �FW W � � � � �� �2 � ��F |𝑐 � + − |2 = �𝑐 � � �FW 2� × ��2 � ≈ �𝑐 × ≈ �𝑐 � � �2F W �1�2 × �2 = � � � � (27) � We get approximate to �(�2 ) if �2F ≫ �2 so the decay width �′ → W+W− is Γ � � W � �2 � F + � − � →� � = F � 2 � 48𝜋 � � × (1 − � ) × �4 (1 + 20�2 + 12�4 ), with �(1, �2 , �2 ) = − 4�2 � � � (28) � � � The total decay width of �′, Γ�F is calculated by summing all the decay width above The result obtain as: Γ� F = 3Γ(� ′ → + (ߥ ߥߥΓ(� ′ → 𝑙𝑙ߥ ) + 2Γ(� ′ → 𝑢ߥ 𝑢 ) + 2Γ(� ′ → �ߥ� ) + Γ(� ′ → � ߥ�) +Γ(� ′ → �ߥ�) + Γ(� ′ → �ℎ) + Γ(� ′ → W + W − ) (29) The branching ratios is (� →X�) ��(�′ → X�) = Г F (30) Г�F If we assume � �F = 1.7 TeV, 𝑆� = 0.3, �2 = 2, ��(� ′ → 𝑙𝑙) ≈ 0.1, ��(�′ → �ℎ) ≈ 0.001, ��(�′ → WW) ≈ 10−8 Conclusion & discussion the �′ and W1, W2 decays are performed in the 2-2-1 model The decay width of W1 is the same as in SM, about 2.108 GeV The Z’ decays mostly lepton, therefore, we can seek for its signal in the lepton interactions Decay width of particle has a huge effect on the transfer functions of force carriers The larger the decay width of a particle is, the shorter its lifetime gets For that reason, detecting them in a low energy scale is a difficult task However, the signals of those kinds of particles can be found by LHC because its scale is about a few TeV In this model, there is the mixing between exotic quarks and the existence of FCNC, therefore, will not be considered In the future, we will work with these problems Conflict of Interest: Authors have no conflict of interest to declare 11 Acknowledgements: This research is funded by Vietnam National University Ho Chi Minh City (VNU-HCM) under grant number C2017-18-12 Vo Quoc Phong et al TẠP CHÍ KHOA HỌC - Trường ĐHSP TPHCM REFERENCES Chuan-Hung Chen, & Takaaki Nomura (2017) Phenomenology of an 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)� Model at the LHC Phys Rev D 95, 015015-015026 Patrignani, C et al (2016) Particle Data Group Chin Phys C 40, 100001-101809 David Griffiths (2008) Introduction to element particles Addison-Wesley Publishing Company Bardin, D., & Passarino, G (1999) The Standard Modelin the Making Clarendon Press, Oxford Boucenna,S M., Celis, A., Fuentes-Martin, J., Vicente, A., & Virto, J (2016a) Phenomenology of an 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)� Model With lepton-flavour non-universality JHEP 1612, 059-112 Boucenna, S M., Celis, A., Fuentes-Martin, J., Vicente, A & Virto, J (2016b) Phys Lett B 760, 214-219 TÍN HIỆU TƯƠNG TÁC YẾU MỚI TRONG MƠ HÌNH 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)� Võ Quốc Phong, Nguyễn Thị Trang Bộ mơn Vật lí Lí Thuyết, Trường Đại học Khoa học Tự nhiên – ĐHQG TPHCM Corresponding author: Võ Quốc Phong – Email: vqphong@hcmus.edu.vn Ngày nhận bài: 29-10-2018; ngày nhận sửa: 24-12-2018; ngày duyệt đăng: 25-3-2019 TÓM TẮT Theo mơ hình 𝑆𝑈(2)1 ⊗ 𝑆𝑈(2)2 ⊗ 𝑈(1)F (mơ hình 2-2-1), phân 1rã W± (giống W± mơ hình chuẩn) W±, �′ �ẽ đượ𝑐 𝑛�ℎiê𝑛 𝑐ứ𝑢 Bề rộng phân rã W± 2,1 GeV, phù hợp với SM liệu thử nghiệm Bề rộng phân rã W2± lớn mà đóng góp cho phân rã kênh chứa quark ngoại lai Hơn nữa, nghiên cứu cho thấy tỉ lệ rã nhánh lepton Z’ chiếm phần lớn Từ khóa: phân rã yếu, mở rộng mơ hình chuẩn, mơ hình chuẩn 12 ... follows In Sect.2, a short review of the 2-2-1 model In Sect.3 and 4, We show and calculate the channels of W and �′ which are the new signals of weak interaction at the 1-TeV scale Finally, in Sect.5,... discuss these decay Review on 2-2-1 model In this model, the SM gauge symmetry is extended to the 2-2-1 model, including the particles of the SM and some new particles The SM particles belong to the. .. ��; �� �� =