INTRODUCTION TO PHYSICS OF NEUTRINOS

Solutions to Homework 2 - 26/04/2021

r

④⑨ 1, -_ it 8^2,4 + gY-MARY

→ fp'=i¥r^dn¥'+gut 'rmA,

'

Y'

=iy-utrmquyli-gyutyml.UA#+i1gUdrUt)UY--iYrmdrYtiY-utVdpUYtg4rmAµ4+i4J%UtUY

using utu= I → qutu= - utdyu

a fp'= i 4-V44 + GINA,µ=iÑNDrY =L,

→ IUCN) invariant Lagrangian .

② Far = of Av - Ou Am - ig [Ar , Av ] ④④

→ Frit'

= d,Ai T - d✓A-IT

'

- ig [Air, At tb]

=D,Ai T'- drAj T

'

-

cg AiAi[T"

,Tb]

= of Ait'

- OrAj T'

-1g f-"'

AiAi T'

Ñfµ=dµAi_dvAñtgfa"AjAr#-We know that A

,→ A'm = UA,U*+ ilg Ud,Ut

From ④ →

For'

=D,Aj - duAi - ig [Ai,Ai]

= da IU Av Ut + ig udvut)-of (ut,ut+ ily U donut)

- igf.UAµ letting Ud, Ut, UAunt + ily adult]

= die.

Aunt + U-E-tUA.ro, ut

-

UIA leAndru* - ily died, ut*

÷÷÷÷÷÷÷÷.

- i/gudvdrut-igf.UA, Ut, UAV UH

+ i/g[udµht, Udvut] ⑦= UFµUt + did Aunt + Utd, Ut

ti/gdnudvut-ilgdvuda-U-vUA.at-UI¥ + uA¥et-

udvutUAyut-tuqietytvut-UA-kttiyguq.netudvut

- ilgudvutu gut= UfµUt +0µW Aunt + Uqut UAunt- dull Anti - U diet uAµU++ ily dnudvut + ignorant u diet- ily duh faut - ilgudvutuq.at

Now we use : Utu =L → d,Utu = - utdµU

→ Fµ'-

- Ufwut + dµU Aunt - Nut fall Aunt- dull Aunt + nutdvUAµUt+ ily dare dunt -iguutq.udvut-ilgdvhduut-iyguutdvuq.se/-=UFuUt .

→ Fm transforms in the adjoint of IUCN) .

④We notice that

Lyn =- 12171 Fm FM)

→ I'yµ= -tztrlffn f- "Y = -tgtrlufmutuft.ru 'T=- Lz TrlFor FN) = Lyn

,due to the

cyclic property of the trace .

③ Am → Af = UA, ut + ily U diet1 (s + i data )Aµ (1- idiotb) thy 11tidala)qq.IT=

ftp.tioataANH-iobtb/tgIdaOcTct----Am-iArOaTa+idaTaAut1g0µA To

= An ti Oa A} Ita ,Tb] + Ig goat= AIT - f" 'O'AµbT, + Eg dado Tc

↳ Aji = Aj - f-"'gaAf + 1g dude .

We also easily find that

F)i = Ffv - fan oaf,!

,

④ A,→ Afn = U Aunt :(ttioata) Aul s - ios Tb)

= An tidalTa,AM

→ JA,= - fabcga.fi to

→ DA; = - fiiagiA,j

I;] ftp.T.pt"

Ai = - f- ftp.y/Fwifji)f-"'

Ai

=- YF,Égq%÷- f ""Ai

= - hzfjj ( dad,, dbi - of, iµ fbi/f"'

Ai

→ Ji o - f"'

fbµ Ai

T.ua?ddihm1Faewehhaey →y

'= UY = & + ida Tal4

→ 84= 1 data Y

L , = it 04 - i get AN)%q-µdY = - EN Tay

⑧Rutting everything togetherJ? =

- fah ' FbµAi -4-8^1^4 .

⑤ The equations of motion are obtainedbyvarying the adieu ant. the fields .

We find :

i 8^1221 = 0,

i DIE or = 0,

D'

Fei = g I 8^1-921 .

⑥ A → A'= U Aut = eioata A e- iostb

BHer-hampbell-Hamsdetffonmlo.FIe-✗= It [ ✗it ] + Efx, [ ✗it ]] + - - .

with ✗ = ldata,I = A

,we find

⑦A-'= At [ Wata

,A] * qt:[Hata ,[iobtb , A ]) + - - -

= A + ( da Abltaitb ) - tqdadb Ac [ Ta , [Ts , Tal) f- -

= A - Oa Ab Gabo To - Ig dad, A- a Ebcd [Ta, Td ] t - --

= A - da Ab Eabctc + Lzdadb Ac Ebcd Eade Te

= A - Eabccla Abt - lzdadbtc (dab Ice - deb do a) Te

= Actc - Eabcdatbtc - 1g @'

Act - data Oc To)

→ Ac = Ac - Eabcclatb - I @ Zdbo - Obdc)Ab

Ac = Ab (Jbc - Eabcda - 1-2/02dbc - Oboe) t --1*0

A • =

OcbAbkl1tiOaLa-tqHaLaYt-YobAbIAb@bc-Eb.caadat tqf Odbc - Oba/ * - -)= ④

→ The adjoint of JULY homeforms as avectorof $0137 . Do you

know why ?

-