b gio dc v o to vin hn lm khoa hc

v cng ngh vn

vin vt l

trn nh thm

vt cht ti trong mt s m hnh 3-3-1m rng

Chuyn ngnh: Vt l l thuyt v Vt l tonM s: 62 44 01 03

lun n tin s vt l

Ngi hng dn khoa hcGS. TS. ng Vn Soa

H Ni - 2014

Li cm n

Ti xin by t lng bit n n GS. TS. ng Vn Soa hng dnti hc tp, nghin cu trong sut thi gian lm nghin cu sinh v gipti hon thnh lun n ny. Xin cm n GS. TS. Hong Ngc Long,TS. Phng Vn ng, TS. Th Hng, TS. L Th Hu, ThS. CaoHong Nam - Vin Vt l v TS. Nguyn Huy Tho, TS. H ThanhHng - Trng i hc S phm H Ni 2 gip v c nhiu nggp i vi kt qu ca lun n.

Ti xin cm n Trng i hc Phm Vn ng ni ti ang cngtc c nhng h tr v ng vin cn thit trong thi gian ti lmnghin cu sinh. Xin cm n Vin Vt l l c s o to to iukin thun li v gip ti trong qu trnh lm nghin cu sinh v bov lun n.

Cui cng, ti xin dnh s bit n su sc ti gia nh ng vin,ng h v h tr v iu kin v mi mt ti c th yn tm nghincu v bo v thnh cng lun n ny.

ii

Li cam oan

Lun n ny l kt qu m bn thn ti thc hin trong thi gianlm nghin cu sinh ti Vin Vt l. C th, chng mt l phn tngquan gii thiu nhng vn c s c lin quan n ni dung ca lunn. Trong chng hai ti s dng kt qu nghin cu m ti thchin cng vi thy hng dn GS. TS. ng Vn Soa v GS. TS. HongNgc Long. Chng ba ti s dng cc kt qu nghin cu cng vi TS.Phng Vn ng - Vin Vt l v ng nghip l TS. H Thanh Hng- Trng i hc S phm H Ni 2. Chng bn l bin lun nghavt l da trn cc kt qu nghin cu.

Cui cng ti xin cam oan v khng nh rng, y l cng trnhnghin cu ca ring ti. Cc kt qu c trong lun n "Vt cht titrong mt s m hnh 3-3-1 m rng" l kt qu mi, khng trng lpvi bt k lun n hay cng trnh no cng b.

H Ni, ngy 30 thng 4 nm 2014Tc gi lun n

Trn nh Thm

iii

Mc lc

Li cm n ii

Li cam oan iii

Cc k hiu chung vi

Danh sch bng vii

Danh sch hnh v viii

M u 1

1 Vt cht ti v s m rng ca m hnh chun 5

2 Axion trong m hnh 3-3-1 v thc nghim tm kim 152.1 Axion trong m hnh Peccei-Quinn . . . . . . . . . . . . 15

2.1.1 Vn strong-CP . . . . . . . . . . . . . . . . . . 162.1.2 i xng Peccei-Quinn, bo ton CP v s xut

hin axion . . . . . . . . . . . . . . . . . . . . . . 272.2 Axion trong m hnh 3-3-1 vi neutrino phn cc phi . 32

2.2.1 Tng quan v m hnh . . . . . . . . . . . . . . . 322.2.2 i xng Peccei-Quinn v axion . . . . . . . . . . 352.2.3 Qu trnh r ca axion thnh hai photon . . . . . 37

2.3 Tit din tn x ca qu trnh chuyn ha photon-axiontrong trng in t ngoi . . . . . . . . . . . . . . . . . 382.3.1 Yu t ma trn . . . . . . . . . . . . . . . . . . . 382.3.2 S chuyn ha trong in trng tnh . . . . . . . 402.3.3 S chuyn ha trong t trng tnh . . . . . . . . 422.3.4 S chuyn ha trong ng dn sng . . . . . . . . 45

2.4 Tm tt kt qu . . . . . . . . . . . . . . . . . . . . . . . 47

iv

3 Vt cht ti trong m hnh 3-3-1-1 v thc nghim tmkim 503.1 M hnh 3-3-1-1 . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.1 Fermion trung ha v cc ht lepton sai . . . . . 503.1.2 i xng chun 3-3-1-1 v W -parity . . . . . . . 553.1.3 Th v hng v khi lng . . . . . . . . . . . . 59

3.2 Vt cht ti v thc nghim tm kim . . . . . . . . . . 663.2.1 Mt tn d ca boson chun X0 . . . . . . . . 673.2.2 Mt tn d ca fermion trung ha NR . . . . . 683.2.3 Thc nghim tm kim vt cht ti NR . . . . . . 70

3.3 Tm tt kt qu . . . . . . . . . . . . . . . . . . . . . . . 72

4 Kt lun 744.1 Cc kt qu chnh ca lun n . . . . . . . . . . . . . . . 744.2 Cc hng nghin cu tip theo . . . . . . . . . . . . . . 75

Danh sch cc cng b ca tc gi 79

Ti liu tham kho 80

Ph lc 89

A Tm yu t ma trn 90

B Kim tra cc d thng U(1)N 92

C Ngun gc ca W -parity 95

v

Cc k hiu chung

Trong lun n ny ti s dng cc k hiu sau:

Tn Vit ttVt cht ti (Dark Matter) DMM hnh chun (Standard Model) SMLin hp in tch-Chn l (Charge conjugation-Parity) CPCP trong tng tc mnh Strong-CPMy gia tc hadron ln (Large Hadron Collider) LHC

Trung tm nghin cu ht nhn Chu u(Conseil Europen pour la Recherche Nuclaire) CERNSc ng lc hc lng t (Quantum Chromodynamics) QCDChn l W hay chn l lepton W -parityM hnh chun siu i xng ti thiu(Minimal Supersymmetric Standard Model) MSSM

vi

Danh sch bng

2.1 S ph thuc ca b rng r v thi gian sng caaxion theo khi lng ca n. . . . . . . . . . . . . . . . 38

3.1 Tch L ca cc a tuyn trong m hnh. . . . . . . . . . 523.2 S lepton ca cc ht trong m hnh. . . . . . . . . . . . 533.3 Tch B ca cc a tuyn trong m hnh. . . . . . . . . . 543.4 Cc a tuyn trong m hnh 3-3-1-1 vi tch N tng ng. 563.5 R-parity ca cc ht trong m hnh 3-3-1-1 gm hai loi

l cc ht lepton sai v cc ht thng thng. . . . . . . 58

vii

Danh sch hnh v

2.1 Tit din tn x ton phn (cm2) ca qu trnh chuynha photon thnh axion trong in trng tnh ng vixung lng q = 104 103eV . th trn v vi 300im v th di v vi 3000 im. . . . . . . . . . . . 43

2.2 Tit din tn x ton phn (cm2) ca qu trnh chuynha photon thnh axion trong t trng tnh ng vi xunglng q = 104 103eV . th trn v vi 300 im v th di v vi 3000 im. . . . . . . . . . . . . . . . 46

2.3 Tit din tn x ton phn (cm2) ca qu trnh chuynha photon thnh axion trong ng dn sng vi xunglng q = 105 104 eV. . . . . . . . . . . . . . . . . . 47

3.1 Cc ng gp chnh cho qu trnh hy X0 thnh W+W. 673.2 Cc ng gp chnh cho qu trnh hy ca NR. . . . . . 68

viii

M u

L do chn ti

Trong nhiu thp k qua, vic tm kim cc ht mi trong vt l htc bn v ang thu ht rt nhiu nh vt l, nhm tm hiu v giithch cu trc cng nh bn cht ca V tr. Nhng thnh cng v cngngh quan st ca th k 21 em li cho chng ta nhng hiu bitsu hn, nhng thc cht vn ch l mt phn rt nh hiu bn chtca V tr. Theo thc nghim quan st hin nay, V tr hin ti cha68.3% nng lng ti, 26.8% vt cht ti (Dark Matter - DM), ch c4.9% l vt cht thng thng (vt cht m chng ta quan st c)[1]. Trn thc t c hai quan nim v DM. Dng th nht l DM cto ra t cc ht vt cht thng thng, chng ta gi chng l vt chtti dng baryonic (baryonic DM). i tng ch yu ca DM dng nyl cc ngi sao khng pht ra bc x v tri trong khng gian V tr.Cc ngi sao ny khng c s lin h vi h thng cc sao trong V tr,chng c gi l MACHO (Massive astrophysical compact halo object).Cc ng c vin cho dng DM ny l cc ngi sao ntron hay h en.Dng th hai ca DM l dng vt cht khng bt ngun t cc dng vtcht thng thng, chng c gi l non-baryonic DM. Cc ng c vincho non-baryonic DM c cho l cc ht WIMPs (weakly interactingmassive particles), l cc ht c khi lng nhng tng tc rt yu vivt cht thng thng (cc ht ch c tng tc hp dn m khng ccc tng tc khc). Cc nh thin vn hc ch yu nghin cu cc ngc vin ca DM l baryonic DM, trong khi cc nh vt l ht c bnth tm kim DM l cc ht WIMPs. Trong lun n ny, chng ti tptrung nghin cu DM da trn quan im ca vt l ht c bn.

Trn quan im ca vt l ht c bn, cc ht DM l cc ht trungha, khng b r hoc thi gian sng ca chng phi ln (tc lthi gian sng ca DM phi ln hn tui ca V tr). Hin ti, cc ht

1

WIMPs cha c tm thy trong cc my gia tc v cng cha c bngchng no cho ta xc nh cc thng tin v spin cng nh khi lng cachng. Chnh v vy, nghin cu bn cht ca DM v tm kim chngl mt trong nhng vn v ang c cc nh khoa hc trn thgii, k c cc nh vt l l thuyt v thc nghim quan tm.

Mt khc, m hnh l thuyt m t cc tng tc ca cc ht c bntrong V tr c thc nghim ng h nht hin nay l m hnh chun(Standard Model - SM). Tuy nhin, trong SM khng tn ti ng c vintha mn tnh cht ca DM. Do , chng ta cn phi m rng SM chng xut hin cc ng c vin ca DM. Do tnh cht v spin ca DMl khng xc nh v ph khi lng ca DM l rng nn cc ng cvin ca DM l rt phong ph. Chng c th l ht v hng, ht vct hay ht fermion.

Chng ti mun nhn mnh, khi m rng SM th vng khng giantham s xut hin trong m hnh s rng hn. Tuy nhin, da vo cc sliu thc nghim v mt v thi gian sng ca DM, chng ti c thgii hn c vng khng gian tham s xut hin trong m hnh. Davo vng khng gian tham s va tm c v cc tng tc ca chng,chng ti c th d on c v kh nng tm kim DM mt cch trctip hoc gin tip.

V vy, ti chn ti "Vt cht ti trong mt s m hnh 3 3 1m rng" nghin cu v bn cht v kh nng tm kim DM. M hnhm rng chng ti nghin cu l cc m hnh SU(3)CSU(3)LU(1)Xc thm cc i xng mi.

Mc ch nghin cu

Kho st vai tr DM ca axion trong m hnh 3-3-1 vi neutrinophn cc phi. Nghin cu tng tc ca axion vi photon trongtrng in t ngoi v trn c s a ra phng n c li nht thu axion trong thc nghim.

Xy dng m hnh 3-3-1-1 v kho st vai tr DM ca fermion trungha cha trong m hnh.

2

i tng nghin cu

Axion trong m hnh 3-3-1 vi neutrino phn cc phi. Fermion trung ha trong m hnh 3-3-1-1.

Ni dung nghin cu

Nghin cu ngun gc xut hin axion trong vic gii quyt vn strong-CP v tnh bn ca n thng qua qu trnh r thnh haiphoton trong m hnh 3-3-1 vi neutrino phn cc phi v kh nngpht hin axion trong thc nghim.

c im ca m hnh 3-3-1-1. Vai tr DM ca fermion trung ha trong m hnh 3-3-1-1 v thc

nghim tm kim.

Phng php nghin cu

L thuyt trng lng t. L thuyt nhm. S dng quy tc Feyman tnh bin tn x v b rng r. S dng cc d liu thc nghim mi nht c cng b trn tp

ch Particle Data Group.

Dng phn mm Matlab R2008a tnh s v v th.

ng gp ca lun n

Nhng kt qu nghin cu ca lun n l bng chng quan trng gpphn khng nh s tn ti ng c vin cho DM l axion v fermiontrung ha trong cc m hnh m rng m hnh chun c gi thit thmmt s i xng mi. xut phng n c li nht tm kim ngc vin cho vt cht ti trong thc nghim.

3

B cc ca lun n

Trong lun n ny ngoi phn m u v ph lc, ni dung chnhc chng ti trnh by trong 4 chng:

Chng 1: Gii thiu tng quan v vt cht ti v s m rng ca mhnh chun.

Chng 2: Trnh by l thuyt chung v vn strong-CP v ngungc xut hin axion thng qua qu trnh ph v i xng Peccei-Quinn gii quyt vn strong-CP. Tip theo, chng ti xt c th cho mhnh 3-3-1 vi neutrino phn cc phi v kho st vai tr ng c vinvt cht ti ca axion trong m hnh ny, nghin cu tnh bn ca axionthng qua qu trnh r ca n thnh hai photon v nghin cu s chuynha photon thnh axion trong trng in t ngoi bao gm in trngtnh, t trng tnh v ng dn sng v xut cho thc nghim cchng c li nht thu axion.

Chng 3: Xy dng m hnh 3-3-1-1 da trn m hnh 3-3-1 c gnvi i xng chun mi U(1)N . Kho st mt s thuc tnh DM ca ccht lepton sai c trong m hnh, t xc nh ht no l ng c vincho vt cht ti.

Chng 4: Tm tt cc kt qu chnh ca lun n v cc hng nghincu tip theo.

4

Chng 1

Vt cht ti v s m rng ca mhnh chun

Hin nay ngun gc v mt DM trong V tr l mt cu hi trungtm trong c vt l thin vn v vt l ht c bn. C rt nhiu ccbng chng thc nghim chng t s tn ti ca DM trong V tr.Bng chng th nht chng t s c mt ca DM trong V tr l vntc ca cc thin h. Gi s mt vt cht ca cc thin h l ngu, vn tc ca cc thin h s ph thuc vo khi lng v khongcch theo h thc

v2 =GNM(r)

r(1.1)

Bn ngoi thin h (ni m nh sng khng bc x) khi lng M(r) lhng s th vn tc v2 s gim t l nghch vi khong cch. Tuy nhin,cc php o t thc nghim [2] chng t vn tc v bn ngoi thinh vn l hng s. Ngha l, khi lng s ph thuc vo khong cchbn ngoi thin h M r. iu ny chng t tn ti mt dng vt chtm chng ta khng nhn thy bn ngoi thin h.

Mt bng chng th hai v DM l thu knh hp dn. Cc nh vt lthin vn dng thu knh hp dn tm kim DM. Thu knh hpdn l hin tng nh sng s b gy khc khi m tia sng i qua ccvt th c khi lng ln. Chnh v vy, khi nh sng i qua h en hoccc ngi sao ntron (vt th l ng c vin baryonic DM) th nh sngb gy khc. Cc nh thin vn hc nghin cu cc bc nh chp butri v hiu ng sng chng t s gy khc ca nh sng trong khnggian V tr. T nhng s liu thc nghim , chng t s tn ti cacc vt th trong V tr m chng ta khng quan st c. Chng ta

5

gi chng l DM.S xut hin ca cc tia X pht ra t cc m kh nng trong cc

thin h Elip chng t s tn ti ca DM. Da trn cc m hnh ngnhit vi nhit kT = 3keV , Fabricant v Gorenstein tin ontng khi lng t tm thin h n bn knh 392 kpc l 5.8 1013Mvi M l khi lng Mt Tri. Tuy nhin, khi lng kh nng ch c2.8 1012M. Khi lng quan st c ch c mt phn trm tngkhi lng. Chi tit v bng chng ny c ch r trong ti liu [3].

Mt bng chng khc na chng t s tn ti ca DM l s liu quanst thc nghim PLANCK. Theo thc nghim quan st hin nay, Vtr hin ti cha 68.3% nng lng ti (dark energy), 26.8% DM, ch c4.9% l vt cht thng thng (vt cht m chng ta quan st c) [1].

Chng ta khng ch c nhng bng chng trc tip chng t s tnti ca DM m cn c nhng bng chng gin tip khc cng chng ts tn ti ca DM. T nhng nghin cu v phn vt cht v vt chttrong V tr a ra kt lun vt cht nhn thy ch ng gp mtphn rt nh vo trong V tr. Tuy nhin, s liu [1] chng t = 1.iu ny chng t dng vt cht quan st c ch ng gp mt phnnh vo mt vt cht trong V tr. Phn ln vt cht v nng lngng gp vo mt V tr l cha quan st c .

T nhng bng chng quan st trn chng t s tn ti ca DM trongV tr. Tuy nhin, tm kim v nghin cu DM trong V tr vn l mtcng vic cha c li gii. Vt cht ti l ht g? Lm cch no c thpht hin ra chng? Khi lng v spin ca chng nh th no? Tt ccc cu hi vn ang c cc nh vt l ht c bn, c thc nghimv l thuyt ang rt quan tm. Theo quan im vt l ht c bn thngi ta phn loi DM theo vn tc chuyn ng ca chng. C th:

Vt cht ti nng (hot dark matter) l cc ht vt cht c khi lngnh v chuyn ng nhit vi vn tc rt ln (tng i tnh). Theoquan im nh vy th neutrino l ng c vin cho vt cht tinng.

Vt cht ti lnh (cold dark matter) l cc ht c khi lng vchuyn ng vi vn tc nh (phi tng i tnh). Cc ng c vinca ht vt cht ti lnh l neutralino, boson Higgs, axion,...

Vt cht ti m (warm dark matter) l nhng ht DM c tnh cht

6

trung gian gia vt cht ti nng v vt cht ti lnh.

Trc y, cc ng c vin hng u cho vt cht ti lnh l cc htc khi lng v khng tng tc vi vt cht thng thng. Ngi tacng cho rng, phn nng lng b mt i trong cc my gia tc chnhl tn hiu ca DM. Tuy nhin, khoa hc hin nay m rng ng kdanh sch cc ng c vin v cung cp nhiu du hiu mi c th tmkim DM. Cc s liu thc nghim v tm kim DM mt cch gin tipcho thy ph khi lng ca vt cht ti lnh l rt rng. Cc thng tinv spin ca vt cht ti lnh cng khng c xc nh. Chnh v vy,cc ng c vin ca vt cht ti lnh l rt phong ph. Do , tronglun n ny, chng ti s tp trung nghin cu cc ng c vin ca vtcht ti lnh. Chi tit, chng ti s nghin cu cc iu kin cn thit m bo ht trung ha l bn. Cc ng c vin ca DM chng tikho st cng phong ph. Chng c th l ht v hng v cng c thl ht fermion. Chng ti tp trung ch yu kho st mt ng c vinca WIMPs l ht nh (c eV ) v mt ng c vin ca WIMPs l htnng (c TeV ).

Chng ti mun nhn mnh, mt trong nhng iu kin quan trngca DM l ht trung ha v bn vng. Mt khc, trong l thuyt htc bn, m hnh thnh cng nht hin nay l SM, khi m hu ht cctin on ca n c thc nghim khng nh vng nng lng200 GeV. Theo , ph ht trong SM gm cc fermion c spin 12 , ccboson chun spin 1 v boson Higgs spin 0. Fermion l cc ht cu tonn vt cht thng thng gm cc quark (up, down, charm, strange,top, bottom), ba th h lepton mang in (electron, muon, tauon) vcc neutrino tng ng. Cc boson chun l cc ht truyn tng tcbao gm: photon () truyn tng tc in t, 8 gluon g truyn tngtc mnh, cc boson chun W v Z truyn tng tc yu. Ht cui cngtrong SM c gi thit l ht Higgs sinh khi lng cho tt c cc httrong m hnh thng qua c ch Higgs [4]. Vic tm thy ht Higgs trongnm 2012 vi khi lng trong khong t 125 GeV n 126 GeV bi mygia tc LHC (CERN) l mt thnh cng na ca SM [5, 6]. Tuy nhin,khng ht no trong cc ht ni trn c th ng vai tr l ng c vincho DM. Hu ht cc ht u c khi lng v khng bn, vi thi giansng rt ngn so vi tui ca V tr. Ch cn li cc ht nh cn tnti trong V tr bao gm: electron, cc quark u, d v ba neutrino, trong

7

cc ht mang in c th tn ti trong cc t hp bn nh proton,ntron. Electron c th cho ng gp vo DM ch khi chng lin kt vicc proton to thnh cc t hp trung ha nhng t hp ny cho mt nng lng B nh hn rt nhiu so vi mt DM o c hin nay.Vi cc neutrino th gii hn trn khi lng neutrino t vt l ht v vtr hc cho thy mt tn d ca neutrino ' imi/47eV 0.012[4]. iu ny khng nh cc ht ng gp vo DM phi l cc ht mitrong cc m hnh m rng SM.

Ngoi vn DM, SM cn cha mt s vn khc cn phi cgii quyt xut pht t cc d liu thc nghim c hin nay nh vn khi lng v s dao ng ca cc neutrino, s phn bc thang nnglng ph v i xng in yu vi thang Plank, vn vi phm CPtrong tng tc mnh (strong-CP), s lng t ha in tch ca ccht, gii thch s th h fermion, s khng thng nht ba hng s tngtc, cha gii thch c cc qu trnh vt l xy ra vng nng lngcao hn 200 GeV,... Cc bng chng trn, chng t chng ta cn phim rng SM.

Trong lun n ny, chng ti tp trung vo hng nghin cu mrng SM gii quyt vn DM. Trc tin, chng ti quan tm ns xut hin ca DM gn lin vi vic gii quyt bi ton strong-CP.Chng ti s chng minh c s tn ti ca DM thng qua c ch phv i xng Peccei-Quinn. Nh chng ta bit, Lagrangian ca scng lc hc lng t (Quantum Chromodynamics - QCD) tn ti mts hng vi phm CP c dng

L =

16pi2F aF a, (1.2)

vi

F a =1

2F

a, (1.3)

v l tens phn xng c , , , = 0, 1, 2, 3; 0123 = 1.iu c bit y l s hng ny l mt vi phn ton phn nn

ngi ta cho rng n khng gy ra hiu ng vt l. Nhng thc t, vy l ton t vi phm CP (tr ring CP = 1) nn n cho ng gpvo mmen lng cc in ca ntron. T thc nghim o mmen nyngi ta thy c gi tr cc nh 109. Gi tr cc nh ny l mt

8

iu khng t nhin i vi cc lp lun l thuyt. Do , n c gi lvn strong-CP. Ngi ta phi tm cch xy dng mt l thuyt hpl gii thch vn ny. Mt trong cc cch gii quyt ph hp nhtc chp nhn hin nay do Peccei-Quinn xut, c lin quan n sph v i xng U(1) chiral dn n s xut hin mt ht gi v hngmi c kh nng ng vai tr DM. Nm 1970, Peccei-Quinn [7] ch rarng vn strong-CP gii quyt c bng cch a ra mt ht gi vhng nh c gi l axion [8] v vic axion xut hin trong l thuytnh th no c trnh by chi tit chng 2. Hin nay, khi lngaxion c gii hn bi thc nghim [9], cc gii hn c a ra bithin vn hc v v tr hc [10], phm vi gii hn trong khong t 106

eV cho n 103 eV. Nu axion c khi lng gn gii hn di c 105

eV th n s l ng c vin cho DM. Nu axion trong V tr ng gpmt t l no vo DM chng c th c pht hin bi cc my daxion [11]. Hin nay thc nghim vn ang tm kim axion sinh ra tMt tri [12] v axion tn d t thi k V tr sm [13].

Ngoi axion vn cn c cc ng c vin khc cho DM xut hin trongcc m hnh m rng SM ang c nghin cu hin nay nh bn nghnh siu i xng nh nht trong cc m hnh siu i xng (LightestSupersymmetric Particle - LSP), axino bn ng hnh siu i xngca axion [14, 15, 16, 17], gravitino bn ng hnh siu i xng caht truyn tng tc hp dn graviton, mt s ht mi xut hin trongcc m hnh 3-3-1 [21] hay m hnh thm chiu [18],... Trong lun nny, chng ti ch tp trung nghin cu DM l axion v fermion trungha trong cc m hnh 3-3-1 c thm mt s i xng mi. y l lpcc m hnh m rng SM theo hng m rng nhm i xng in yuSU(2)L

U(1)Y ca SM. C th, nhm ny c m rng thnh nhm

SU(3)LU(1)X v cng vi nhm i xng mu SU(3)c to thnh nhm

chun SU(3)cSU(3)L

U(1)X . Cc m hnh 3-3-1 gii quyt c

mt s vn m SM cha gii quyt c. V d: (i) bi ton s thh c gii quyt t iu kin kh d thng v iu kin tim cn tdo [19], (ii) gii thch c s lng t ha in tch [20], (iii) gii thchs phn bc khi lng gia cc th h quark do cc th h nm trongcc a tuyn khc nhau, (iv) axion v majoron xut hin mt cch tnhin trong mt s phin bn 3-3-1 [21], cho cc ht nh c th l ngc vin DM. H qu ca cc m hnh 3-3-1 l d on cc qu trnh vt

9

l thang nng lng khng qu cao, do d dng kim chng bngthc nghim.

Trong lp cc m hnh 3-3-1 khng siu i xng, cc ht ng c vinca DM xut hin mt cch khng t nhin. Tc l m bo cho ccng c vin ca DM l bn th chng ta cn phi c cc iu kin rngbuc gia cc tham s trong m hnh. Tuy nhin, nhiu iu kin a rakhng th m bo ng n mi bc ca khai trin nhiu lon. Chnhv vy, chng ta cn phi a thm vo cc i xng mi loi b cctng tc khng mong mun. i xng a vo c th l i xng ginon hoc i xng lin tc. Trong lun n ny, chng ti s kho stc hai cch a thm i xng vo cc m hnh 3-3-1. T , chng tichng t s tn ti ca DM. C th:

Trong chng 2 chng ti s nghin cu chi tit ngun gc v vai trDM ca axion trong m hnh 3-3-1 vi neutrino phn cc phi gn vii xng Z11 Z2. Nhn chung cc i xng gin on c a vonhm mc ch loi b cc s hng tng tc khng mong mun nhngvn gi li c cc s hng sinh khi lng cho cc ht ph hp vithc nghim. Theo [21], cc m hnh c ph ht cng phong ph nh lpm hnh 3-3-1 th cng c kh nng tn ti i xng ZN ph hp vi Nc gi tr ln. Ngi ta chn N c gi tr ln loi b nhiu nht c thcc s hng phc tp v khng cn thit trong Lagrangian. Tuy nhin,vi trng hp Z13 th ngi ta phi xy dng ph fermion phc tphn m bo tn ti i xng Peccei-Quinn mt cch t nhin trongm hnh. V vy, i xng Z11 c chn l ph hp nht. y chngti ln lt gn cc tch Z11 Z2 cho cc trng c trong m hnh saocho Lagrangian cui cng s bt bin. Khi , Lagrangian ny cng tng bt bin vi i xng U(1)PQ vi cc tch Peccei-Quinn c gnph hp. M hnh ny ch cn mt trng v hng ph v i xngPeccei-Quinn v sinh axion. Tip theo, chng ti kho st thi gian sngca axion thng qua qu trnh r ca axion thnh hai photon. Chngti cng tho lun v xut phng n c li nht pht hin axionthng qua qu trnh chuyn ha ca axion trong trng in t ngoi.

Nh cp trn, ngoi axion th ht mi trong cc m hnh3-3-1 cng c th ng vai tr l ng c vin DM. Tht vy, chng tis chng t rng cc ht ny tng ng vi mt lp ht mi, l di ixng parity c th ng vai tr DM. gii thch r hn ta nhc li

10

m hnh 3-3-1 vi nhm chun SU(3)C SU(3)LU(1)X , trong hainhm cui c m rng t i xng in yu ca SM, cn i xngQCD vn c gi nguyn. Cc neutrino phn cc phi hoc cc fermiontrung ha mi c th c sp xp trong cc tam tuyn hoc phn tamtuyn lepton theo i xng SU(3)L,

LeLcR

hocLeLN cR

,tng ng l m hnh 3-3-1 vi neutrino phn cc phi [22] hoc mhnh 3-3-1 vi fermion trung ha [23] (xem thm ti liu [24]).

Cc ht mi xut hin trong cc m hnh 3-3-1 nhng khng c trongSM (chng hn nh cc ht v hng, fermion v boson chun) c thchnh l cc ng c vin cho DM. iu ny c ch ra t cc tngtc chun, Lagrangian Yukawa ti thiu v th v hng ti thiu, ihi cc ht mi (tng t nh cc ht lepton sai s c nh nghatrong chng 3) ch tng tc theo cp, ging vi trng hp cc htbn ng hnh siu i xng trong cc m hnh siu i xng. Kt quny t ng n t bn thn cu trc ca nhm i xng chun 3-3-1[24, 22]. Nhng n lc u tin trong vic xc nh cc ng c vin choDM trong cc m hnh 3-3-1 c th c tm thy trong ti liu [28, 29].Tuy nhin, tnh n nh v mt tn d ca chng vn cha cgii quyt trit . S bn ca DM trong cc m hnh 3-3-1 do cc ixng khc gy ra c tho lun u tin trong ti liu [26, 27]. Vi mcch ny, i xng s lepton c a vo sao cho cc ht bileptonnh nht c th m bo tnh bn [26]. Lu rng, tt c cc tngtc khng mong mun trong Lagrangian Yukawa v th v hng uvi phm tng minh s lepton [30], nhng tng tc ny b chn do ixng ny (ngoi tr tng tc ca hai lepton v mt tam tuyn v hngch vi phm s lepton v, nhng li gy ra ph khi lng neutrino khngph hp. Tuy nhin, trong m hnh ca chng ti th tng tc ny bkh do s vi phm s lepton ton phn). Khi ta khng cn thm ixng Z2 na. Mt vn khc l, s lepton b vi phm do tng tchiu dng 5 chiu c trng cho khi lng neutrino.

Trong [27], s sp xp cc ht mi lm mt i c tnh bilepton cachng v i xng s lepton khng loi b c cc tng tc khngmong mun. V vy, i xng Z2 c a vo bng tay, trong cc

11

tch Z2 c gn loi b cc s hng khng mong mun. i xngny c xem nh mt cch gii quyt tnh n nh ca DM [27]. Tuynhin, v Z2 tc dng ln cc a tuyn ca m hnh phi b ph v mtcch t pht bi trung bnh chn khng ca Higgs, nn khng c l dono gii thch ti sao cc ht DM v hng khng mang s lepton likhng th nhn trung bnh chn khng v phn r. Ngoi ra, trong [27],mt i xng lin tc U(1)G tc dng ln cc ht thnh phn, khnggiao hon vi i xng chun tng t nh tch lepton trc y cs dng thay cho i xng Z2 nhm m t DM. Chng ta nh rng,cc tng tc ca boson chun vi fermion, v hng hoc t tng tcgia cc boson chun l cc h qu v b gii hn bi chnh i xngchun. Chng lun lun tn ti m khng b loi b hoc thm vo bicc tng tc khc. chn cc tng tc v chn khng khng mongmun, r rng c nhiu i xng khc c c ch nh U(1)G hoc tchlepton l cc cch gii quyt tng ng ca bo ton tng tc chun.Tuy nhin, tt c cc i xng lin tc ny u c th gp phi nhngvn khc.

Cc i xng lin tc ni trn c gi thit l i xng chnh xcc trng cho tnh n nh ca DM. Do , chng c th c xem nhl i xng t nhin rt gn t cc i xng cao hn bao gm nhmi xng ca 3-3-1 (v chng khng giao hon vi i xng chun) tcdng mc cy ln Lagrangian, cc tng tc khng mong mun bchn tng minh. Hay ni cch khc, Lagrangian ti thiu ca l thuythin nhin cha cc i xng cao hn bao gm i xng chun, l ixng s b ph v t pht xung cc i xng d tng ng. y tacn nhc li mt tnh cht c trng ca cc m hnh 3-3-1. l tchlepton (hay thm ch bt k loi i xng U(1)G nu c a vo clp, b qua i xng tch lepton) s tc dng nh i xng chun dca i xng cao hn no v i xng ny phi b ph v t pht m bo cc boson chun nhn khi lng ln lm cho l thuyt phhp thc nghim. Mt khc, i xng tch lepton hoc ngay c i xnglin tc tng qut c bit thc t b vi phm do cc d thng.Do , cc i xng ny s khng m bo c s bn ca DM. ivi bi ton bn ca DM, tng t nh R-parity trong siu i xng,s t nhin hn khi tm kim mt i xng chnh xc, l i xng dgin on v khng b ph v, c ngun gc t mt i xng lin tc

12

v c lp d thng no , biu din theo s lepton v cc i xngcn thit khc, chng hn nh tch baryon hoc U(1)G. Chng ta cnphi nhn mnh rng, trong s cc i xng lin tc phn tch thtch lepton c l l t nhin nht bi cc l do sau: (i) tt c cc tngtc khng mong mun trong cc m hnh 3-3-1 thng thng s b loib bi chng vi phm s lepton [30]; (ii) i xng gin on c trngcho s bn ca DM chng hn nh s lepton hoc baryon c th b phv theo nhiu cch m bo sinh khi lng ph hp cho neutrino vbt i xng baryon. Chng ta s thy rng, cch lm trn tng t vivic m rng l thuyt SU(5) thnh l thuyt SO(10) vi tch B Ltr thnh tch chun nh x. Trong lun n ny, tch lepton s c sdng khc so vi i xng U(1)G.

Bng cch kho st tnh cht s lepton khng tm thng v W -parity (tng t nh R-parity trong siu i xng) trong mt m hnh3-3-1 c th [23], chng ti ch ra c l thuyt c th cha cc ng cvin cho DM mt cch t nhin. chi tit hn, chng ti xt m hnh3-3-1 vi fermion trung ha (NR), l m hnh khc vi m hnh trongti liu [27]. Cc fermion trung ha ny khng mang s lepton nh c nghin cu trc y trong m rng seesaw TeV ca SM [31] vtrong m hnh 3-3-1 vi cc i xng v [23]. Chng ti kho st ixng s lepton, ng hc ca n v cc i xng khc, m kt qu dnn mt m hnh 3-3-1-1 mi. Chng ti chng t rng tn ti mt ixng d khng b ph v (c lp d thng) ca m hnh 3-3-1-1 c cch ging nh i xng R-parity trong siu i xng, di i xngny hu ht cc ht mi u mang tch l. iu th v l m hnh c thcha rt nhiu loi ng c vin cho DM, chng hn nh n tuyn vhng, fermion v boson chun, l cc ht thng xut hin trong ccphin bn m rng khc ca SM v ging vi kt lun trong ti liu [27].Tuy nhin, cc ng c vin ny c th c khi lng ln thang TeV,khc vi cc ng c vin ca DM nh trong nhiu SM m rng quenthuc. Trc cng b [27] v cng trnh ca chng ti, cc nghin cutrc y cho cc m hnh 3-3-1 ch xt cho n tuyn v hng [29] vcc ht siu i xng nh nht tng ng vi phin bn m rng 3-3-1siu i xng. L do m chng ti cp n cc i xng nh 3-3-1-1v W -parity l: cc ng c vin cho DM c tnh n nh ng hccha h c nghin cu trc y. Hin tng lun DM trong m hnh

13

ca chng ti s khc so vi nhng m rng trc. M hnh c th lmvic tt hn di cc gii hn thc nghim so vi m hnh 3-3-1 thngthng vi neutrino phn cc phi do W -parity. Phng php c cp trn s khng ph hp na nu ta p dng cho cc m hnh 3-3-1khc, chng hn nh m hnh 3-3-1 trong ti liu [27], m hnh 3-3-1 vineutrino phn cc phi [22] v m hnh 3-3-1 ti thiu [24]. Trong thct, tt c cc ht bao gm c nhng ht mi trong cc m hnh sbin i tm thng di W -parity. V vy, i xng parity ch thc sc ngha trong lp cc m hnh 3-3-1 cha i xng v [23].

14

Chng 2

Axion trong m hnh 3-3-1 v thcnghim tm kim

2.1 Axion trong m hnh Peccei-Quinn

Ngi ta chng minh c s hng (1.2) nhn ng gp t hai ngunchnh: i) t trung bnh chn khng ca l thuyt c gi l "instantons";ii) t s nh ngha li pha ca cc trng c trong l thuyt c lin quann mt loi i xng U(1) chiral c gi l i xng Peccei-Quinn, khiu U(1)PQ. T ngi ta a ra c mt s hng gii quyt vn strong-CP. V d, nu trng fermion c khi lng bng khng, thngi ta c th ty nh ngha li pha ca trng m bo cho nggp ton phn lun bng khng vi mi ng gp bt k t instantons.Tuy nhin, thc nghim cho thy rng cc quark lun c khi lngnn cch x l trn khng hp l. Mt hng gii quyt khc, c chol hp l nht hin nay do Peccei-Quinn xut nm 1977 [32]. Theo, t gi thit l thuyt ban u cha cc fermion khng khi lng,Peccei-Quinn a vo cc trng v hng sinh khi lng cho cc quarkthng qua ph v i xng U(1)PQ. Tip theo, s dng hai iu kin lthuyt phi tha mn l iu kin cc tiu th ca trng v hng viu kin thc ca ma trn khi lng quark, ngi ta thu c tngng gp t instantons v nh ngha li pha ca trng quark lunbng khng. Ni cch khc, khi l thuyt tha mn CP bo ton vvn strong-CP c gii quyt trn vn. Mt iu th v khc i vil thuyt trn c Weinberg pht hin l s ph v i xng U(1)PQc lin quan n s xut hin ca mt ht c khi lng rt nh gi laxion [33]. Ht ny c xem l mt trong cc ng c vin cho DM v

15

hin nay ang c thc nghim rt quan tm v tm kim. y chng ti khng i vo xy dng v tnh ton chi tit mt m

hnh c th v thc t no m ch tp trung vo cc vn c bn sauy:

Gii thch r hn s xut hin cc ngun vi phm CP, minh hatnh ton trong mt m hnh n gin nht.

Gii thch li s kh s hng vi phm CP theo hng gii quytca Peccei-Quinn.

2.1.1 Vn strong-CP

ng gp t chn khng vo s hng vi phm CP trong QCD

Nh chng ta bit trong QCD tn ti php bin i U(1) chiral, lmt i xng vi l thuyt c in nhng khng l i xng i vi lthuyt lng t. C th, php bin i i xng U(1) chiral ( U(1)5) cdng [34]:

qD(x) qD(x) = ei5qD(x), (2.1)trong , qD(x) l biu din spinor Dirac bn thnh phn, 5 xt trongc s chiral c dng cho.

5 =

I2 00 I2

, (2.2)

qD(x) =

qq

, (2.3)trong , q, q l hai biu din spinor Weyl phn cc tri hai thnh phn.

Xt s hng ng nng ca fermion:

Lfermion = iqD(x)DqD(x), (2.4)

y

qD(x) = q+D(x)

0, (2.5)

= 0, 1, 2, 3, (2.6)

16

l cc ma trn Dirac cn D l o hm hip bin tng ng c nhngha theo cc trng chun ca l thuyt

D = iA, (2.7)vi A = AaT

a, Aa l cc trng chun tng ng vi cc vi t Ta ca

i s Lie tng ng vi nhm chun.Vi l thuyt c in, theo nh l Noether, xt i xng pha nh

x (x) ca Lfermion c dng:

qD(x) qD(x) = ei(x)5qD(x),qD(x) qD(x) = qD(x)ei(x)5, (2.8)

s cho tng ng mt dng bo ton. C th xt tc dng c in:

S =d4xLfermion =

d4xiqD

DqD.

S =d4xLfermion =

d4xiqD

DqD

=d4xiqD(x)e

i(x)5D[ei(x)5qD(x)

]. (2.9)

V {5, } = 0 5 = 5 ta suy ra c:ei(x)5 = (cos i5 sin) = cos i5 sin

= cos + i5 sin

= (cos + i5 sin) = ei(x)5. (2.10)

Theo nh ngha o hm hip bin trong (2.7), ta c:

( iA)[ei(x)5qD(x)

]=

[ei(x)5qD(x)

] iAei(x)5qD(x)= (e

i(x)5)qD(x) + ei(x)5qD(x) iAei(x)5qD(x)= ei(x)5 [i(x)5] qD(x)+ ei(x)5DqD(x). (2.11)

Thay (2.10), (2.11) vo (2.9) ta c:

S =d4x iqD(x)

ei(x)5

[ei(x)5 (i(x)5) qD(x) + ei(x)5DqD(x)]=

d4xiqD

DqD +d4xiqD

((x)5) qD

17

= S +d4x [(qD(x)

(x)5qD(x))

(x)(qD(x)5qD(x))]= S

d4x (x) (qD(x)

5qD(x)) . (2.12)

Ch kt qu c c trong (2.12) l dod4x [qD(x)

(x)5qD(x)] =0 v y l mt vi phn ton phn.

Trong (2.12) do (x) chn c bt k nn suy ra tc dng bt binphi ng vi mi cch chn (x). V vy s hng sau phi bng 0:

(qD(x)5qD(x)) = 0,

JA(x) = 0; JA(x) = qD(x)5qD(x)). (2.13)Nh vy vi l thuyt c in, iu kin tc dng c in bt bin tngng vi dng JA phi bo ton.

Nu xt trong l thuyt lng t, i lng bt bin l tch phnng theo tt c cc trng thnh phn ca l thuyt.

Z =D[Aa]

DqDDqDe

iSfermion. (2.14)

Ngi ta thy vi php bin i U(1)5, cc o D[qD], D[qD] khngbt bin, v phn bin i ny c chuyn thnh bin i ca tc dngS. Chnh phn bin i ny gy ra s khng bo ton ca dng trc J A.S khng bo ton ny gi l d thng trc (axial anomaly). Ta c thch ra c theo lp lun di y.

Xt ring phn ly tch phn theo trng fermion, t:

Z[A] =DqDDqDe

iSfermion. (2.15)

Khai trin cc trng qD, qD theo cc trng thi ring n(x) ca tont D/, c m t bng h thc:

D/n(x) = nn(x). (2.16)

Khi ta khai trin c cc trng Dirac theo cc trng thi ring, cth l:

qD(x) =nann(x),

qD(x) =n+n (x)bn(x), (2.17)

18

trong an, bn(x) l cc h s khai trin.Cc n(x) tha mn iu kin trc chun:

+i j d4x = ij. (2.18)

Cc o tch phn ng chuyn thnh:

D[qD] D[qD] ndanmdbm. (2.19)Khi thc hin php bin i U(1)5 tc dng ln cc trng, cc bin an,bm cng bin i theo, c th:

qD(x) qD(x) = ei5qD(x)' (1 i5)qD(x) =

nann(x). (2.20)

Ngi ta biu din c cc h s a theo cc h s ban u theo biuthc sau:

an =m

n(1 i5)mamd4x

mCnmam. (2.21)

Khi lin h gia hai o tch phn trc v sau khi i bin cdng:

ndan ndan =1

detCnmndan. (2.22)

Da vo cc nh ngha trn, ngi ta xc nh c:

1

detCnm' ei

n

n5nd

4x. (2.23)

kh phn k trong tch phn (2.23) do c v hn s n, ngi ta sdng mt hm chnh tha mn cc iu kin bt bin chun, bt binLorentz v tha mn:

f |x= = 0; f |x=o = 0, (2.24)m bo tch phn phim hm c gi tr hu hn. Chn hm c dng:

f = f(D/2), (2.25)trong :

D/2 = DD = D2 i

2F

, (2.26)

19

l ton t c trng thi ring n. Vit li nh thc trn theo dng nhthc mi:

1

detCnm' ei

n 5f(D/

2)nd4x. (2.27)

Ly tng theo tt c cc trng thi n, ngi ta xc nh c:

1

detCnm' ei

d4k(2pi)4

d4xTr5f(D/2), (2.28)

tch phn d4k

(2pi)4 xut hin do khai trin n trong khng gian xung lng.

S dng cc tnh cht ca 5, khai trin hm f(D/2) theo khai trinTaylor, sau thay vo (2.28) s hng cui cng cn li l:

1

detCnm= exp

i d4k(2pi)4

d4xf(D/2)18FF(4i)

.(2.29)

Ly tch phn theo d4k, ch D2 = DD = (k A)2 v i bind4k d4k ta c:

1

detCnm' i

32pi2

d4x FF i

32pi2

d4x FF

, (2.30)

vi F F.Khi t (2.15) ta c:

Z[Aa] =D[qD] D[qD]e

iSfermion =

ndan

mdbme

iSfermion

=

mdbm

n(detCnmda

n)e

iSfermion

=

mdbm

ndane

iSfermion+ i32pi2d4x F F

=mdbm

ndane

iSfermion. (2.31)

T (2.31) suy ra tc dng S bin i mt lng:

S S = S + S = S 32pi2

d4x FF

. (2.32)

Tng ng vi n Lagrangian cng bin i mt lng:

S =d4xL L =

32pi2FF

, (2.33)

20

Tng t cho i bin b theo bin i:

qD qD = [(1 + i5)qD] 0 = qD(1 i5), (2.34)do 0 v 5 phn giao hon. Kt qu l qD, qD bin i ging nhau dotc dng ca 5. V vy bin i ca qD, qD khng kh nhau m ngcli chng c gi tr bng nhau.

Kt qu cui cng Lagrangian bin i mt lng gp i gi tr ca(2.33):

L = 16pi2

FF. (2.35)

So snh vi nh ngha dng trc:

L = JA, (2.36)ta c biu thc cho dng trc trong l thuyt lng t:

JA =

1

16pi2FF

=1

16pi2Tr(FF

). (2.37)

Nhn xt: i vi l thuyt lng t, i lng bt bin l Z chkhng phi tc dng S. S c th thay i khi i bin tch phn phimhm, nhng Z lun khng i v hu hn.

Nh vy biu thc (2.37) cho ta thy dng trc trong l thuyt lngt khng cn bo ton na. S khng bo ton ny gi l d thng.

Ngi ta ch ra c v phi (2.37) l mt vi phn ton phn:

Tr(FF) = 8

[(

1

2AAA ig

3AAA)

] K. (2.38)

Theo suy lun thng thng, s hng ny bng khng v khng chong gp vo tc dng do gi tr cc trng bng khng ti v cng.Nhng t Hooft [35] chng minh rng: vi trng hp QCD cha cctrng chun khng giao hon, chn khng c cu trc phc tp hnnhiu. Nh ta bit, cc trng chun tng ng nhau mt phpbin i chun (vt l khng thay i). t Hooft ch ra trong thct, biu thc (2.38) cng c vt l khc nhau khi thc hin php ccphp bin i chun khc nhau. Theo (2.38), tc dng S bin i mtlng:

S =1

16pi2

d4xK, (2.39)

21

trong K ph thuc vo cc trng chun A. Xt trng hp lthuyt khng giao hon n gin nht SU(2), ta c A sai khc nhaumt php bin i chun sau (chn chun Aa0 = 0):

Ai = Aai

a

2 A = Ai1 + i

(~

)1, (2.40)

vi i = 1, 2, 3 l ch s 3 thnh phn khng gian trng chun. Khi nu iu kin bin ti v cng ti Aa bng khng cng phi c tngng iu kin bin ti cc gi tr trng chun Aai sai khc nhau mtbin i chun (2.40). V vy iu kin bin ti v cng Ai = 0 ch lmt trng hp ring ca iu kin bin Ai = 0. iu kin ny tngng vi iu kin chn Ai = 0 hoc Ai = i

()1. Lp trngchun tha mn iu kin ny gi l cc trng chun thun gauge (puregauge). Vi iu kin ny th (2.39) khc khng v bin i chiral U(1)5khng phi l mt i xng ca l thuyt.

Vi cc lp lun trn ta thy trng chun bng khng ti v cngtrong cc iu kin chun khc nhau. Ngi ta chng minh c miiu kin chun ny tng ng vi mt cu hnh chn khng (vacuumconfiguration) khc nhau. Ngi ta phn lp cc cu hnh chn khngkhc nhau tng ng vi 1 khi r (chn khng tng ngvi trng thi v cng). Mi chn khng c c trng bi mt slng t n sao cho

n ei2pin, (2.41)khi r (chn khng), vi n = 0,1,2, .... iu kin ny tngng vi A A ti v cng. S nguyn n cn c gi l swinding c trng cho cc topo khc nhau ca khng gian.

Ngi ta c th chn cc n nh sau [60]:

1(~x) =~x2 d2 + 2id~.~x

~x2 + d2; n = [1]

n . (2.42)

Khi cc cu hnh chn khng khc nhau phn bit nhau bi ch sc trng n, k hiu trng thi chn khng tng ng l |n. Theo nhngha ny th 1|n = |n+ 1 v

k|n = |n+ k, (2.43)

22

khng tha mn iu kin bt bin chun k. V vy trng thi chnkhng thc s phi l t hp ca cc trng thi ny. Trng thi chnkhng thc s c trng bi s , gi l chn khng , c nh nghal:

| = nein|n, (2.44)

tha mn iu kin bt bin chun sai khc mt h s pha ei:

1| =n=+n=

ein1|n =n=+n=

ein|n+ 1

= ein=+n=

ei(n+1)|n+ 1 = ei|.

Ngi ta chng minh c n l s winding tha mn iu kin:

n =ig3

24pi2

dr3Tr[ijkA

inA

jnA

kn], (2.45)

vi Ain tng ng vi php bin i chun n trong (2.42).Ngi ta chng minh c trong chun A0a = 0 ch c J

0 6= 0 v gitr:

J0 =4

3igijkTr[A

iAjAk]. (2.46)

Biu thc n trong (2.45) ly ti thi im t bt k. Ngi ta chngminh c ti hai thi im t = v t = + s ny c sai khc nhaumt lng khc khng:

= n|t= n|t= = g2

32pi2

d4x F a F

a . (2.47)

Nu k hiu trng thi chn khng ti hai thi im cui v unh sau:

|+ =nein|nt=+ =

nein|n+,

| =nein|nt= =

nein|n. (2.48)

Xc sut chuyn gia hai trng thi ny l:

+| =(m

+m|eim) (

nein|n

)

=m,n

ei(mn) +m|n =,n

ei +n+ |n, (2.49)

23

trong k hiu li

= m n = g2

32pi2

d4x F a F

a , (2.50)

nh trong (2.47).S dng nh ngha thng thng cho tch phn ng tnh bin

chuyn chn khng +|, ta c:

+| =

d[A]e

iSeffitive[A]

( 1

32pi2

d4x F a F

a

), (2.51)

vi

Seffective = SQCD[A] +

32pi2

d4xF a F

a . (2.52)

Nh vy, Lagrangian b bin i mt lng do tc dng ca chn khngtrong QCD. V vy trong phn tip theo ta mc nh trong Lagrangian c sn s hng ny do ng gp ca chn khng.

ng gp t php bin i U(1) chiral vo s hng vi phm CP trong QCD

thy r c nh hng ca php bin i pha khi tc dng lncc trng fermion sinh ra s hng vi phm CP, ta xt mt Lagrangian

n gin nht gm mt quark dng qD =

qq

trong biu din spinor4 thnh phn, v mt trng chun U(1) c tenx cng trngF = F F. Cc spinor hai thnh phn q, q l cc spinor Weyltri.

Xt s hng ng nng ca trng fermion:

L1 = iD = iqD

DqD, (2.53)

vi

qD = q+D0 =

(q q

) 0 11 0

=

(q q

) 0 11 0

= ( q q ) , (2.54)

=

0 0

, (2.55)24

ta c s hng ng nng trong biu din hai thnh phn,

L1 = i(q q

) 0 0

D qq

= iqDq + iqDq. (2.56)

cho gn trong cc biu thc cha spinor hai thnh phn, ta nhngha

iDq = q/, (2.57)

iDq = q/. (2.58)

Suy ra s hng ng nng vit dng spin 2 thnh phn nh sau:

L1 = qq/+ qq/. (2.59)

Tng t cho s hng khi lng, lin h gia cch vit 4 thnh phnv hai thnh phn cho cc trng fermion c m t theo h thc:

L2 = m = mqDqD = m(qq + qq). (2.60)

Trong biu din Dirac thng thng khi lng m lun c chuynv dng s thc. Vi trng hp tng qut ta c th vit biu thc caL2 di dng khi lng phc:

L2 = mqq +mqq. (2.61)

Nh vy t (2.59) v (2.61) ta thu c Lagrangian tng qut c dngsau:

L = 14g2

FF + qq/+ qq/ +mqq +mqq, (2.62)

y khi lng c gi tr phc

m = |m|ei. (2.63)S hng khi lng mqq + mqq trong biu din hai thnh phn

trong Lagrangian (2.62) cng c th vit c theo bn thnh phn:

mqq +mqq = (Re m+ iIm m)qq + (Re m+ iIm m)qq

= Re m(qq + qq) + iIm m(qq qq)= Re mqDqD + iIm mqD5qD. (2.64)

25

Khi lng vit di dng Dirac phi l s thc, do ta phi loi shng th hai trong (2.64). lm c iu ny ta chn mt php bini i xng U(1) kh nhng phi m bo Lagrangian bt bin.

Bin i i xng vit theo biu din 2 thnh phn nh sau:

q q = ei2 q ; q q = qei2 , (2.65)cn theo dng 4 thnh phn:

qD qD = ei2 5q ,

qD qD = qDei2 5. (2.66)

T y ta thy y chnh l i xng U(1) chiral. i xng ny cPeccei-Quinn dng x l vn strong-CP nn thng c gi li xng Peccei-Quinn, k hiu U(1)PQ. Khi s hng khi lng sbin i nh sau:

mqq +mqq = |m|eiq ei2 ei2 q + (|m|ei)(ei2 q)(ei2 q)= |m|(qq + qq). (2.67)

Biu thc (2.67) cho thy, qua php bin i i xng (2.65) thnhphn khi lng phc chuyn sang thc tc cho ta khi lng l sthc.

Thng thng khi thc hin mt php bin i i xng m La-grangian vn bt bin th cc hiu ng vt l khng thay i. Nhng y li c s thay i hiu ng vt l do nh hng ca chn khng trongQCD, nh hng ny sinh ra mt s hng Leff vi phm CP. S hngny c xc nh theo phng php tch phn ng cho l thuyt ci xng chiral U(1) gi l U(1)PQ.

Dng phng php tch phn ng vi phim hm:

Z =[dA]

[dq] [dq] eiS, (2.68)

trong S =d4x L l tc dng ca l thuyt.

T (2.68) ly tch phn theo dq, dq th S Seff , L Leff . Khi :Z =

[dA]

eiSeff . (2.69)

Thc hin php bin i i xng (2.65) tc q q , q q, sau taly tch phn theo q, q th S v L li bin i nh sau:

S S = Seff + Seff . (2.70)26

L L = Leff + Leff , (2.71)trong

Leff =1

32pi2 Tr(FF

). (2.72)

S hng Seff c tr ring CP bng -1 nn gi l s hng vi phm CP.T ta c th ni, s vi phm CP ph thuc vo .

Nh vy ban u Lagrangian QCD c m t bi s hng khilng v s hng tng tc chun, s hng ng nng xem nh hinnhin. Nhng ngay sau , t d thng trc trong QCD ngi ta thyxut hin mt tham s mi trong Lagrangian:

L =

16pi2F aF a, (2.73)

vi

F a =1

2F

a. (2.74)

T gi tr thc nghim, da vo gii hn trn ca mmen lng ccin neutron ta c gi tr gii hn trn ca 109. n y ny sinhra cu hi: ti sao tham s li c gi tr nh nh vy. iu ny lthuyt khng gii thch c (theo l thuyt ng l phi c gi trln nhng thc nghim li cho nh). Do , ngi ta gi l vn strong-CP.

2.1.2 i xng Peccei-Quinn, bo ton CP v s xut hinaxion

C mt s cch gii quyt vn strong-CP nhng cch tt nht cPeccei-Quinn a ra bng cch gi thit l l thuyt c thm i xngchiral ton cc U(1)PQ, gi l i xng Peccei-Quinn.

Xy dng l thuyt gii thch nh

T (2.52), ban u ta c s hng vi phm CP nh sau:

Leff = L+ig2

32pi2F aF

a. (2.75)

27

Xt Lagrangian n gin nht gm mt trng fermion , mt trngv hng v mt trng chun A c tenx cng trng F a =A

a Aa. Khi Lagrangian:

L =1

4F aF

a + iD + [G (

1 + 52

) +G(1 5

2)]

+ ||2 2||2 h||4, (2.76)trong , s hng th nht l ng nng ca trng chun, s hng thhai l ng nng ca trng fermion , s hng th ba l s hng tngtc gia trng v hng vi trng fermion , s hng th t l ngnng ca trng v hng , hai s hng cn li l th v hng V ()ging nh m hnh chun nhng khc ch y l n tuyn cnSM l a tuyn.

c ph v i xng t pht xy ra y ta xt 2 < 0.Di tc dng ca i xng U(1)PQ:

= ei5, = e2i. (2.77)S dng phng trnh tch phn ng ta c Lagrangian bin i mt

lng l vi phn ton phn Leff :

L L = L+ Leff , (2.78)lm cho S bin i mt lng Seff ,

Seff S eff = Seff + Seff , (2.79)trong

Seff = i2g2

32pi2

d4xF aF

a d4xLeff , (2.80)

vi

Leff =ig216pi2

F aFa. (2.81)

Kt qu l Lagrangian hiu dng ton phn Leff trong (2.75) ditc dng ca php bin i U(1)PQ c dng mi sau:

Leff Leff = L+ig2

32pi2F aF

a +i2g232pi2

F aFa

= L+ig2( 2)

32pi2F aF

a. (2.82)

28

So snh (2.75) v (2.82) ta c th ni, qua i xng U(1)PQ th:

= 2, (2.83)ngha l qua i xng U(1)PQ th s vi phm CP ca l thuyt by gili ph thuc vo . L thuyt khng vi phm CP tng ng vi = 0.Peccei-Quinn gii thch = 0 bng 2 iu kin:

1. Khi lng Dirac phi nhn gi tr thc.

2. Cc tiu th V () dn n:

arg(ei

G < >) = = 0, (2.84)

trong c nh ngha trong tch phn phim hm di y.

Tht vy, xt tch phn phim hm sau:

Z(J, J) =

qeiq

(dA)q

d

d

d

exp[d4xL(, ,A) + J]. (2.85)

Da vo (2.85) ta tnh c trung bnh chn khng ca trng vhng theo cc tham s , c nh ngha nh sau:

1

Z

ZJ

|J=J=0 =< >= ei, (2.86)

trong , l cc hng s thc cn xc nh theo iu kin cc tiuth.

n y ta biu din di dng phc theo cc bin v hng mi, , , :

= ei(+ + i). (2.87)

T (2.86) v (2.87) suy ra

< > = 0,

< > = 0. (2.88)

Khi

= arg(ei

Gei) = arg[ei(+)G] = arg[ei(

+)G], (2.89)

29

v l hng s thc khng lin quan n gc nn trong (2.89) ta b quan.

Thay (2.87) vo (2.85) v tnh ton chi tit nh trong [32], ta thuc biu thc trung bnh chn khng ca hai trng v hng mi ,:

< > =d

d [A0 +

nFn cosn],

< > =d

d 2

nGn sinn. (2.90)

Kt hp (2.88) v (2.90) ta cd

d [A0 +

nFn cosn] = 0, (2.91)

d

d 2

nGn sinn = 0. (2.92)

T ta suy ra

= 0 hoc = pi. (2.93)

T iu kin cc tiu ca th V () = 2+h()2k|| cos suyra nhn gi tr bng 0 ( = 0). Nh ni trn, khi = 0 th = 0dn n l thuyt s bo ton CP.

Kh s hng vi phm CP

Nu = 0 th s hng vi phm CP trong Lagrangian hiu dng (2.82)bin mt ngha l l thuyt s bo ton CP. Do vn t ra l phigii quyt nh th no cho = 0.

gii quyt vn ny ta xt Gei

< > trong (2.84) v vit tchphn thc v phn o nh sau:

Gei

< >= |G|ei arg(G) ei

(ei) = (|G|)ei[arg(G)++]. (2.94)Suy ra

= arg(Gei < >) = arg(G) + + . (2.95)

Nh lp lun trn

= 0. (2.96)

30

Kt hp (2.95) v (2.96) ta suy ra:

arg(G) + + = 0. (2.97)

By gi ta thay vo Lagrangian v ch xt trung bnh chn khng catrng . Thay

< >= ei, (2.98)

G |G| ei argG = |G| ei(+). (2.99)Khi s hng khi lng trong Lagrangian l:

Lmass =

[G(

1 + 52

) +G(1 5

2)

]

=

[|G|ei(+)ei(1 5

2) + |G|ei(+)ei(1 + 5

2)

]

= (|G|)[ei

(1 5

2) + ei

(1 + 5

2)

], (2.100)

vi

5 =

I2 00 I2

. (2.101)Ta c

1 52

=

0 00 I2

; 1 + 52

=

I2 00 0

, (2.102)

Lmass = (|G|)[ei

(1 5

2) + ei

(1 + 5

2)

]

= (|G|)ei

I2 00 0

+ ei 0 00 I2

= (|G|)

I2ei 00 I2e

i

= (|G|)

I2(cos i sin ) 00 I2(cos

i sin )

= (|G|) I2 cos 0

0 I2 cos

+ iI2 sin 0

0 iI2 sin

31

= (|G|) cos I4 +

iI2 sin 00 iI2 sin

= (|G|)

cos + i I2 0

0 I2

sin

= (|G|) [cos + i5 sin ]= (|G|) ei5. (2.103)

T (2.103) ta c nhn xt sau:* khi lng l thc th i lng phc ei5

phi bng 1 tc l = 0.

* Khi = 0 dn n s hng i(2)32pi2 FaF

a s mt i, khi Lagrangian s bo ton CP.

Tm li: Lagrangian bo ton CP th ta phi a trng vhng vo ph v i xng Peccei-Quinn. V trung bnh chn khng< >= ei 6= 0, khc khng l v phi lun khc khng Lmass tnti.

Nh vy, phi tn ti trng v hng trong l thuyt, m =ei(+ + i) dn n mt thnh phn trong khai trin ca l axion.

2.2 Axion trong m hnh 3-3-1 vi neutrino phncc phi

2.2.1 Tng quan v m hnh

Trong [21], cc tc gi nghin cu mt s h qu mi xut hinkhi a i xng gin on vo cc m hnh 3-3-1 ban u. Mt trongs cc h qu quan trng y l i xng Peccei-Quinn xut hin mtcch t nhin trong Lagrangian c in ca cc m hnh 3-3-1. C th,ngi ta a ra i xng Z2 loi b tt c cc s hng bc 3 khngmong mun trong th v hng. Tuy nhin trong trng hp ny, cchiu ng hp dn vn to ra cc s hng hiu dng trong Lagrangiangy ra s ph v i xng Peccei-Quinn v cho ng gp ln vo khilng axion, nh hng n tnh bn ca n. V vy, m bo axionbn chng ti a vo m hnh i xng gin on ZN . Trong chngny chng ti chng t rng, i xng Peccei-Quinn l kt qu t nhintrong m hnh 3-3-1 vi cc neutrino phn cc phi sau khi a vo mhnh i xng Z11

Z2. Sau khi i xng Peccei-Quinn b ph v dn

32

n s xut hin axion, chng ti tin hnh nghin cu tnh bn caaxion thng qua qu trnh r ca n thnh hai photon.

Trong m hnh 3-3-1 vi neutrino phn cc phi cc lepton c spxp vo cc tam tuyn, thnh phn th ba ca tam tuyn l cc neutrinophn cc phi:

aL =

aLlaLNaL

(1, 3,1/3), laR (1, 1,1), (2.104)

vi a = 1, 2, 3 l ch s th h. y neutrino phn cc phi vit theok hiu mi NL (R)c. Hai th h quark u tin c sp xp vocc phn tam tuyn cn th h quark th ba c sp xp vo mt tamtuyn:

QiL =

diLuiLDiL

(3, 3, 0) , i = 1, 2,

Q3L =

u3Ld3LUL

(3, 3, 1/3),uiR (3, 1, 2/3) , diR (3, 1,1/3) , DiR (3, 1,1/3)UR (3, 1, 2/3) , u3R (3, 1, 2/3) , d3R (3, 1,1/3) .

(2.105)

M hnh ny xut hin cc quark mi khng c trong SM l U , Di. Vvy, chng c gi l cc quark ngoi lai (exotic quark). in tch cacc quark ngoi lai U v Di cng ging nh cc quark thng thngqU =

23 v qDi = 13 . ph v i xng t pht sinh khi lng cho

cc boson chun v cc fermion, m hnh i hi phi c ba tam tuynv hng, c th l

=

0

0

(1, 3,1

3

), (2.106)

=

+

0

+

(1, 3, 2/3) , (2.107)

33

=

0

0

(1, 3,1/3) . (2.108)Tng tc Yukawa sinh khi lng cho cc fermion c th c vit

di dng chung nht nh sau:

(LY + LY ) = G1QiLu3R +Gij2 QiLdjR +Ga3Q3LuaR+Gia4 QiL

daR +G3a5 Q3LdaR +Gia6 QiLuaR

+habfaLebR + hab

ijk(faL)i(fbL)cj(

)k + h.c.,(2.109)

trong ta quy c ly tng theo cc ch s lp. hon chnh ph ht chng ti a thm vo mt trng v hng

n tuyn (1, 1, 0) cho php tn ti mt i xng gin on cnthit, i xng ny m bo cho axion khng nhn thm cc ng gpln t hp dn vo khi lng ca n. Biu thc tng qut nht ca thv hng trong m hnh m bo iu kin ti chun ha v bt binchun c tch thnh hai phn nh sau:

VH = 2

2 + 22 + 2

2 + 22 + 1

4 + 24 + 3

4

+4(+

) (+

)+ 5

(+

) (+

)6

(+

) (+

)+ 7

(+

) (+

)+8

(+

) (+

)+ 9

(+

) (+

)+ 10 (

)2

+11 ()(+

)+ 12 (

)(+

)+13 (

)(+

), (2.110)

v

VNH = 2

+ + f1++ f2

+ + 14(+

)2+15

++ 16++ 17+

+12ijk (f3ijk + f4ijk + f5ijk)

+ijk (18ijk + 19ijk + 20ijk)

+ijk (21ijk + 22ijk + 23ijk)

+24(+

) (+

)+ 25

(+

) (+

)+26

(+

) (+

)+ 27

(+

) (+

)+H.c, (2.111)

34

trong VH v VNH ln lt l phn cha cc s hng lin hp hermintianv phn cha tt c cc s hng khng tha mn iu kin hermintian(non-hermintian). Nh vy chng ti c tt c cc thnh phn cnthit xy dng i xng gin on cho m hnh.

2.2.2 i xng Peccei-Quinn v axion

i xng gin on ZN c th a vo mt cch t nhin khi l thuytc s trng trong ph ht [21]. gii quyt vn strong-CP, trctin chng ti gn tch Z11 ph hp cho cc trng. Cc tch ca Z11c nh ngha l e2pii k11 ,{k = 0,1, ... 5} v cc trng bin itheo i xng Z11 nh sau:

1, faL 11 faL, 11 , daR 12 daR, 3, (eR, u3R) 13 (eR, u3R), QiL 4QiL,

diR 14 diR, 5, uaR 15 uaR, Q3L 0Q3L. (2.112)Ngoi i xng Z11 chng ti a thm vo i xng Z2 tc dng ln

cc trng qua php bin i sau:

(, , dR, u3R) (, , dR, u3R), (2.113)

cc trng cn li bin i tm thng di i xng Z2. Khi thv hng VNH bt bin di i xng Z2 ch cn li s hng duy nht.

i xng Peccei-Quinn xut hin trong m hnh, chng ti gncc tch Peccei-Quinn thch hp cho cc quark sao cho tch Peccei-Quinnca quark tri v quark phi lun tri du nhau. C th, cc quark bini di i xng U(1)PQ nh sau:

uaL eiXuuaL, uaR eiXuuaR,u3L eiX

uu3L, u3R eiX

uu3R,daL eiXddaL, daR eiXddaR,diL eiX

ddiL, diR eiX

ddiR. (2.114)

Khi cc lepton bin i di i xng U(1)PQ nh sau:

eaL eiXeeaL, eaR eiXeReaR,aL eiXaL, aR eiXRaR, (2.115)

35

trong Xe, XeR, X v XR ln lt l cc tch Peccei-Quinn ca cclepton tri, lepton phi, neutrino tri v neutrino phi. Da vo cc shng tng tc Yukawa v s hng phi bt bin di i xngU(1)PQ ny, chng ti thu c cc h thc

Xd = Xu, Xd = Xu, X = XeR, Xe = XR. (2.116) thun tin cho vic tnh ton ta chn Xd = Xd, dn n

Xd = Xd = Xu = Xu = Xe = XeR = X = XR. (2.117)T h thc (2.117) ta thy cc tch Peccei-Quinn ca cc lepton triv phi, cc neutrino tri v phi ngc du nhau. Nh vy, tt c ccfermion trong m hnh u c tch Peccei-Quinn tha mn c trngchiral ca i xng ny. Vy chng ti ch ra c s tn ti ca ixng Peccei-Quinn trong m hnh.

Cc trng v hng trong m hnh bin i di i xng Peccei-Quinn nh sau:

e2iXd, 0 e2iXd0, , 0 e2iXd0,+ +, 0 e2iXd0,+ +, 0 e2iXd0, , 0 e2iXd0. (2.118)

Th tng qut bt bin di i xng 3-3-1, i xng Peccei-Quinn vi xng Z11 Z2 l

V (, , ) = VH + ijkijk+H.c. (2.119)

T th v hng (2.119), chng ti thu c ma trn khi lng Higgsbng cch khai trin c th cc trng v hng ban u 0, 0, 0 v quanh cc gi tr trung bnh chn khng nh sau:

0 =12(v +R + iI) ,

0 =12(v +R + iI) ,

=12(v +R + iI) ,

0 =12(v +R + iI) . (2.120)

36

Thay (2.120) vo th v hng (2.119) v cho ha cc ma trn khilng xut hin trong th v hng, chng ti thu c cc tr ringv trng thi ring khi lng cho cc ht v hng. T y chng ting nht c cc trng thi ring ny vi cc trng Higgs vt l, ccGoldstone b hp th bi cc boson chun c khi lng v axion. ivi axion, trng thi ring khi lng c xc nh nh sau [21]:

a ' 11 + V 2

(I V I) , (2.121)

y V = vv . Ta c th chn v ' 1010GeV , v ' 103GeV dn nv

v 1 m bo cho axion bn [21].T phng trnh (2.121) chng ti thy rng, ch c hai ng gp vo

trng thi ring khi lng ca axion l I v I. Nhng V =v

v 1

nn V I coi nh bng 0, ch cn I cho ng gp chnh vo trng thiring khi lng ca axion. V l n tuyn nn I khng tng tcvi vt cht thng thng, cn I ch tng tc vi cc quark ngoi laido tnh cht ca m hnh 3-3-1. iu ny dn n axion cng ch tngtc vi cc quark ngoi lai nhng hng s tng tc rt b do t l viV . Tuy nhin, axion c th r thnh hai photon khi xt n bc mtvng ca l thuyt nhiu lon. nghin cu tnh bn ca axion chngti s tnh ton chi tit cho qu trnh r ca n thnh hai photon.

2.2.3 Qu trnh r ca axion thnh hai photon

Tng tc ca axion vi photon c xc nh thng qua Lagrangianhiu dng

La =ca

32pi2vPQa(x)FF

, (2.122)

trong h s tng tc hiu dng ca nhn ng gp t cc b nhbc cao. Trong m hnh ny, cc b nh mt vng ch nhn ng gpt cc quark ngoi lai. Do vy hng s ca c cho bi [21]

ca = 23vPQ

XQQ

2Q ' 0.44, (2.123)

vi Q = U,Di.

37

ma(eV ) 103 104 105 106

(sec1) 9.4 1064 1067 1070 1073(sec) 1.06 1063 1066 1069 1072

Bng 2.1: S ph thuc ca b rng r v thi gian sng ca axion theo khi lngca n.

Bin cho qu trnh r axion thnh hai photon c cho nh sau:

< f |M | i >= ica32pi2vPQ

(k1)k1k2(k2). (2.124)

B rng r ca axion l

(a ) = 164pi

ca32pi2vPQ

2m3a. (2.125)Lu rng, gi tr ca bin thin rt rng do ph thuc bc ba vokhi lng ca axion. Axion nhn khi lng nh qua d thng chiral,m2a 4QCD/f 2PQ, trong 109 < fPQ < 1012 GeV t thc nghim [36].Vi QCD ' 217 MeV, ta c ma O(105) eV.

T nh gi trn ta tnh s cho khi lng ca axion trong khong106eV ma 103eV v ch rng [GeV ]1 = 16.61025 sec1 [37]. Sph thuc ca b rng r v thi gian sng ca axion vo khi lng can c cho bi bng 2.1.

T bng 2.1, chng ti nhn thy rng, thi gian sng ca axion lrt di ( 1072 sec), v vy axion c th l ng c vin tt cho DM. Mtiu ng quan tm y l, thi gian sng ca axion ln hn nhiu sovi tui V tr ( 1017 sec), v vy m dao ng ca axion vn tn ticho n by gi.

2.3 Tit din tn x ca qu trnh chuyn ha photon-axion trong trng in t ngoi

2.3.1 Yu t ma trn

Trong m hnh bt k, nu Lagrangian tn ti nh tng tc ca mtht nh vi hai photon th ngi ta c th xy dng th nghim sinh ht da vo s tng tc ca photon vi trng in t ngoi (photonbn c in). Axion l mt trong s cc ht nh vy. Trong nhng nm

38

gn y, t khi pht hin ra c ch tng tc in t ca axion, cc nhvt l rt c gng tm kim n trong phng th nghim. Sikivie l nhvt l u tin nghin cu qu trnh chuyn ha axion thnh nng lng

in t trong bung cng hng [39]. ng cho rng, phng php ny cth thu c cc dng axion n t cc thin h trong V tr nu axionl DM cu thnh nn cc thin h . Cc xut thc nghim nhmthu axion trong phng th nghim nh tng tc ca n vi trng int c cp [40, 41] v mt s kt qu mi c cng b gny [42, 43].

C th tm tt th nghim tm kim axion trong in t trng nhsau: Mt photon vi nng lng ban u qo t chm tia laser (hoc tiaX) tng tc vi mt photon bn c in ca trng in t to raaxion c nng lng po v xung lng p =

p20 m2a. Chm photon b

chn li bi vch ngn cn axion th chui qua vch ngn tng tc viphoton bn c in ca trng in t th hai to ra mt photon thcc nng lng qo, ngi ta pht hin tn hiu ca axion qua so snh nnglng po v qo cng vi cc ting n in t thu c t thit b ghi. bit chi tit chng ta c th xem thm trong [40, 42]. Cc cng btrc y [44, 37] tnh cc tit din tn x vi phn v tnh s choqu trnh chuyn photon thnh axion trong trng in t ngoi. Tronglun n ny, chng ti tnh tit din tn x ton phn cho qu trnhchuyn photon thnh axion trong trng in t ngoi, c th l trongin trng tnh, t trng tnh v ng dn sng.

Trc ht chng ti nghin cu yu t ma trn ca qu trnh chuynha ny. Nh ta bit, khi lng v cng tng tc ca axion vicc ht thng thng t l nghch vi ln ca gi tr trung bnh chnkhng v c a ra bi Peccei v Quinn. Axion tng ng vi cc htgi v hng nh l cc boson Goldstone xut hin di thang ph vi xng t pht ca i xng UPQ(1). H axion-photon c th c mt bi Lagrangian sau [10, 39],

L = 14FF

+ g

4pi

afaFF

+1

2a

a 12m2a

2a

[1 +O(2a/v2)

], (2.126)

vi = e2

4pi~c , a l trng axion, ma l khi lng ca n, F =12F

v fa l hng s phn r ca axion. Hng s ny c xc nh theo khi

39

lng ca axion ma nh sau [39, 42]: fa = fpimpimumd[ma(mu+md)]

1.Axion khng tng tc vi photon bc cy. Tng tc ny ch xuthin khi xt n b nh mt vng, tng ng vi gin tam gic, trong hai nh l tng tc ca photon vi fermion mang in, nh cnli l tng tc ca axion vi fermion. Hng s tng tc ph thuc votng m hnh c th v c cho bi: g = 12

(NeN 53 mdmumd+mu

), y

N = Tr(QPQQ2color) v Ne = QPQQ

2em. K hiu Tr quy c ly tng trn

tt c cc fermion Weyl phn cc tri - phi. QPQ, Qem v Qcolor ln ltl tch Peccei-Quinn, in tch v mt trong cc vi t ca nhm SU(3)c.Trong m hnh Dine - Fischler - Srednicki - Zhitnitskii [72]: g(DFSZ)'0.36 v m hnh Kim - Shifman - Vainshtein - Zakharov [73] ( axionkhng tng tc vi cc quark v lepton nh): g(KSVZ)' 0.97.

Xt s chuyn photon () vi xung lng q thnh axion (a) vi xunglng p trong trng in t ngoi. i vi qu trnh ny, cc tng tclin quan nm s hng th hai ca biu thc (2.126). S dng quy tcFeynman (xem Ph lc A) chng ti c c biu thc yu t ma trnca qu trnh chuyn photon thnh axion nh sau:

p|M|q = ga2(2pi)2

q0p0

(~q, )q

Vei~k~rF class d~r, (2.127)

y ~k ~q ~p, ga g pifa= gma(mu + md)(pifpimpimumd)

1 v(~q, ) l vect phn cc ca photon.

Biu thc (2.127) lun ng i vi trng in t ngoi ty . Sdng biu thc ny, chng ti tnh tit din tn x cho qu trnh chuynphoton thnh axion trong in trng tnh, t trng tnh vi th tchcha trng lx ly lz (thay v a b c trong [44]). ng thi chngti cng xt cho trng hp ng dn sng. y cc k hiu sau cs dng : q |~q|, p |~p| = (p2o m2a)1/2 v l gc gia ~p v ~q.

2.3.2 S chuyn ha trong in trng tnh

Trong phn ny, chng ta xt trng in t ngoi l in trng unm trong th tch lx ly lz. Chng ti s dng h ta vi trc xsong song vi hng ca in trng, ngha l F 10 = F 01 = E. Yu t

40

ma trn c cho bi:

p|Me|q = ga(2pi)2

q0p0

(~q, )01qFe(~k), (2.128)

y i lng c trng cho in trng ngoi l

Fe(~k) =Vei~k~rE(~r)d~r.

Ch s trn e xut hin trong M e k hiu cho qu trnh chuyn ha trongin trng. i vi in trng u vi cng E chng ti c [44]:

Fe(~k) = 8E sin

(1

2akx

)sin

(1

2bky

)sin

(1

2ckz

)(kxkykz)

1. (2.129)

Thay (2.129) vo (2.128) chng ti tm c tit din tn x vi phn(DCS) cho qu trnh chuyn ha photon thnh axion

de( a)d

=g2aE

2

2(2pi)2

sin(12lxkx) sin(12lyky) sin(12lzkz)kxkykz

2 (q2y + q2z) .(2.130)

T (2.130) ta thy rng, nu photon dch chuyn theo hng intrng, ngha l q = (q, q, 0, 0) th tit din tn x vi phn trit tiu.Nu xung lng ca photon song song vi trc y, ngha l q = (q, 0, q, 0)th phng trnh (2.130) tr thnh

de( a)d

=32g2aE

2q2

(2pi)2

[sin(

lx2p sin sin)

sin(ly2(q p cos )

) sin(lz

2p sin cos)

]2

(p2 sin2 sin cos(q p cos ))2 . (2.131) y l gc gia trc z v hnh chiu ca ~p ln mt phng Oxz.

Xt trng hp c bit vi ' 0, t (2.131) chng ti c:de( a)

d=

2g2aE2l2xl

2z

(2pi)2(1

1 m2aq2

)2 sin2qly2

11 m2a

q2

.(2.132)

41

Trong gii hn m2a q2 v ly m1a , phng trnh (2.132) tr thnhde( a)

d' g

2aE

2l2xl2z

16pi2. (2.133)

T (2.133) chng ti thy rng tit din tn x vi phn khng ph thucnng lng ca photon vo.

Trong trng hp tng qut, vi bt k th tit din tn x tonphn c tnh theo cng thc e(q) =

d(de/d), trong tit din

tn x vi phn cho bi (2.131). Lu rng, trong trng hp ny titdin tn x vi phn ph thuc kh phc tp vo xung lng ca photonvo v bin thin rt nhanh theo xung lng q. Do , thay v tnh trctip biu thc gii tch cho tit din tn x ton phn chng ti tinhnh kho st s bng phn mm Matlab R2008a. Theo , th biudin s ph thuc ca tit din tn x ton phn e(q) theo xung lngq ca photon vo. Cc tham s khc c chn nh trong [37, 44], c thlx = ly = lz = 1m, E = 100kVm v ma = 10

5eV . Hnh 2.1 biu din thny trong khong xung lng q = 104 103 eV tng ng vi s imgieo l 300 im v 3000 im. th cho thy tit din tn x tonphn c gi tr kh ln ( 1029cm2). Trong trng hp ' 0, tnhs cho biu thc (2.133) cho thy tit din tn x vi phn khng phthuc vo xung lng photon vo v c gi tr d

e(a)d ' 1.8.1037cm2.

Tit din tn x vi phn trong trng hp ny khng lin quan n ccthang nng lng c trng ca m hnh. Gi tr ca n ch ph thucvo vic chn la cc tham s nh khi lng ca axion, th tch chatrng v cng in trng.

Chng ti nhn mnh rng, trng hp xung lng ca photon vovung gc vi in trng E nh xt trn l trng hp ti u nhtcho thc nghim.

2.3.3 S chuyn ha trong t trng tnh

Tip theo chng ti nghin cu s chuyn ha photon thnh axion trongt trng u to bi ng dy (solenoid) c bn knh R v di h.Gi s t trng hng theo trc z (l trc ng dy), tng ng vi ccthnh phn tenx cng trng F 12 = F 21 = B. Yu t ma trn

42

1 2 3 4 5 6 7 8 9 10x 104

0

0.5

1

1.5

2

2.5

3

3.5

4x 1029

q(eV)

CROS

SSE

CTIO

N(cm2

)

1 2 3 4 5 6 7 8 9 10x 104

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 1029

q(eV)

CROS

SSE

CTIO

N(cm2

)

Hnh 2.1: Tit din tn x ton phn (cm2) ca qu trnh chuyn ha photon thnhaxion trong in trng tnh ng vi xung lng q = 104 103eV . th trn vvi 300 im v th di v vi 3000 im.

43

cho qu trnh chuyn ny l:

p|Mm|q = ga(2pi)2

q0p0

(~q, )21qFm(~k), (2.134)

ch s trn m xut hin trong Mm k hiu cho qu trnh chuyn hatrong t trng. y i lng c trng cho t trng l

Fm(~k) =4piBR

kzk2x + k

2y

J1(Rk2x + k

2y

)sin

(hkz2

), (2.135)

vi J1 l hm Bessel cu loi mt.Thay (2.135) vo (2.134) chng ti tm c tit din tn x vi phn

cho qu trnh chuyn ha photon thnh axion [37, 44]:

dm( a)d

=g2aq

2

2(2pi)2(1 q

2z

q2)F 2m(

~k)

=2g2aB

2R2q2

k2z(k2x + k

2y)J21 (R

k2x + k

2y)

sin2(hkz2

)(1 q2z

q2). (2.136)

T (2.136) chng ta nhn thy rng, nu photon dch chuyn theohng ca t trng, ngha l q = (q, 0, 0, q) th tit din tn x viphn trit tiu. Nu xung lng ca photon song song trc y, ngha lq = (q, 0, q, 0) th phng trnh (2.136) tr thnh [37, 44]

dm( a)d

= 2g2aR2B2J21

Rq

1 cos

1 m2aq2

2 +1 m2a

q2

sin2 cos2

1 cos

1 m2aq2

2 +1 m2a

q2

sin2 cos2 1

q2

sin2hq2

1 m2aq2

sin sin

(1 m2a

q2) sin2 sin2 2

1 . (2.137)44

Trong gii hn ' 0, m2a q2 v R m1a ta c:dm( a)

d' 1

2pi2g2aV hB

2, (2.138)

y V l th tch ca ng dy.T (2.138) chng ti thy rng tit din tn x vi phn t l bc hai

vi cng t trng B, bc mt vi th tch V v chiu di h ca ngdy. V vy, khi t trng ngoi l trng c in th thc nghim cth tng xc sut tn x bng cch tng cng t trng hoc thtch V hoc di h ca ng dy.

Chng ti tin hnh tnh s cho tit din tn x ton phn t cngthc (2.137) theo cc gi tr tham s c chn nh sau [37, 44]: R =h = 1m = 5.07 106eV 1 v B = 9 Tesla = 9 195.35 eV2. Tit dintn x ton phn trong khong xung lng q c th c biu din hnh 2.2.

T th biu din tit din tn x ton phn ca qu trnh chuynha photon-axion trong in trng tnh (hnh 2.1) v t trng tnh(hnh 2.2), chng ti nhn thy rng, tit din tn x ton phn ca qutrnh to ra axion trong t trng ( ' 1019cm2) ln hn nhiu so vitrng hp in trng ( ' 1029cm2). Tnh s cho tit din tn x viphn (2.138) chng ti cng c kt qu d

m(a)d ' 1.1 1032cm2.

2.3.4 S chuyn ha trong ng dn sng

Trong phn ny chng ti xt s chuyn ha photon thnh axiontrong trng in t bin thin tun hon ca ng dn sng vi modesng TE10 v c bit chng ti quan tm n qu trnh cng hng khitn s trng ngoi bng khi lng axion.

Nghim khng tm thng ca mode sng TE10 [38]:

Hz = Ho cos

(pix

lx

)eikzit,

Hx = iklxpiHo sin

(pix

lx

)eikzit,

Ey = ia

piHo sin

(pix

lx

)eikzit. (2.139)

y sng in t truyn theo trc z. Nu xung lng ca photon

45

1 2 3 4 5 6 7 8 9 10x 104

0

0.5

1

1.5x 1019

q(eV)

CROS

SSE

CTIO

N(cm2

)

1 2 3 4 5 6 7 8 9 10x 104

0

0.5

1

1.5x 1019

q(eV)

CROS

SSE

CTIO

N(cm2

)

Hnh 2.2: Tit din tn x ton phn (cm2) ca qu trnh chuyn ha photon thnhaxion trong t trng tnh ng vi xung lng q = 104 103eV . th trn v vi300 im v th di v vi 3000 im.

46

song song vi trc x th tit din tn x vi phn c cho nh sau:

d( a)d

=8g2aH

20 l

2xq

2

pi4(1 +

q)

(q p cos ) pi2l2x

2

cos lx2 (q p cos ) sin ly2 (p sin cos) sin lz2 (p sin sin + k)

[(q p cos )2 pi2l2x ]p sin cos(p sin sin + k)

2

.

(2.140)

Chng ti tnh s cho tit din tn x ton phn da vo biu thctit din tn x vi phn (2.140) vi H0 = B, = ma = 105 eV. Cctham s cn li c chn ging nh trng hp in trng v t trngtnh. Hnh 2.3 biu din s ph thuc ca tit din tn x ton phn vo xung lng q. Min xung lng c chn vi cc gi tr thp hn,q = 105 104 eV. Chng ti thy rng, tn ti mt cng hng chnhtng ng vi ' 1017cm2 ti q = 5.1 105 eV. Gi tr cng hngny ln hn rt nhiu so vi trng hp trong in trng tnh v ttrng tnh.

1 2 3 4 5 6 7 8 9 10x 105

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1x 1017

q(eV)

CROS

SSE

CTIO

N(cm2

)

Hnh 2.3: Tit din tn x ton phn (cm2) ca qu trnh chuyn ha photon thnhaxion trong ng dn sng vi xung lng q = 105 104 eV.

2.4 Tm tt kt qu

Cc kt qu chnh trong chng 2 c tm tt nh sau:

47

Phn th nht, gii thch chi tit s tn ti s hng vi phm CPtrong cc m hnh c i xng chun khng giao hon, c gi lvn strong-CP. Vn ny c gii quyt hp l nht thngqua c ch ph v i xng Peccei-Quinn mt cch t pht. Phnny chng ti cng gii thch chi tit s xut hin ca axion thngqua c ch ny.

Phn th hai, chng ti kho st trng hp ring cho i xngPeccei-Quinn v c ch ph v i xng Peccei-Quinn trong mhnh 3-3-1 vi neutrino phn cc phi. Bng cch gn thm ixng Z11

Z2 vo m hnh, chng ti chng minh c i xng

Peccei-Quinn xut hin rt t nhin vi ph ht fermion vn cgi nguyn. Sau khi ph v i xng t pht ht axion xut hin,trong ng gp chnh vo trng thi ring ca axion l phn oca trng Higgs , n tuyn v hng c a vo ph vi xng Peccei-Quinn. Ngoi ra, axion ch nhn ng gp rt nht Higgs v khng nhn ng gp t cc Higgs cn li trong mhnh. V vy, axion khng tng tc vi vt cht thng thng vtng tc rt yu vi cc quark ngoi lai. y l mt thuc tnhca DM. Tuy nhin, axion vn c th r thnh hai photon. Khost s cho qu trnh r ny trong gii hn thc nghim (khi lngaxion nh), chng ti thy thi gian sng ca axion ln hn 1060s,rt ln so vi tui ca V tr hin ti (1017s). iu ny khng nhaxion c kh nng l ng c vin cho DM.

Phn cui cng ca chng ny, chng ti xut ba phng ntm kim axion trong thc nghim qua nghin cu s chuyn haca axion trong in trng tnh, t trng tnh v ng dn sng.Trong tng trng hp c th, chng ti thit lp biu thc tnhtit din tn x ton phn ca ba qu trnh ny. Sau , chng tis dng phn mm Matlab R2008a tnh s v v th biu dins ph thuc ca tit din tn x ton phn vo xung lng caphoton vo. Kt qu tnh s cho thy, trong c ba trng hp uc kh nng thu c tn hiu ca axion trong thc nghim. Cth, tit din tn x ton phn trong in trng tnh 1029cm2, trong t trng tnh 1019 cm2 v trong ng dn sng ' 1018cm2. Nh vy trng hp chuyn ha trong in trngtnh cho tit din tn x ton phn nh hn rt nhiu so vi hai

48

trng hp cn li. c bit trong trng hp ng dn sng tn timt nh cng hng chnh cho tit din tn x ton phn rt ln 1017cm2 ti q = 5.1 105 eV. y l trng hp tt nhttrong c ba trng hp thu axion trong thc nghim.

49

Chng 3

Vt cht ti trong m hnh 3-3-1-1v thc nghim tm kim

3.1 M hnh 3-3-1-1

3.1.1 Fermion trung ha v cc ht lepton sai

Nhm i xng chun ca m hnh 3-3-1 l SU(3)CSU(3)LU(1)X ,trong SU(3)C l i xng ca QCD cn SU(3)L U(1)X l s mrng t i xng in yu ca SM. Ton t in tch l tch cn likhng b ph v ca i xng chun Q = T3 (1/

3)T8 +X, y Ti

(i = 1, 2, 3, ..., 8) l vi t ca nhm SU(3)L, X l tch ca U(1)X . Siutch ca SM l Y = (1/3)T8 +X.

Cc fermion di i xng 3-3-1 c sp xp nh sau:

aL =

aLeaL

(NaR)c

(1, 3,1/3), eaR (1, 1,1), (3.1)

QL =

dLuLDL

(3, 3, 0), Q3L =u3Ld3LUL

(3, 3, 1/3) , (3.2)uaR (3, 1, 2/3) , daR (3, 1,1/3) , (3.3)UR (3, 1, 2/3) , DR (3, 1,1/3) , (3.4)

trong a = 1, 2, 3 v = 1, 2 l ch s th h hay ch s v. Cc s trongngoc ln lt l cc s lng t ca SU(3)C , SU(3)L v U(1)X . NaR vU, D tng ng l fermion trung ha mi (chng l n tuyn ca SMtng t nh cc neutrino phn cc phi quen thuc) v cc quark ngoilai. in tch ca cc quark ngoi lai Q(U) = 2/3 v Q(D) = 1/3

50

ging nh cc quark thng thng. Nh cp, s lepton ca NaRbng khng L(NaR) = 0 khc vi R. V vy, m hnh c gi l mhnh 3-3-1 vi fermion trung ha.

Ta bit rng, m hnh 3-3-1 vi neutrino phn cc phi L(aR) 6= 0c thang seesaw rt cao ( gii thch cc khi lng nh ca neutrinothng thng) t 1010 n 1014 GeV [45, 46]. Thang nng lng ny lqu cao so vi cc my gia tc thng thng (c TeV) nn rt kh kimchng thc nghim. Tuy nhin, m hnh 3-3-1 vi fermion trung ha[31, 23] cho thang seesaw c TeV do L(NR) = 0 v v vy m hnh cth c kim chng. Ngoi ra, m hnh 3-3-1 vi fermion trung ha chota gii thch ma trn trn neutrino t nhin [47, 23]. S c mt ca NRthay cho R s dn n mt lp cc ht mi, l di mt i xng chnl v c th cung cp ng c vin cho DM.

ph v i xng chun SU(3)L U(1)X v i xng U(1)Q vsinh khi lng cho cc fermion, m hnh 3-3-1 i hi ba tam tuyn vhng [22],

=

+102+3

(1, 3, 2/3), (3.5)

=

01203

(1, 3,1/3), (3.6)

=

01203

(1, 3,1/3). (3.7)

i xng chun SU(3)LU(1)X b ph v t pht thng qua hai bc.Bc th nht SU(3)L U(1)X b ph v v nhm i xng in yuca SM v cc fermion mi nh cc quark ngoi lai U , D v cc bosonchun mi s nhn khi lng. Cc boson chun mi l trng gn vimt vi t trc giao vi siu tch yu v hai trng mang in X 0/0, Y

tng ng vi vi t T4 iT5 v T6 iT7. Bc th hai, nhm i xngSU(2)L U(1)Y b ph v v U(1)Q v sinh khi lng cho cc fermionv cc boson chun ca SM, chng hn nh W, Z, ea, ua, v da.

S lepton (L) ca cc thnh phn trong tam tuyn lepton tng ngl (+1, +1, 0) v n khng giao hon vi nhm i xng chun SU(3)L,

51

khng ging nh trng hp ca SM, bi v:

[L, T4 iT5] = (T4 iT5) 6= 0,[L, T6 iT7] = (T6 iT7) 6= 0. (3.8)

Do vy, cc boson chun X v Y mang s lepton bng 1. iu ny cngxy ra i vi m hnh 3-3-1 vi cc neutrino phn cc phi v m hnh3-3-1 ti thiu. y l tnh cht c trng ca cc m hnh 3-3-1. Vvy, nu s lepton l mt i xng ca l thuyt th n c th c ngnht nh l tch tn d ca mt i xng cao hn c bo ton,

G SU(3)L U(1)L, (3.9)vi U(1)L l mt i xng mi c xut ng kn i s gia slepton v i xng chun 3-3-1. [Nguyn nhn l cho L tng ngl mt vi t ca nhm SU(3)L th tch L ca mt a tuyn hon chnhphi c tng bng 0. iu ny l khng ng. Ngoi ra, tch L v Xphi khc nhau cho cc n tuyn fermion. Ngha l, s lepton v intch ca mt ht nhn chung khng trng nhau]. S lepton l t hp cacc vi t cho ca nhm SU(3)L U(1)L v c xc nh bi

L =23T8 + L, (3.10)

y T8 l vi t ca SU(3)L v L l tch ca U(1)L [30]. Tch L ca ccht trong m hnh c cho bi bng 3.1 (lu rng cc trng thnhphn 03,

02 v

01 phi c s lepton bng 0 bi v cc trng ny c trung

bnh chn khng khc khng sinh khi lng cho mt s ht v phv i xng chun). Hn na, chng ta c th d dng ch ra rng cc

Multiplet aL eaR Q3L QL uaR daR UR DR L 2

31 1

3

1

30 0 1 1 1

31

3

2

3

Bng 3.1: Tch L ca cc a tuyn trong m hnh.

tng tc thng thng ca l thuyt u bo ton L [16]. S lng tlepton L c cho bi bng 3.2.

Chng ta c th thy rng, cc ht trong SM vn gi nguyn s leptonban u. Tuy nhin, hu ht cc ht mi chng hn nh NR, U , D, X,Y , 3, 3, v 1,2 mang s lepton khc vi t nhin ca n c quy

52

Particle aL ea NaR ua da U D +1

02

+3

01

L 1 1 0 0 0 1 1 0 0 1 0Particle 2

03

01

2 03 W Z Z

X0 Y

L 0 1 1 1 0 0 0 0 0 1 1

Bng 3.2: S lepton ca cc ht trong m hnh.

nh bi SM. C ngha l, NR nm trong tam tuyn lepton nhng li cs lepton bng 0; U , D l cc quark nhng li c s lepton khc 0; ccboson chun X, Y v cc ht v hng 3, 3, v 1,2 c s lepton khc0. V vy m hnh ny chnh l m hnh vi phn ln cc ht mi mangs lepton sai (wrong-lepton particles) hay cn gi l m hnh vi cc htlepton sai.

Trong lun n ny, chng ti gi s rng i xng G l i xng chnhxc th dn n i xng U(1)L ca s lepton cng s l i xng chnhxc. Tuy nhin, v c ba tam tuyn v hng u mang tch di ixng G v s nhn gi tr t trung bnh chn khng, v vy i xng Gphi b ph v t pht. Gi tr trung bnh chn khng ca cc trng vhng trung ha l nghim ca phng trnh cc tiu th v c xcnh nh sau [22]:

01 = 03 = 0, (3.11)02 6= 0, 01 6= 0, 03 6= 0. (3.12)

Ngoi ra, vi trung bnh chn khng nh (3.11) v (3.12) chng ta cth vit trung bnh chn khng cho ba tam tuyn v hng,

= 12(0, v, 0)T ,

= 12(u, 0, 0)T ,

= 12(0, 0, )T . (3.13)

Tt c cc ht trong m hnh (ngoi tr L v NR) s nhn khi lng mc cy tng t nh cc m hnh 3-3-1 vi cc neutrino phn ccphi [22]. Chng ta gi s l thang c trng cho ph v i xng bc u tin SU(3)LU(1)X v SU(2)LU(1)Y ; u, v l thang ph vi xng bc hai SU(2)LU(1)Y v U(1)Q. ph hp vi yu cu

53

thc nghim th,u2, v2 2. (3.14)

Vi i xng G, mc d s lepton L l t hp ca hai tch b ph v Lv T8 nhng n vn bo ton (L = L = L = 0). Ph v ixng ca G thnh s lepton nh sau:

G = SU(3)L U(1)L U(1)L, (3.15)gi s tn ti ca 8 Goldstone boson c cha trong ba tam tuynv hng , , . Tuy nhin, cc Goldstone boson ny s tng ng vicc Goldstone boson ca ph v i xng chun SU(3)LU(1)X v ixng U(1)Q.

Hn na, s baryon khng giao hon vi i xng chun s dn ni xng chun chnh xc b ph v nh sau,

G SU(3)L U(1)B U(1)B, (3.16)bi v s baryon U, D cha c bit nn n c th bt k (trnghp ny gip ta loi b cc tng tc khng mong mun c vi phms baryon do tch B bo ton). Cc ht v hng 01 v 03 s mang sbaryon vi gi tr trung bnh chn khng c xc nh nh trn. n gin ta ly B(U) = B(D) = 1/3 = B(ua) = B(da) v a n ixng baryon U(1)B l i xng chnh xc sau ph v [30],

B = B. (3.17)S baryon ca cc ht trong m hnh c cho bi bng 3.3.

Multiplet aL eaR Q3L QL uaR daR UR DR B 0 0 1

3

1

3

1

3

1

3

1

3

1

30 0 0

Bng 3.3: Tch B ca cc a tuyn trong m hnh.

Chng ta ch mt s cc trng hp: (i) Nu 01 6= 0 hoc 03 6= 0th L s b ph v cng vi T8 v L. (ii) Bo ton L trong m hnh nykhng cn l h qu ca l thuyt nh trong SM, bi v i xng U(1)Lkhng b rng buc v dn n cc tng tc khng mong mun vi phmtng minh s lepton L chng hn nh tng tc Yukawa v th Higgs[46, 30]. (iii) Nu s baryon ca U, D c chn mt cch hp l thnhn xt (i) v (ii) cng ng cho s baryon.

54

Cch lm nh trn cng c th p dng cho m hnh 3-3-1 ti thiu[24] v m hnh 3-3-1 vi neutrino phn cc phi [22]. Tuy nhin, ccht lepton sai trong cc m hnh ny s khc vi m hnh ang xt lchng c s lepton bng 2 (mang s lepton 2 n v). Chng gi l ccht bilepton. Ngoi ra, vn DM trong nhng m hnh ny so vi ccm hnh ang xt s hon ton khc, nh chng ta s thy bn di.

3.1.2 i xng chun 3-3-1-1 v W -parity

Nh chng ta bit, s lepton L v L tha mn (3.10) c ara trong cc m hnh 3-3-1 trc y [30]. Tuy nhin, c tnh ng lcca chng cha c hiu mt cch y . Bi bo [23] ln u tin cho gii thch s b v bn cht ng lc khng tm thng ca s leptontrong m hnh 3-3-1. Trong lun n ny, ng lc gn vi s lepton sc phn tch chi tit v y . V T8 l tch chun ca nhm SU(3)L,s lepton L v do L hin nhin phi l tch chun. Tht vy, nu Lv L l cc tch ton cc th T8 LL cng l tch ton cc. iu nymu thun vi i xng chun SU(3)L. V s lepton l tch chun, ccd thng tng ng vi L v L xut hin. L thuyt nh vy khng thti chun ha c do ng nht thc Ward-Takahashi b vi phm. Do chng ta cn phi loi b d thng L v L. Tt c nhng phn tchtrn cho L v L cng ng vi trng hp s baryon nu n c xemnh l mt tch chun.

Gii php kh d thng gn vi s lepton l kho st m hnhvi i xng chun N B L ( y B l s baryon c nh ngha trn). ng thi ta a vo ba neutrino phn cc phi bin i nhn tuyn ca nhm 3-3-1:

aR (1, 1, 0). (3.18)Neutrino phn cc phi aR c s lepton v s baryon nh thng thng,L(aR) = L(aR) = 1 v B(aR) = B(aR) = 0. S thm vo aR khd thng hp dn [Gravity]2U(1)N (ch d thng hp dn ca L vNR khng trit tiu nhau). Chng ta c th kim tra rng l thuyt lc lp (trit tiu) i vi tt c cc d thng (xem ph lc B cho tnhton chi tit).

Nhm i xng chun ca m hnh l

SU(3)C SU(3)L U(1)X U(1)N . (3.19)55

V vy, ta gi m hnh thu c l m hnh 3-3-1-1. Cc a tuyn cam hnh 3-3-1-1 v cc tch N tng ng c cho bi bng 3.4. Ch, ph v tch B L chng ta phi thm mt trng v hng phcbin i nh n tuyn ca nhm 3-3-1,

(1, 1, 0), (3.20)vi B() = B() = 0, L() = L() = 2. i xng 3-3-1-1 s b ph vt pht v cc ht trong m hnh nhn khi lng ng thng qua bntrng v hng , , , .

Multiplet aL eaR aR Q3L QL uaR daR UR DR N = B L 2

31 1 2

30 1

3

1

3

4

32

3

1

3

1

32

32

Bng 3.4: Cc a tuyn trong m hnh 3-3-1-1 vi tch N tng ng.

Tch chun B L c xc nh nh sau:

B L = 23T8 +N. (3.21)

y l tch khng b ph v sau khi SU(3)L U(1)N b ph v bi, , . S m rng m hnh 3-3-1 vi fermion trung ha thnh m hnh3-3-1-1 cng c th p dng cho cc m hnh 3-3-1 thng thng nhm hnh 3-3-1 ti thiu v m hnh 3-3-1 vi neutrino phn cc phi.Cch m rng ny kh tng t nh vic m rng l thuyt SU(5) thnhSO(10) vi tch B L c xem nh l mt tch chun nh x.

Hin tng lun ca m hnh 3-3-1-1 rt phong ph [48]. Tuy nhin,trong lun n ny chng ti ch tp trung vo mt h qu l i xnggin on cn d sau ph v i xng chun 3-3-1-1. i xng ginon ny s cho mt s ht mi bn vng v chng c th l ng cvin ca DM. V vy, ng lc gn vi s lepton v baryon s khngc quan tm nghin cu. y, s lepton c hiu nh l mt hqu ca bo ton tch tng ng vi nhm G = SU(3)L U(1)L. Thas u tin l phin bn i xng ton cc ca i xng chun. Nghal, khi tnh ton s lepton, tt c cc s lng t gn vi i xng toncc SU(3)L cho cc a tuyn ca m hnh ging nh i xng SU(3)Lnh x. Do , nu T8 v L c trng cho s lepton th chng cxem nh l mt tch ton x v khng nn nhm ln vi cc tch chun

56

SU(3)L U(1)X . Tng t, s baryon B cng c xem nh l tchton cc thng thng. Thc t, cc m hnh 3-3-1 lun lun bo tons baryon. V vy, s lepton L hay L tc ng ln m hnh l tngng vi tch N v s c nhc n thay cho N trong cc tho lundi y. Tm li, m hnh 3-3-1 vi cc fermion trung ha v tch L(cng thm cc neutrino mi phn cc phi v n tuyn v hng) chiu nh l m hnh 3-3-1-1, trong tng tc chun ng vi tch Nkhng c xt n.

Mc d l thuyt bt bin vi cc i xng U(1)L v U(1)B v L, Bkhng b ph v bi cc gi tr trung bnh chn khng ca , , . Tuynhin, B, L nn b ph v theo mt cch no gii thch bt ixng vt cht v phn vt cht ca V tr, ng thi cung cp khilng cho neutrino. Nh c ch ra, bn cht ca L trong m hnhny l tch nh x v n l kt qu ca T8. L thuyt vi L nh x cth a v l thuyt vi tch N = B L. Chng ta phi ph v honton tch N cng nh cung cp khi lng ln cho boson chun miZN tng ng ht ny khng b pht hin bi cc my gia tc hinti. iu ny c th t c nu ta a vo n tuyn v hng nh cp, vi gi tr trung bnh chn khng,

= 12. (3.22)

Vi nhng v hng v trung bnh chn khng cho, ta c thchng minh c rng parity (tng t parity trong MSSM) l i xnggin on cn d v khng b ph v sau ph v i xng B L =(2/3)T8 + N [hay SU(3)L U(1)N ]. Parity ny l mt i xngchnh xc ca m hnh 3-3-1-1 v do khi k n spin s c dng:

P = (1)3(BL)+2s = (1)23T8+3N+2s. (3.23)

P vn bo ton tt c cc trung bnh chn khng c cho trn. Xemph lc C cho mt chng minh chi tit v P . R-parity ca cc ht trongm hnh c cho bng 3.5.

Chng ta thy rng tt c cc ht c s lepton sai khc so vi t nhinthng thng ca chng nh c quy nh bi SM bi 1 n v, v dL(NR) = 0, L(U) = 1, L(X) = 1, L(3) = 1 th c gi l cc htlepton sai vi tch R-parity l. Ngc li, cc ht thng thng l ccht trong SM v cc ht mi c tch R-parity chn. Nh vy R-parity

57

+1 (ordinary or L e u d W Z 1,2 1,2 3 bilepton particle) R Z ZN1 (lepton sai NR U D 3 3 1,2 X Y

particle)

Bng 3.5: R-parity ca cc ht trong m hnh 3-3-1-1 gm hai loi l cc ht leptonsai v cc ht thng thng.

c ngun gc t i xng chun 3-3-1-1 v l mt i xng t nhin cacc ht lepton sai. Ht lepton sai nh nht (LWP) l bn vng do ixng R-parity v c kh nng l ng vin ca DM. Nh c cp,chng ta c th c mt s cc vi phm i vi L v B (v d L = 2 bph v bi ) lm cho m hnh ng v hin tng lun trong khi vnbo ton parity cho DM. Thng qua nghin cu ny, chng ta c th ccc ht l R-parity khng phi l siu ht nh trong cc l thuyt siui xng. Ta c th ch ra tng minh rng mi tng tc ca l thuytch lin kt cc ht l R-parity theo tng cp v km theo cc ht thngthng do i xng chun 3-3-1 [23]. Xt mt s v d v cc h qu:

1. Tng tc Yukawa

LY = heabaLebR + habaLbR + habcaRbR+ hUQ3LUR+hDQL

DR + huaQ3LuaR + hdaQ3LdaR

+hdaQLdaR + huaQL

uaR +H.c. (3.24)

Ta thy rng cc ht v hng l parity 3, 3 v 1,2 khng tngtc vi cc fermion thng thng m ch lin kt vi cp ht chnl eN , N , uU , dU , dD, uD v bo ton i xng 3-3-1. Khng ctng tc vi phm tch L (dn n vi phm R-parity) chng hnnh aLbR, caLbL, Q3LuaR, QL

daR, Q3LUR,... Hn na,do R-parity c bo ton bi cc gi tr trung bnh chn khngca 03 v

01 (trung bnh chn khng bng 0) nn cc quark thng

thng v quark ngoi lai khng trn ln vi nhau. iu ny dnn trong m hnh s khng c s xut hin ca dng trung hathay i s v mc cy. Vn ny trong m hnh 3-3-1 vi ccneutrino phn cc phi [22] hon ton c loi b bi parity. Hnna, NR khng trn vi L v R do i xng parity.

58

2. Tng tc boson chun:Cc boson chun mang tch l parity X, Y khng lin kt vi ccboson chun ca SM [49]. Do i xng R-parity, boson chun Xkhng trn vi hai boson Z v Z v v vy vi phm CPT mccy c loi b [51]. Do , cc vn gn vi m hnh 3-3-1 vineutrino phn cc phi [22] c gii quyt bi i xng parity.

Chng ta lu rng, trong MSSM, spin ca cc ht thnh phn trongcng mt siu a tuyn khng giao hon vi siu i xng (tng t nhtrong m hnh 3-3-1-1 khng siu i xng, s lepton khng giao honvi i xng chun). Tuy nhin, R-parity tng ng vi spin, s leptonv s baryon (R-parity khng giao hon vi cc vi t siu i xng) cbo ton v khng b ph v ngay c khi i xng ton cc b ph vcng vi s ph v siu i xng. Mt iu cn nhn mnh y l siuth MSSM bo ton s lepton v s baryon khng phi l h qu t ngca l thuyt mc ti chun ha. iu ny khc vi SM. R-parity cngun gc khc so vi i xng chun ca m hnh 3-3-1-1. iu nydo tnh cht ca tch lepton vn nm trong nhm SU(3)L U(1)L,tch baryon U(1)B v spin. c bit, i xng G hay U(1)L ca s leptonb ph v do ph v i xng chun. Bo ton i xng L trong mhnh ny c th c m bo bi R-parity. Ph v tch N = B Lc th xut hin thang TeV hoc thang rt cao. V vy, chng ta giR-parity trong m hnh ny l W -parity v y W ngha l lepton sai(wrong-lepton).

Ty thuc vo khng gian tham s ca m hnh, LWP c th l htvector (X0), v hng (01 hay

03), hoc fermion (N

0R). Tt c cc ht

ny phi trung ha in nu chng l ng c vin cho DM. Trc khixem xt cc trng hp ny, trong phn tip theo chng ta s i ngnht cc ht v hng vt l v khi lng ca chng vi s ch ncc ht lepton sai.

3.1.3 Th v hng v khi lng

Nu n tuyn v hng c gi tr trung bnh chn khng c thang ca ph v i xng 3-3-1 th n s tng tc vi cc v hng thngthng , , thng qua th v hng. Hin tng lun ng vi ixng BL b ph v vi s hin din ca boson chun ZN s xut hin

59

ng thi vi vt l mi ca m hnh 3-3-1 thang TeV [48]. Nu trungbnh chn khng rt ln so vi , ngha l trng Higgs cho ph vi xng U(1)N rt nng, th s tch khi th hiu dng ca , , nng lng thp. Ngoi ra, ZN cng tch khi ph v khng trn ln vicc boson chun ca m hnh 3-3-1. Trng hp ny s c xt chi tittrong lun n. Lu rng, tnh cht ca W -parity trong c hai trnghp khng thay i.

Chng ta c th khai trin trng nh sau:

=12( +R + iI). (3.25)

Phn o (I) ca l boson Goldstone ca ZN , trong khi phn thc(R) l boson Higgs mi mang s lepton bng 2 v chn di W -parity.Khi lng ca ZN t l vi trung bnh chn khng ca trng vhng .

nng lng thp, th hiu dng do ng gp ca , , c dngnh sau (th phi bo ton i xng 3-3-1 v W -parity):

V = 21+ 22

+ 23 + 1()2 + 2()2 + 3()2

+4()() + 5()() + 6()()

+7()() + 8()() + 9()()

+(fmnpmnp +H.c.), (3.26)

vi 1,2,3 v f c th nguyn khi lng, 1,2,3,...,9 khng c th nguyn.Cc s hng c dng , ()2, ()() vi phm L (hay L) b cmdo i xng parity. Lu rng, hng s tng tc f bo ton tt c cci xng t nhin ca m hnh, v vy s khng c l do thch ng loi b s hng kiu ny [50] (trong ti liu, v d xem bi bo th nhtca [29], s hng ny thng b loi b). Hn th, chng ti s ch rarng s c mt ca n m bo cho tt c cc boson Higgs mi nhnkhi lng mong mun v cho mt ph Higgs ph hp vi l thuyt hiudng nng lng thp.

ng nht cc ht v hng vt l chng ti s khai trin cctrng trung ha v W -parity phi c bo ton nh c cp:

=

+1

12(v + S2 + iA2)

+3

,

60

=

12(u+ S1 + iA1)

212(S 3 + iA

3)

,

=

12(S 1 + iA

1)

212( + S3 + iA3)

, (3.27)

y S 1,3 v A1,3 l l di W -parity trong khi S1,2,3 v A1,2,3 chn di

W -parity. Do , hai loi ht ny khng trn vi nhau. Tng t chocc ht v hng mang in 1 v 2 khng trn vi 3 v 2. iu nyc th thy r trong cc kt qu di y. iu kin cc tiu th cxc nh bi

v21 + v31 +

1

2v24 +

1

2vu25 +

12fu = 0, (3.28)

u23 + u33 +

1

2uv25 +

1

2u26 +

12fv = 0, (3.29)

22 + 32 +

1

2v24 +

1

2u26 +

12fuv = 0. (3.30)

Cc ht gi v hng A1, A2 v A3 trn vi nhau do f 6= 0. Mt thp ca cc trng ny l ht gi v hng vt l (A) vi khi lng

A =u1A1 + v1A2 + 1A3

u2 + v2 + 2,

m2A = f2

u2v2 + u22 + v22

uv. (3.31)

Chng ta thy rng f < 0 nu u, v, > 0. Hai t hp trng cn litrc giao vi A v c khi lng bng khng. Chng c ng nht vicc boson Goldstone ca Z v Z tng ng:

GZ =1(u1A1 + v1A2) + (u2 + v2)A3

(u2 + v2 + 2)(u2 + v2),

GZ =uA1 + vA2

u2 + v2. (3.32)

Khi lng ca A t l vi nu f nm thang (f ). gnng u, v, chng ta c GZ ' A3 v A ' (vA1 + uA2)/

u2 + v2.

61

Cc ht v hng S1, S2 v S3 trn ln v c xc nh bi La-grangian khi lng:

(S1 S2 S3)

3u

2 fv22u

125uv +

f22

126u +

fv22

125uv +

f22

1v2 fu

22v

124v +

fu22

126u +

fv22

124v +

fu222

2 fuv22

S1S2S3

.(3.33)

Ma trn khi lng ny s cho mt trng thi vt l nh v c ngnht vi boson Higgs (H) ca SM. V f , hai trng thi vt l cnli (H1,2) c khi lng t l vi thang . Xt bc chnh (f , u,v), cc trng thi vt l va cp vi khi lng tng ng c xcnh nh sau:

H1 ' vS1 + uS2u2 + v2

, m2H1 ' f2

(u

v+v

u

),

H2 ' S3, m2H2 ' 222,H ' uS1 + vS2

u2 + v2,

m2H '42(5u

2v2 + 3u4 + 1v

4) (2uv(f/) + 6u2 + 4v2)222(u2 + v2)

.

(3.34)

Mt t hp ca S 1 v S3 l trng vt l S

= (S 3 + uS1)/

u2 + 2

vi khi lng m2S =(129 fv2u

)(u2+2). Trng thi trc giao GS =

(uS 3 + S 1)/u2 + 2 l trng Goldstone khng khi lng. Tng

t, mt t hp ca A1 v A3 l trng vt l A

= (A3uA1)/u2 + 2

vi khi lng m2A =(129 fv2u

)(u2+2). Trng thi trc giao GA =

(uA3+A1)/

u2 + 2 l trng boson Goldstone khng khi lng. D

dng kim tra c rng GS v GA l cc boson Goldstone tng ng

vi hai thnh phn thc ReX v o ImX ca trng boson chun X. Vvy, t hp ca chng c ng nht l boson Goldstone ca trngchun X:

GX =12(GS + iG

A) =

1 u3u2 + 2

. (3.35)

ng thi, chng ta cng c cc trng vt l v hng phc trung hal t hp ca S v A (trc giao vi GX). Khi lng trng ny c

62

cho nh sau:

H =12(S + iA) =

u1 + 3u2 + 2

,

m2H =

(1

29 fv

2u

)(u2 + 2). (3.36)

H l trng vt l v hng trung ha duy nht mang tch W -parity lv ng vai tr nh DM. Lu rng khi lng H lun nm thangnng lng . bc chnh ( u, v) ta c H ' 3 (l n tuyn vhng ca SM) v GX ' 1.

Chng ta cng c hai ht v hng vt l mang in, H3 mang tchW -parity l cn H4 mang tch chn:

H3 =v2 +

3

v2 + 2, H4 =

v2 + u1

u2 + v2, (3.37)

vi khi lng tng ng

m2H3 =

(1

27 fu

2v

)(v2 + 2