Ứng dụng các nhóm đối xứng mở rộng trong nghiên cứu vật lý cơ bản

ĐAI HOC Q U Ố C CilA IIÀ NỘI
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ỨNG DỤNG CÁC NHÓM ĐỐI XỨNG MỎ RỘNG
TRONG NGHIÊN cứu VẬT LÝ HẠT c ơ BẢN
Ma so : ọ 1. »0.08
Chù nhiệm (10 tai : I S. IIÌI 11uy Bmiiịí
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ĐẠI HỌC QUỐC' GIA HÀ NỘI
A - HÁO CÁO TÓM TẢT TÌNII 1IÌNII
VÀ KKI QUA I IIỊÍC HIKN t)KTÀI NĂM 2000
1. Tên (tê tà i :
ÚNG DỤNG CÁC NHÓM ĐÔÌ XÚNG MỞ RỘNG
TRONG NMIIKN Cl'il I VẬT I V HẠT c ơ HẤN
Mã số : QT. OO.OK
2. Chủ nhiệm d ề tà i: TS. I là I luy Hằnti.
3. Các cán bộ tham gia:
IJ(ÌS. IS. Iloànu Nuục 1.01112. TS. Nu II yen Anh Kỳ, TS. Đặn Sỉ Vãn Soa.
ThS. c'ao Tlìị Vi Ha, ThS. t)ặiiL! Oìnli ’lơi, IS. Nuuyéii Mạn ('hunu.
4. M ục tit'll và Iiội thing Iighien cứu :
Nuhiên cứu các hệ I|ii;i vậl lý cùa các mõ hình ly ihuyôl mớ rộnu mâu LÌuiiín
được xây clựnu 11ôn cơ sứ cát' nhóm dối xúnu.
Otiii ra các kêl luận càn ihiêi cho ill ục nuhiệm nil Am pliál hiện các hạl mơi
ilưực lion iliián III lý llniycl.
5. Tình liình thực liiẹn (lê tai :
Xiìy ilựnu lý ihiiyêì các uronu lác là nội limit: chính cú a Vill ly ciíc hạl C1Í
bán. Máu diiiiỊÍn cúa c;k' UKíim lác diện lu yêu ra tlòi tlúnli ilau 1111)1 Imov IHMKII
II'OIIU vậl lý và dã rát (hành CỎI1U VC phiurnu iliện lliụv nuhicm . T u\ nhicn 111) vãn

kliònti Ilá lời (.lươc inõl sỏ vãn lie lý iluiỵẽi quail tIOI1 u VÌI ilo ill) cán ilưụv m ớ
rònii. Các mẩu clnian m ớ rộ liu uàn dãy dược xây clụnu liên cơ sớ siêu doi XIÍI1U
hó a m ill! cluiâii dan li iluov các nh à vặl lý I|iian lãm nuhièn cứu sì co nh iêu LI LI
điếm đáim cluí ý. Đi ilieo hướne million cứu này duinu lõi dã dạl clươc các kóI
C|uá chính sail dày:
- Tính loán liốl diện tán xạ vi phàn của các quá trình và đưa ra các (.lánh uiá
số cũnu như các điều kiện 111 ực nuhiộm cần ihiốl giúp cho việc lìm kiêm các hai
axion iron ti mầu chuân và các hạ! Sc| Liaik II'OIIU mầu cliLiân siêu dối \ihm lõi
ihicu (M SSM).
- Nuhiõn cứu sự vi pliam iloi \ứnu ( l J ironu máu cluiân siêu dõi \IÍI1!Ì loi
ihiổii và dua ra cách xác định một troim các ihiim sô cúa mãn nà) .
Câu trúc nên mộl mầu 331 siêu đôi xứng với neutrino xoắn pliái và
nghiổn cứu sector Higgs Irong mẫu 331 tối thiếu.
Gần dây biốn dạng lượng lử tủa đại số Lie dã xâm nháp vào nhiều lĩnh
vực của vật lý dặc biệt là lý ilmyôì trường lượng lử và hạl cơ hán. Trong đề tài
này chúng la đưa ra sự liên hệ của biên dạng lổng quát'với độ chuyên dời
trong mẫu chu ấn có dối xứng phái trái, xây dựng nôn dại số siêu đôi xứnu biín
dạng N = 2 khi Iham sô' trở lliành toán tử.
Đánh giá chung:
- Đồ lài ihực hiện dúniỊ liên độ và dạt chái lượng cao. Các kêl I|iiá cú a dồ
lài đã dược cônu hô' a 8 hài báo dăng trên các lạp chí IroniỊ, nụoai nước và liên
Tuyển lặp háo cáo Hội ntiliị vậl lý lý lliuyốl loàn quốc lần lliứ XXVI.
- Đồ lài dã góp phần dào lạo 3 học viên cao học, 3 nghiên cứu sinh.
6. Tình hình sử dụng kinh plứ của dê tài:
- Được (.luyộl và cấp : 7.000.000 đổng.
- Đã cho các khoán sau :
+ Chi phí tlniê mướn :
+ Chi phí hoại đón ụ chuyên mòn:
+ Công lác phí và hội Ill’ll ị :
+ Cung ứng vãn phòng:

+ Chi khác:
4.400.000 (.lồng.
1.1 ()().()()() ílònu.
91 ().()()() dỏng.
300.000 dỏng.
2X0.000 đổng.
Xác Iiliận cúa
Iỉam Chù nhiệm Klioii Vậl lý
Clui Irì (le lài
K 'lM
TS. Hà H uy Bàng
Xác nhận cùa Trường Đại học Khoa học Tụ nhiên
V IE T N A M N A T IO N A L U N IV E R S IT Y
— - 0 O0 — -
Project:
THE APPLICATIONS OF GENERALIZED SYMMETRY GROUPS
IN STUDYING ELEMENTARY PARTICLE PHYSICS
Code Q 1. 00.08
D irector o f project: Dr. I la I III V Bang
Ha noi. 2001
VIETNAM NATIONAL UNIVERSITY
REPORT ON THE SITUATION
AND THK KKSUIXS OK I N K I’ROJKCT
/. Title o f the p roject:
Ti ll' APPLICATIONS OF GF.NF.RAI.IZI-D SYMMETRY GROUPS
IN STUDYING ELEMENTARY PARTICLE PHYSICS
Code : QT. 00.08
2. Director of project: Dr. Ha Huy Banu.
3. Participants:
- Ass. hoi. Ur. I Ioann Niioc Limn. Dr. Nmiyen Anil Ky, Dr. Danu Van

S(ta. Vki.'. ('an Till Vi Ha. Msc. Danu Dinli 'I’lli, Dr. Nuuycn.Mau Clumu.
4. Aim and content of researchers:
Si Lilly inu physical conscqucnces (>r beyond standard mode! based on
extended symmetric 1>roups. Dcrivinu, conclusions for cxpcricncc.s It) rind new
particles I hill prcilicled Imm theories.
5. Scheme for the project:
Consuuclinu llic theory 1)1' interactions is main aim (>!' panicle physics,
[lie slandaixl model 1)1 llic clcclrowcak inlcralions has been considered ail
extremely succcsslul theory from the phenomenological point C)!' view.
I Imvcvcr, this model contains a larue numher of unanswered questions and ils
ucnciali/alion is ncccssai). [lie ILXCIII proposed models based oil
Mipersymincll'ic standard model have ihc nicc lea lures and have been obtained
mail) allcnlion.N.
Ill ÍÌIÌ.S direction, n r Ill/ve obiainecl lilt' main vcsiths Ơ.V follow s:
- (om p uliim I he differential CIONS-sections OÍ processes and uivine the
numerical evaluations and some cslimcs lor expcrimenlal conditions lor
huiilinu ax ion and squarks in Ihc Minimal Supersymmclric Standard Model
IMSSM ). -
- Sludyinu llic CP-violaiion ill MSSM and simucsiinu llic method ol
cic 1 ill ill tị one 1)1 parameters 1)1' ill is model. Conslniclinu a version ol 331
supcrsymmclric model willi null-handed nculrinos. Investigating in (Jclail the
iliuus sccloi ol'lhc minimal 3-3-1 model.
Recently, quanlum clclbrmalions of Lie aluchra has been shown to lie
deeply moled ill many problems 1)1' physical inlcicsl, especially ill quanlum
field theory and particle physics. In this project wc have given the connection
between llic ucncrali/.ccl dcloi'inaliun and ilie transition Irom I he left-riglil
model lo llic standard model, have consliuclcd a ủ - ilelormalion ol llic N = 2
SUSY algebra.
Conclusion:
llic schedule 1)1 pmjecl is noud. results have been published by s

papers in ihc Journals anti ihc reports ol XXVI national conlcrenec ol
llieorelical physics of Vietnam.
Ihc project has played a pari ill the iraininu course of three graduated
sIlkIciHs and three Ph. D. .students.
6. The using of the fund:
I'olal ÍLind is allowed and obtained: 7.000.000 VND
or :
+ 8 papers :
+ prinlinu:
+ Ice:
4.400.000 VND
2.320.000 VND
280.000 VND
THE DIRECTOR OF THE
PROJECT
V ỊU V '''—
Dr. Ha Huy Bang
15. CÁC CỔNG TRÌNH KIIOA HỌC CÚA TẢI QT.00.08
1. Pholoproduclion ol' axions ill a resonant elcclminagnclic cavity.
1C/00/147. (Tạp chí của Trung lâm quốc lố VỘI lý, Italy).
2. Q uantum groups and beyond ihc Standard Model. I I. H. Bang, c . T.
Vi Ba. ironu, Tuyên lập Báo cáo I lội nuhị vậi lý lý tliuyèt loàn quốc. 21 -
23/8/2000. Đà Lạt.
V A generali/.iĩd Ọ-dclonncđ Heisenberg - VVcyl algebra vvilh complex
dclonruition paramelcrs. H. H. Bang. D. p. Khoi, N. A. Ky. Irong Tuyến
lập Lỉiío cáo I lội nghị vịil lý lý lliuyốt toàn quốc, 21 - 23/8/2000, Đà Lại.
4. The lliges sector ill ihc minimal 3-3-1 Model with the most general
leplon - number conserving potential. N. T. Anh, N. A. Ky and II.N.
1 .oim. Ini. J. Mod. Phys. A 16. No. 4 (2 001 ) 541.
>. Photon - axion conversion cmss-seclions ill an clcclromcignelic Held,

1). V. Soil. II. II. Bang. Ini. .1. Mod. Phys Vol 16. N„ X (2001 ) 1491.
6. Squaiks ilccays into w and z bosons in ihc MSSM wilh complex
parameters. D. T. L. Tliuy. N. c. Cuong and H. 11. Bang. iroiigTuycn lập
IHH> cáo I lội liuhi \ 'til Iv lý llniyủi toàn quốc lần lliứ 26. Hạ Long 2001.
7. Wave function corrections of Squark in the MSSM with complex
parameters. D. T. L. Thuy, L. T. Tuong and H. H. Bang, đã được nhận
đãnu ờ tạp chí Khoa hoc ĐHSP Hà Nội.
K. The ù - đ d onn cd N = 2 supersymmelric algebra. H. H. Bang and
C’.T.Vilia đã dược nhận đãnu ớ Com 111. in Pliys.
Available at: h ttp://w vv.ictp .tr ie s te .it/~ p u b _ off
IC /2 0 0 0 / 1-J7
United Nations Educational Scientific and Cultural Organization
and
International Atomic Energy Agency
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
P H O T O P R O D U C T IO N O F A X IO N S
IN A R E S O N A N T E L E C T R O M A G N E T IC C A V IT Y
Dang Van Soa, Hoang Ngoc Long
The Abdus Salam International Centre, for Theoretical Physics, Trieste, Italy,
Ha Huy Bang
Department of Physics, Hanoi National University, Hanoi, Vietnam
and
Nguyen Mai Hung
Hanoi Training College, Hanoi, Vietnam.
A bstra c t
Photon - axion conversions in a resonant electromagnetic cavity with frequency equal to the
axion mass are considered in detail by the Feynman diagram methods. The differential cross
sections are presented and numerical evaluations are given. It is shown that there is a reso
nant conversion for the considered process. From our results, some estimates for experimental
conditions are given.

MIRAMARE - TRIESTE
September 2000
In the 1970’s, it was shown that the strong CP problem can be solved [1] by the introduction
of a light pseudoscalar particle, called the axion [2], At present, the axion mass is constrained
by laboratory searches [3] and by astrophysical and cosmological considerations [4] to between
10 6 eV and 10 3 eV. Besides that, an axino (the fermionic partner of the axion) naturally
appears in SUSY models [5,6], which acquires a mass from three - loop Feynman diagrams in a
typical range between a few eV up to a maximum of 1 keV [7,8],
A particle, if it has a two - photon vertex, may be created by a photon entering an external
electromagnetic (EM) field. The axion is one of such particles. So far almost all experiments
designed to search for light axions make use of the coupling of the axion to photons. Conversion
of the axions into EM power in a resonant cavity was first suggested by Sikivie [9]. He suggested
that this method can be used to detect the hypothetical galactic axion flux that would exist if
axions were the dark matter of the Universe. Various terrestrial experiments to detect invisible
axions by making use of their coupling to photons have been proposed [10,11], and the new
results of such experiments appeared recently [12,13].
An experimental EM detection of axions is briefly described as follows[14]: The initial photon
of energy q0 from the laser (maybe better from X ray) interacts with a virtual photon from the
EM field to produce the axion of energy p0 and momentum : p = y/pQ — mị. The photon beam
is then blocked to eliminate everything except the axions, which penetrate the wall because of
their extremely weak interaction with ordinary m atter. (Such shielding is straightforward for
a low-energy laser beam.) The axion then interacts with another virtual photon in the second
EM field to produce a real photon of energy (j'0, whose detection is the signal for the product ion
of the axion. Recently, a photon regeneration experiment using RF photons was described in
Ref.[15], This experiment consists of two cavities which are placed at a small distance apart. A
more or less homogeneous magnetic field exists in both cavities. The first, or emiting cavity, is
excited by incoming RF radiation. Depending on the axion-photon coupling constant, a certain
amount of RF energy will be deposited in the second, or receiving cavity. However, the author
studied the problem by using the classical method. The purpose of this paper is to consider
conversions of photons into axions in a resonant EM cavity with frequency equal to the axion

mass using the Feynman diagram methods.
For the axion- photon system a suitable Lagrangian density is given by [4,9]
L = -~Fln,F ^ + + ị d ^ ộ a - ị m 2j l [ l Hr 0(ộị/v2)}, (1)
4 47T Ja z z
where 4>a is the axion field, m a is its mass, Fpv = ỊịEijvpơFpơ, and fa is the axion decay constant
and is defined ill terms of the axion mass m a by [9,12]: f a = f 7Tm nự m umđ[ĩiia{rllu + rnd)}~ •
Interaction of axions to the photons arieses from the triangle loop diagram, in which two
vertices are interactions of the photon to electrically charged feinion and an another vertex
is coupling of the axion with fermion. This coupling is model dependent and is given by:
!h = 2 ( w- - 3 - ^ + ^ t ) where N = T^QpqQĨoIot) alld Ne = QpqQ'L Tr represents the
2
sume over all left - handed Weyl fermions. QPQ, Qem, and Qculor are respectively the Peccei-
Quinn charge, the electric charge, and one of the generators of SU{3)c. In particular in the
Dine- Fischler - Srednicki - Zhitnitskii model [16]: 97(DFSZ)~ 0.36, and in the Kim- Shifinan-
Vainshtein- Zakharov mode] [17] (where the axions do not couple to light quarks and leptons) :
57(KSVZ)~ -0.97.
Now we consider the conversion of the photon 7 with momentum (] into the axion II with
momentum p in an external electromagnetic field. For the abovementioned process, the relevant
coupling is the seconrl term in (1). Using the Feynman rules we get the following expression for
the matrix element
121
where k = q - p, the momentum transfer to the EM field, gal = (J1-j-= +
’7id)(7r/^mTym v ri^ )-1 and £^(q,ơ) represents the polarization vector of the photon. Ex
pression (2) is valid for an arbitrary externa] EM field. In the following we shall use it for
conversions in the resonant EM cavity of the TMno mode with frequency equal to the axion
mass. Here we use the following notations: q = |<j],p = I pi = (]>l — m’-;)1/2 and 0 is the angle
between p and q. The nontrivial solution of the TMno mode is given by [18]
1ĨX \ nil
16ĨT . / 7XX \ f Try
here the propagation of the EM wave is ill the z - direction.

Substitution of (3) to (2) gives us the following expression for the m atrix element
(p\M\q) = —
-
[(ei (q, r)g2 - c2(g, r)qi) Fz + £i (q, r)g0-FJ + £2{q,T)q0Fy] , (4)
(27T)VP0<?0
where Po — 90 + w = mai and
_ 8E ọ ị q x -Px)(<ly -py )cos[ÍQ (qI - p I )]cos[^b(qv - py)}sm[ịd(rh - p z)]
í ' [(<Jx - p i ) 2 - j£][(gy - P y ) 2 - - P z)
. F, =
ub2(qy - p yY
F =
____
(5)
v Lja~(cjj — P j ) '
where a, b, and d are three dimensions of the cavity and LJ is the frequency of the EM field.
Substituting Eq.(5) into Eq.(4) we finaly obtain the DCS for conversions
3
dơ(7 —> a)
díl
(<?*2 + ?y2) ^ 2 + (1 - ế )Ọ 0 2^ 2 + (1 - q- ị ) q 02Fy2
9Ổ7Po
2(2*)2qo
~1<ìx(iyFxFy — 2qoqzFyFz + 2qoqyFrFz].
(6)
Asuming that, the momentum of the photon is paralell to y- axis then Eq. (6) gets .the
filial form
d ơ ( 7 - > a ) _ M g ^E ịq 2 m
dí2" (2tt)2 1 <7 J
(ọ — p COS Ớ) +
771 ab2

tan (p"
cos I (p sin Ớ sin ip" ) COS J (f/ — p COS 6) sin |p sin 8 COS Ip"
[(psinỡsin^ ”)2 - ^j)][(<7 — p COS ớ)2 - Ệ}
where <fi" is given in [19].
From (7) we have = 0 for 0 = ip" = 0 and
for 9 = <p" = and p =
(7)
(8)
q(eV)
Figure 1: DƠS as a Junction of the photon momentum q: the solid curve corresponds k(q) with
VIu = 10-6 pF, the dashed curve kl(q) - with via — 10-3 el',
III order to estim ate photon - axion conversions in the window mass of axions, from Eq.(8),
if Iin = 10Gcm _1/,2g 1/,25_1 and a = b = d = 100 cm then DCS’s depend on the momentum of
4
photons which is shown in Fig.l. The solid curve corresponds to mQ = 10- ® eV and the dashi'd
curve correspond to m a = 10-3 eV. It implies that when the momentum of photons is per
pendicular to the momentum of axions then there exists a resonant conversion at the value
q S3 4.7X 10 2 eV, and DCS’s have values in the narrow range of 5.4X 10“ 19cm2 to 6.2X 10“
At the high values of the momentum q then DCS’s have very small values. This result is much
beter than those in the wave guide ( the reader can see Refs.[14]). This is very convenient for
applying to EM detection of axions. It is to be noted that, the explicit resonance happens only
in this case and in the direction perpendicular to the direction of the photon. In the other cast'
(see for example [20]) the axions are produced mainly in the direction of the photon.
Finally, in this work, we propose only a theoretical ground, other problems connected with
the EM detection of axions in the resonant EM cavity will be investigated in the near future.
Acknowledgments
D .v. Soa would like to thank Prof. A. Masiero for helpful discussions at ICTP, Trieste,
This work was supported in part by the Research Programme oil Nniural Si'R'iues ill I
Vietnam N ational University under the grant number QT.00.08.
References

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[2] S. Weinberg, Phys. Rev. Lett. 40, 223 (1977); F. Wilczek, ibid. 40. 279 (1977),
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and M. s. Turner, The Early Universe, Addison- Wesley Publ. Company, 1990.
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[12] c . Hagman, p. Sikivie, N. s. Sullivan, and D. B. Tanner, Phys. Rev. D 42, 1297 (1990). For
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B 434, 147 (1998); M. Minowa et ai, Nucl. Phys. B 72, 171 (1999).
[14] D. V. Soa and H. H. Bang, Preprint, ICTP, Trieste, Italy, IC /99/184. To be published in
Int. J. Mod. Phys. A.
[15] F. Hoogcveen, Phys. Lett. B288, 195(1992).
[10] M. Dine, w . 1 . ■ liler, and M. Srednicki, Phys. Lett. D 104, 199 (1981); A. p. Zhitnitskii,

Yad. Fiz. 31, 497 (1980) [Sov. J. Nucl. Phys. 31, 260 (1980)].
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c
QUANTUM GROUPS AND BEYOND THE STANDARD MODEL
Cao Thi Vi Da and Ha Huy Bang
Department of Physics, Hanoi National University
934 Nguyen Trai, Hanoi
Abstract
In this paper we study the generalized deformation of su(2) algebras in some
differen t form s. G au ge field m odels w ith q u a n tu m g roups playing the role of gauge
groups are discussed. T h e conn ection betw een th e generalized d eform ation and
the transit,ion from the left-right m odel to the s ta n d a rd m odel is given. T his re su lt
can be extended to o th e r m odels.
1. I n t r o d u c t i o n
S y m m etries, in p articu la r gauge sym m etry, is a fu ndam en tal principle in th e 
oretical physics and any p hysical theory d escribing observation s m ust tak e those
into account. This a ttitu d e h as yielded am azing success, p erhaps the m ost notab le
exa m p le being th e S ta n d a rd M odel (SM ), in w hich it is assum ed that, reality has
a s u ( 2 ) ® U (l) sym m etry.
T his sy m m etry group is a so-called “classical” grou p. T h ere is, however,
a m ore g ene ral n o tion w hich includes th e classical ease, that, of a m atrix pscu-
dogroup, or m o re fam iliarly, a q u antu m group (Q G ). Since the d eform alion of
classical m ech anics into q u a n tum m echanics resu lts in new phy sics, it w ould n ot
be surp ris in g to c onsider a theory w ith a QG sy m m etry.
In p a rtic u lar, different asp ects of q u a n tum defo rm ation s of q u a n tu m field
theory h av e been stu d ie d by several auth o rs [1-5].

It is th e m ain purpose of th e p re sen t p aper to co nsider the g en eralized d e
form ed algeb ra s a n d their c onne ctions w ith beyond th e SM.
2. G e n e r a liz e d D e f o rm e d S U ( 2 ) A lg e b ra s
As w ell-know n , generalized deform ed s u (2 ) algeb ra is given as [6]
= [J+,J-} = F{Jo(Jo+l)) - Ir{JoỰo- I)), (1)
where Jo, J \ , J - are the generators of the algebra and F(x) is any increasing
en tire function-defined for X > — 1 /4 . Since this a lg ebra is c h a racteriz ed by the
fu n ction F, wc use for it th e sym bol S U jr(2). T h e a p p ro p riate basis Il,m ) h as the
11
following properties
Jo\l,m) = (2)
J+\l, m) — \JF(l[l + 1)) - F(m(m + l))|/,m + 1), (3)
J~\l, m) = yjF(l{l + 1)) - F(m(m - 1)) |/, m - 1), (4)
where
and
1 3 5
1 = 0, 2 ’ 11 2 ' 2' 2 ’ 3

(5)
m = -I, -I + 1,-1 + 2, ,1-2,I - 1, /. (6)
The Casimir operator is
c J .J V + F(Jo(Jo + 1)) = J+J- +F(Jo{Jo - 1)), (7)
with its eigenvalues indicated by
c \l,m ) = F {l(l + l))\l,m ). (8)
The usual SU(2) algebra is recovered for
F (x(x + 1)) = x (z + 1), (9)
w hile th e q u a n tu m alg ebra s u ?(2)
l*Aj, J±] = ±J± , (J+, J-\ Ị2JqỊq , (10)
occurs for .
(x(x + l)) = [ l |,[ l + 1|, , (11)

w ith q - num b e rs defined as
— X
(12)
q q - Q 1
The SU/.'(2) algebra occurs in several cases, in which the rlis of the last equa
tion in (1) is an od d function of J0 .
In th e following we use n o ta tio n [zj = F(x).
Now wo can define the 2-dimensional representation T of this group as follows:
TeT* = T t ET = e, (13)
det fT = ad — f (q)bc = 1, ( L4)
T -t = ự t y t
/here
e =
(15)
12
and T l is the transposed matrix of
a b
r - ( : !)■ CO)
, Qx-l
__________
___
.
For the case of f[q) = q corresponding to |x] =
of Ref. [5|.
3- H ig g s M e c lia n ism in G eneralized D e fo rm ed s ta n d a r d M odel
Let UK consider a complex matter doublet <i>‘ (and its conjugate <!>' : <i>1)
living in the fundamental rep of SUf-(2), i.e.
* :=( Í 0 ’ * i;= ^ + *0)- (,7)
U nd er 1.11C QCỈ actio n, these tran sfo rm resp ectively as
>-► uj ® <1>; , '!>' S{uj) ® <1>‘ . ( 18)

It follows that, the q u a n tity <1>1 <t>' = <í>°<pÓ + Ộ 1 <t> is ((Mitral and
invariant T h erefore, I,he a p p rop ria te kinetic energy term for this m ilUcr will I)C
((/-)<!>) lives in the fun d am e ntal, so its co variant deriva tive is given by
l)(j> d<l>
D4>" dệ° + UJ
iỹ ( Ị I'11 1 f 3 1 -,\ _ ig\J2
- p r h r 1 -, - - 13 + qW 3 ) ệ - ~ -w <i>\
r/[2] { 2 p \ [ 2 \ [ 2 \ q r q\2\ p '
T T ( t / r r n i 11 i ' l B - «H' 3 V ” + T r ' f '
7|2J v v [2][3 . 2 . . 2 . 1 r <1 [2]
(in)
S uppo se tlicrc ex ists a m ap V : n (M ) —> Ả: such th a t K (<t>44*) is licrmilciiii
invariant, anil bounded from below. If such a V exists, it plays the role of a
potential, and
s n = ((Z?$)‘|£>$) - v ( $ ‘<i>) (2 0 )
is an in v a riant hcrm itea n action for Even tho u g h this action is tnii.nifVst Iv
SU/.’(2) - invariant, assume that the quantum symmetry is broken spontaneously
viá a Higgs m cchanisrn . T h is is acco m plished by assum ing th a t th e re is a real
non zero c o n s ta n t V such th a t V is m inimized (and vanishes) a t U , v a c u u m lo r
the a ction ab ove occurs a t (4>í(ỉ>) = - í ' 2, w hich one can assum e co rrespo n d s to
(^ * ) = 0 and (ậ°) = (4>°) = -J U.
(An exam plo of a m ap satisfy ing all these con d itions m ight be something like
:= - -IS2 <i>'4> - - V 2), (21)
■ V \ 2 2 /
w ith n 6 1Ỉ).
13
If all the above is possible, then the action will acquire the terms quadratic ill
the gauge fields, and thus they will become massive. Now, just as in the classical
case, assume that the mass eigenstates are \v^ , z and A, where
w 3 = COS Ow z + sin OwA, D = — sin 0\v z + COS ỚVV/1, (22)

where 0W £ 11 is the Weinberg angle. However, in order for A to be intcrprctablc
as the photon (more on this see the next Subsection), it must be massless, which
implies that cannot include a term of the form A<p° in D<j)°, since (<f>°) 7^ 0.
Dy using the explicit form (19) and the definitions of z and A above, this may be
accomplished by requiring
' ' 1 '
tanớvv = f(q)
Hi
[2j[3j
(23)
liy inserting I,he value for Ow from the above relation, the covarianl derivatives
Lake t:he form
I)if) — I
/;</>° = dệ'
If]
> 0\v V|2|
1
-—- — sin 0IV ) Z(ị>
<l\A
w 4> + iq sin Ow AỘ ,
iqsj2
í*(q)\2]coSOw Zệ0+ f(q)\2}w *ệ (2A)
The masses of tlio three remaining gauge fields arc found by evaluating Sn
at, (<l>), giving
g2!'2
q2ư2
25)
» « l. = i ^ W ‘ {w^ w l * / W 3 5 1 5 8 ' <
Requiring this to he equal to rriịv {W + \W~) + - m 2z (Z\Z) determines the
m w

!7"
= COS Ow ,
i m \
which gives T1I.W cosOw ln the classical case.
(2G)
4. C on n e c tion w ith L e ft-R ig h t M o d el
In this scctioa, wc start with Left - Right Model (LR).
The parameter that connects LR and SM representations SU(2)/, ® su(2)yj ©
IJ(1) u -/ andSU(2)/ ® u (l)y is the hypcrcharge-spliting taking continuous values
1
0 < A < - .
2
To l)rcak Uic symmetry, we perform a non-orthogonal GL(2) transformation
in the right-handed Cartan subalgebra of the LR to obtain a generalized basis
(ỹ,To )■■
14
\ t o ) \ o 1 2 A ;V To )
(27)
Thu M l M odel cliooaea A — 0 while in the SM A = 1.
T h e uyfiim etry under consideration is now
SU(2)l 49 U(l)(fl_ L)t ® [SU(2)h & U(1 )(i,-L)„]A . (28)
T 0 is fixed and the question ia how to close the [su(2)«] algebra. [su(2) /j]
should contain the charged acctor of t)ie LR aa A —► 0, but is otherwise unspecified,
la a. sim ple isotropic renorm alization of SU(2) generators
To - (1 - 2A)Tu , f ± = \fl ~ 2 & T ± , (30)
s I J (2) liaugu transform ations would then read
0,1* It -> exp [ig n T ịu ịx ý n ịD ^ iì, (31)
'Ha -> "'t«exp[ - igRT j u ( (32)
w here Utl - i>!I I igu Tj\V^ and we have
| r , 7u| - 0 , |l7, r ± | = ±2AT±. (33)

N ext, Iiaiiii; ( l) wi: ca.il express the generators J±, Jư of tlie generalized do-
J I
'1 \j 1 A'u|y -
Ko
+ 1)
1-1 [j 1
Ko}\j -
Ko
+ 1]'
2 [j -
Ko\j -I-
Ku
•1- 1|
1 1 [j -
Ko]\j -1-
Ko
+ 11'
K + ,
(34)
•A, - /»., . (35)
C o m p ilin g (3-1) and (::■/) gives
1 “ 2A = <36)
1 -
2
q + 1
(37)
This work was suppmlctl in pan hy I he Rcsciiich Programme nil N ill II la I S ú enco
VÌL‘(::am Niiiional Univvrsiiv muliT 1 ho maul numlxT H I.00.ON.
R eference
|l| Miliccio 1{. Uhriaco, Miul. I‘)i\ja. Ictt. A8 (1993) 2213.

|2j 11.11. u .ing , Mud. Hhys. Ictt. A ll) (1995) 1293.
|3] 1’a.ul W atls, J. Gtum. I’liys. 2-1 (1997) 61.
[4] II. II. Bang, A W Cun. M l (12) (1997) 1507.
|5| R .J . LMnkul.sl.cin, h e p -ill/1)00 0135; licp -th /990 6136; h e p -th /0 9 0 821Ơ; hep-
tli/o o o 3189.
|6| D. Uonatso:; et a]., hei>-l!i/95] 2083.
15
A G E N E R A L IZ E D ^ -D E F O R M E D H E IS E N B E R G W'EYL
A L G E B R A W I T H A C O M P L E X D E F O R M A T IO N
P A R A M E T E R
Ha Huy Bang a, Dinh Phan Khoi b,c and Nguyen Anh Ky c
Department, of Physics, Hanoi National University, 3Ậ5 Nguyen Trui, Hanoi.
6Department of Physics, Vink University, 192 Lc Duan,Vinh, Nghcan.
cInstitute of Physics, p .o Box 429, Bo Ho, Hanoi 10000.
A bstract
A generalized <7-defonned Heisc.nberg-Wcyl algebra with a complex deforma
tion pnramoter is proposed. T he energy spectra ot harmonic oscillators rill respond
ing t.0 t.lie case are derived using a geieralized definit ion of deformed num bers (</
num bers). Flirt,her, a realization of SU(n)q in term s of the generalized <1 (lofoniK’d
bosons is also given.
1. IN T R O D U C T IO N
For over fifteen years there has been increasing interest, in the subject, of (Informed
(quantum) groups and related topics. In particular, deformations of tile Heiscnbcrg-
Woyl alfi'l'bra iind g-lioson oscillator realizations have been widely invest,igat.cd. After
('ally formations of t.lií' deformed oscillators, various generalizations of <7-(loforinocl boson
algebras have boon considoroil. For example, L. do Falco (it., al. studied a gcncriiliml
(]-deformed Iloisenberg-Weyl algebra with a complex deformation parameter |lj. They
derived t he spc’ctra of t he corresponding deformed oscillators for two spccial ca.sos of the
<7- commutation relations and alao considered a boson realization of Ui(’ noil (loiunnocl
algebra SU{2).

Ill l.liis paper, wo study a problem similar 1.0 that, considered in ref. [lj. Using the
liK'tliod (if bosonizat ion, we. propose a generalized (y-boson niiilizat.ion ol ;i q (IcluniK’cl
llrisnilifTj' Wcyl algebra and derive t.ho energy spnetra of I lie correspondi Ilf’
<7
(IHuimrd
oscillators. J'!)<• m-iulls obtained hero arc; more general thill 1 those: of of I'd. [
1
| iuKI t an 1)0
reduced 11) [.In lat.U'1 with special choices of the structure function. I'urt.hcnnorc. we also
m ako <1 -(Icioriiied o scillator rculizatio as of s u ( 2)q and s u (•//.), w ith a class oi (Ị- mil 11! XTS.
Tin: paper is organized as iollowa. Aft.or considering ill Sect ắ2 a generalize*I <1 -
deform ed Ilpisonberg W cyl alg e bra w ith a com plex defo rm ation p aram clor, w r investi
gate ill Sect. :i ọ-hannonic oscillators. Finally, Sect,. 4 is devoted to generalized boson
n'filizutio IS of th e q u a n tu m algebras su ( 2 ) q and SU {n)q.
2. G E N E R A L IZ E D <?-DEFO R M E D H E IS E N B E R G -W E Y L A L G E 1ỈR A
1U
In [1], L. clc Falco Pt. al. considered the commutation relation
AB - qDA = F(q, N , ), (1)
and its adjoint, relat ion
B'A* - q'A'B' = F m{q',N, ), (2)
whero A and D are Iion-Hermitian operators, <7 e c, N is the number operator 111 tho
usual Heisenberg-W('vl algebra and F is a complex-valued function. Here, iiislc'iul of
A and B satisfying (1) and (2), we consider other non-Hormitian operators A and B
subject, to commutat ion relations
ẢB - BA = f{q, N , ), (3)
B 'A '- A 'B ' = (4)
w here /(<7, TV, ) is an a rb itra ry com plex-valued fun ction . It is shown th a t (J) and (2)
can be obtained from (3) and (4), respectively, by a suitable re-scale transformation of
A and B.
Applying the met hod of hosonization [2], we take the operators A and B in t ill' forms

A = G (N + l)a, B = a*G{N+l), (5)
where the structure function G(N + 1) ia a complex operator function, <7* and n arc the
usual (i.0., n o n -d e fonnod ) crea tion and an n ih ilation op erators. It, follows from (5) that,
in general ■ _
‘ A' 7^ B, (6)
unlike the noil-deformed ease. Replacing Eqs. (5) in (3) and (4), WR obtain
{N + 1 )G2(N + 1) - NG2{N) = f(q, TV, ), (7)
(N + 1)[G'(N + l)]2 - N[G’(N)}2 = f*(q\ N , ). (8)
Those ('((nations (7) ;mil (8) ap plied to a Fock sta te |n > lead to
(n + 1 )G 2(» + 1) - 7iG 2 (n) = f { q , n , ), (9)
(n + l)[G * (n + l )]2 - n[G» ] 2 = /*(<?*, 77, ). (10)
For a fixed either of Eqs. (9) and (10), conjugated to each other, has tho solution
G(n)
\
ill)
3. ^ -D E F O R M E D H A R M O N IC O SC IL L A T O R S
I.cl 11! i now (Ic'.c'i mine the ('/ harmonic osoli.uui structure. io do that, wo doling I ill
(Ị moiiH’iil nil) (Pu) find 7 -position (x,t) npcralors directly from the operators A and B
|a], H ____
,, . IlillUJ / h
^ x „ S ị , ± - J B + A). (
12
)
Following tile classical analog, we w rite dow n the harm onic oscillator II;unilli>iii;ui a.s
follows
H q = 2 ầ + 2 m ij2xt = ĨJĩ {AB + B A )■ (-13}
111 general, t.his Hamiltonian is non-Iỉermit,ian. As in the non-deformed ease, the eigen
values of I[q can l)(i easily calculated
E n = - y ([” • + 1] -I- [/»]) = ^ ( 2 [n] + /(77.)), ( M )
where

[n] = nG2(„.) = J2f(q,l, ) (15)
1=0
is defined as a (/-number corresponding to /. This new deformed number is more (M'lii'ral
than I,lie OI1C.S used ill ref. [1].
As soon from (14) and (15), thè energy spectrum here is, ill.general, complex In
order 1.0 get, a mil Kpcctruin of Iiq we must. impose a constraint, oil Cl2, i.e., on I. loi
example, Hie! nxprcssion E"=0 f(q> ••■) 'ltXS ,0 *H‘ ron'- Another way to gel a real I'licrgy
spi'clrmn is to replace the hosonisat.ion (5) l>y
A = C { N + l)o, 13 = a}G*(N + 1), (5')
that, is
A' = B. (6')
Thru, instead of (11) and (15) we should have
[»|] = » |G ( » 0 |2 - £ / ( ? . / , ■■■), ! lf)i
(=0
wliioh means again Hint E!=o is ;i r''al number. Now the Hamilton!,Ill (i:{) is
lii'imil.ian and the S|)«’CJ( rum (14) is automatically leal. However, for t.ho sake (if I iliil. we
have to impose loo ('ally a cont.rainỉ. OI1 / making the algebra (3) and (4) loss jM'iiciiil.
rJb obtain the results of ref. [1], we cun chooso
f{q, N,. ) = F (q, N + l, )-F{q,N, ), (17)
Ill'll’ /■’('/. /V ) takes one of the following two particular forms
and
(ii)
(19)
As seen from (15), the (/-number [n] with the choice (17) of / is nothing hilt F(q, V. ).
4. B O S O N R E A L IZ A T IO N O F SU{2),
In this section we give boson realizations of the (/-deformed algebras SU(2),( and
SU[n)q w ith a complex deformation param e ter q. In [1] the au th o rs consid ered a bnson
realization of only SU{2 ) but. not, SU{2)q and wrote down explicit actions of ,/+. ./ . ./.
on the Fock states for |<7i I = I<721 = 1- H ere, let us con sider a n u m b er of independent
copips (or modes) of the algebra (3) generated by, .say, Ai and B„-i = 1,2,3, , :

where f(q, Ni, ) are arbitrary complex-valued operator functions and Ni = n]n, is a
number operator in the Heisenberg-Woyl algebra of ưiođe i with laddc'1 operators II, and
a The bosonizat,ion
where G(Ni) are complex-valued operator functions, would be consistent, with the com
mutation relations (20)—(23) if
Since B,Ai and n]a, commute with each other, they have the same oigrn-voctors. which
can he, for oxample, t he Fock states I Hi > :
[Ai, -4j] — 0,
ẢịBị — B,Ai = f(q, Ni, ),
[Ai,Bj] = 0, I / j,
] = 0,
(20)
(22)
(23)
Ai = G(Ni + l)a,-, Bi = a\G(Ni + 1), t = 1,2,3,
(24)
It, can b(' verified that
[Ni,Ak] = -SikAk,
[Ni,Bk]= 6 ikBk,
\N„ N,] = 0,
a\(ii] = 0.
fjii)
(29)
(30)
BlA l\n, > = [n,]177,- >
19
For a bosonization of SU{2), we expect, the operators
J+ = BxA 2, J_ = B2A u Jz = \(N x - N 2) (33)
could solve the goal. Indeed, if the function /(<7, N , ) is chosen t,0 obey the identity
N [y + 1] - [r + l][y] = [-r - y], (34)

t.lio ojjer.at.ors (33) really gercrate the s u (2), algebra:
[J+IJ-] = [2JZ], [JZiJ+]=*J+, [JZ,J_] = -J_. (35)
A representation of SU(2), can be realized on the basis
(S1)'+m(S2)'- m
\j,m >= |»1 > 177-2 >
as follows
{[j + ™]!b' - ml!)
Jz\j, m > = ni\j,m. >,
r|0>,
where
or equivalently,
J+ \j, m > = sj[j + m + l][j -m]\j, m + 1 >,
J-\j,m >= \J[j - m + 1 }\j + m]\j,m - 1 >,
™ = 2 (n i “ n2), j = ^(ni + n2)
0=0,1,2, ;
m = j + m, n2 = j - 111-
(36)
(37)
(38)
(39)
(40)
(41)
More generally, ws consider the quantum group SU(n)q. As is known that SU(n)q
can bo generated by 3(n — 1) Cartan-Chevalley generators Hi and xf- (i = 1,2, — 71. — 1)
subject, to the commutation relations
[Hu x±] = ± c ikx± , . (42)
[x+,x;} =ỏlk[Hk} (43)
and the Serre relat ions
w lirri’
[ ^ , [ ^ , ^ 1 1 = 0, \i-k .\ = 1,

[ X ^ ị X ^ X ệ Ỵ ị ^ O , | i - A - |^ l ,
C'ik — ^14-1 ,A:
is I.lie Curtail matrix for SU(n).
Thr .lonlan-Scliwinprr realization of SU(n)q is choM’11 a.s follows
A ? = BjAi+1,
20
(44)
(45)
(46)
(47)
x r = AB +
1
,
Hị = Ni - Ni+U
(48)
(49)
where
-4, - G(N, + l)fl„
B, = ajG'(/V, + 1).
(50)
(51)
It. can ho verified that.
K A ] — —ỏik-Ak,
[Ni,Bli] = SlkBk,
[Ni,Nk} = 0
(52)
(53)
(54)
and the realizul.ion (47)-(51) f'lillfils the commutation relations (42) (46) of SU(n),r It.
is easy to KCifi I.hat. SU(ii) is a limit of SU(n.),j at <1 = 1.

5. C O N C L U S IO N
We suggested in this paper a (/-deformed Heiaenberg-Weyl algebra, at. a complex (In
formation parameter and with u generalized structure function. Tin: energy spectra of
harmonic oscillators corresponding 1.0 the case is derived due to ail (!xt.(!iul(id notion of
the (j-numbeia. Tilia new definition of l.he (/-numbers is an intermediate sttip hut, of
independent, interest. Further, a realization of SU(2)^ and SU(n)y ill l.cnn of tlir.M' (fcn-
eralizecl deformed bosons are given. The results in tilia paper are ail extension of those
in [1] as we consider a genera! form of the structure function. By choosing suitable for HIM
of the latter we can find the previously obtained results. The next aliip ia to exl.ciid llid
prese nt investig ation s lo th e case of q u a n tu m su p e ra lgdjias . It is really ail iiil.cr(!sii[ig
problem which is now being considered.
T h is work w as s u p p o rte d in p a rt bv th e R esearch Program m e on N a tu ra l Sciences 1)1
V ietnam N atio n al U niversity u n d er the g ran t n u m ber QT.00.08.
I. L. do Fulco, A. Jannussis, n Mignani and A. Sotiropoulou, M ud. Pliyb. Lett A
9, 3331 (1991).
. 2. A. Jttm iussia, Ci. lỉro dim aa, I). S om lus and V. Zibis, Lett. N novo C iniento ;10, 123
(1981).
3. A. J. Miicfarlano, J. Phys. A 22, 4581 (1989).
4. L. c. Dic'tliiiihani, J. Pliys. A 22, L873 (1989).
5. c. Sill) and H. Fu, J. Pliya. A 22, L983 (1989).
G. w . Cluing, K. Cluing, s. Nam and c . Uni, II Nuovo Ciment.o Li 109, 891 (199-1).
R E F E R E N C E S
21
International Journal of Modern Physics A, Vol. 16, No. 4 (2001) 541-556
© World Scientific Publishing Company
THE H IG G S SE C T O R IN THE
M INIM AL 3-3-1 MODEL W ITH THE
M OST G E N ER A L LEPT O N N U M B E R
C O N SER V IN G POTENTIAL
NGUYEN TUAN ANH

Institute of Physics, NCNST, PO Box 429, Bo Ho, Hanoi 10000, Vietnam
NGUYEN ANH KY
institute of Physics, NCNST, PO Box 429, Bo Ho, Hanoi looon, Vietnam
and
Department of Physics, Chuo University, Bunkyo-ku, Tokyo 112-8551, Japan
E-mail: mifiky@ioj).ncst.ac.vri
HOANG NGOC LONG
Institute of Physics, NC NST, PO Box Ự29, Bo Ho, Hanoi 10000, Vietnam
and
LA P T IĨ, B .p. 110, F-74941, Annecy-le-Vieux Cedex, France
Received 13 December 1909
Revised 5 June 2000
The Higgs sector of the minimal 3-3-1 model with three triplets and one sextet is in
vestigated in detail under the most general lepton-number conserving potential. The
mass spectra and m ultiplet decomposition structure are explicitly given in a systematic
order and a transparent way allowing them to be easily checked and used in further
investigations. A previous problem that arose from the inconsistent signs of /2 is also
automatically solved.
PACS numbers: 11.15.Ex, 12.60.Fr, 14.80.cp.
1. Introduction
The S tandard Model (SM) combining the Glashow-Weinberg Salam (GWS) model
with the Q C D under the gauge group S U c(3) 0 SU/,(2) 0 u ,v (l) is one of the
greatest achievements of physics in the 20th century. Many predictions of the SM
have been confirmed by various experiments. However, this model works well only
at the energy range below 200 GcV and gradually loses its prediction power at
higher energies. Therefore, any extension of the SM to fit the theory with the
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