❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦
■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛
❋ís✐❝❛ ❛❧é♠ ❞♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❡♠
t❡♦r✐❛s ❝♦♠ ❞✐♠❡♥sõ❡s ❡①tr❛s
Pr✐s❝✐❧❛ ▼❛ss❡tt♦ ❞❡ ❆q✉✐♥♦
❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ■♥st✐✲ t✉t♦ ❞❡ ❋ís✐❝❛ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ❈✐ê♥❝✐❛s
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ●✉st❛✈♦ ❇✉r❞♠❛♥
❈♦♠✐ssã♦ ❊①❛♠✐♥❛❞♦r❛✿
Pr♦❢✳ ●✉st❛✈♦ ❇✉r❞♠❛♥ ✭■❋✴❯❙P✮ Pr♦❢✳ ❘♦❣ér✐♦ ❘♦s❡♥❢❡❧❞ ✭■❋❚✴❯◆❊❙P✮ Pr♦❢✳ ❆❞r✐❛♥♦ ❆♥t♦♥✐♦ ◆❛t❛❧❡ ✭■❋❚✴❯◆❊❙P✮
❋■❈❍❆ ❈❆❚❆▲❖●❘➪❋■❈❆
Pr❡♣❛r❛❞❛ ♣❡❧♦ ❙❡r✈✐ç♦ ❞❡ ❇✐❜❧✐♦t❡❝❛ ❡ ■♥❢♦r♠❛çã♦
❞♦ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦
❆q✉✐♥♦✱ Pr✐s❝✐❧❛ ▼❛ss❡tt♦ ❞❡✳
❋ís✐❝❛ ❛❧é♠ ❞♦ ♠♦❞❡❧♦ ♣❛❞rã♦ ❡♠ t❡♦r✐❛s ❝♦♠ ❞✐♠❡♥sõ❡s ❡①tr❛s✳ ❙ã♦ P❛✉❧♦✱ ✷✵✵✼✳
❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✲ ❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❙ã♦ P❛✉❧♦✳ ■♥st✐t✉t♦ ❞❡ ❋ís✐❝❛✳ ❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❋ís✐❝❛ ♠❛t❡♠át✐❝❛
❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ●✉st❛✈♦ ❇✉r❞♠❛♥
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ❋ís✐❝❛
❯♥✐t❡r♠♦s✿ ✶✳ ❋ís✐❝❛ ❞❡ ♣❛rtí❝✉❧❛s❀ ✷✳ ❋ís✐❝❛ t❡ór✐❝❛❀
✸✳ ❋ís✐❝❛ ❞❡ ❛❧t❛ ❡♥❡r❣✐❛❀
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢✳ ●✉st❛✈♦ ❇✉r❞♠❛♥✱ ♣❡❧❛ ♦r✐❡♥t❛çã♦ ♠✉✐t♦ ❛t❡♥✲ ❝✐♦s❛ ❡ s❡♠♣r❡ ♣r❡s❡♥t❡ ❞❡s❞❡ ♦ ❝♦♠❡ç♦ ❞♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❚❛♠❜é♠ ♣❡❧❛s ✈❛❧✐♦s❛s ❞✐s❝✉ssõ❡s ❡ ♣♦r t❡r s❡♠♣r❡ ❛❝r❡❞✐t❛❞♦ ❡♠ ♠✐♠ ❝♦♠♦ ♣❡sq✉✐s❛❞♦r❛✳ ❆ ❡❧❡ ♠❡✉s s✐♥❝❡r♦s ❛❣r❛❞❡❝✐♠❡♥t♦s✳
❆♦ ❈♦♥s❡❧❤♦ ◆❛❝✐♦♥❛❧ ❞❡ ❉❡s❡♥✈♦❧✈✐♠❡♥t♦ ❈✐❡♥tí✜❝♦ ❡ ❚❡❝♥♦❧ó❣✐❝♦ ✭❈◆Pq✮ ♣❡❧❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s ❝♦♥❝❡❞✐❞❛✱ ❡ à ❋✉♥❞❛çã♦ ❞❡ ❆♠♣❛r♦ à P❡sq✉✐s❛ ❞♦ ❊st❛❞♦ ❞❡ ❙ã♦ P❛✉❧♦ ✭❋❆P❊❙P✮ ♣❡❧♦ ❛✉①í❧✐♦ ❛tr❛✈és ❞♦ ♣r♦❥❡t♦ t❡♠át✐❝♦✳
❆♦ ♠❡✉s ♣❛✐s✱ ●✐s❡❧❛ ❡ ◆❡❧s♦♥ ♣♦r t♦❞♦ ❛♠♦r ❡ ❝❛r✐♥❤♦ ❡♠ t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞❡ ♠✐♥❤❛ ✈✐❞❛✳ ❚❛♠❜é♠ ♣❡❧♦ ❛♣♦✐♦ t♦t❛❧ ❞❛❞♦ ❞❡s❞❡ ♦ ♠♦♠❡♥t♦ ❞❡ ❡s❝♦❧❤❛ ♥❛ ♣r♦✜ssã♦ ❞❡ ❢ís✐❝❛✳ ❆♦s ♠❡✉s ✐r♠ã♦s ▲í❣✐❛ ❡ ●✉✐❧❤❡r♠❡ ♣❡❧❛ ♣❛❝✐ê♥❝✐❛ ❡ ❜♦❛ ✈♦♥t❛❞❡ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❞✉r❛♥t❡ ❛ ❡t❛♣❛ ❞❡ ❝♦♥❝❧✉sã♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆♦s ♠❡✉s ❛✈ós ❉❛r❧② ❡ ❲❛❧❞②r ♣♦r t♦❞♦ ❝❛r✐♥❤♦ s✉♣♦rt❡ ♦❢❡r❡❝✐❞♦s ❛♦ ❧♦♥❣♦ ❞❡ t♦❞♦s ❡st❡s ❛♥♦s✳ ➚ ❙✐♠♦♥❡✱ ▲❛✉r❛✱ ▲ú❝✐❛✱ ●❡♦❣✐❛✱ ❍❡♥r✐q✉❡✱ ▼❛r✐❛ ❏ú❧✐❛ ❡ ❇❡❛tr✐③ ♣❡❧❛ ❛♠✐③❛❞❡ ❡ ❛✉①í❧✐♦ ❝♦♥st❛♥t❡s✳
❆ t♦❞♦s ♦s ♠❡✉s ❛♠✐❣♦s q✉❡ ♠❡ ❛❥✉❞❛r❛♠ ❞✉r❛♥t❡ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ❘♦❞r✐❣♦ ❆♠♦r✐♠ ❡ ▲✉❝❛s ❱✐❛♥✐ ♣♦r ❡st❛r❡♠ ❛♦ ♠❡✉ ❧❛❞♦ ♣❛r❛ ❛✉❧❛s✱ ❛❧♠♦ç♦s ❡ ❝❛❢és ❞✉r❛♥t❡ t♦❞♦ ♦ ♠❡str❛❞♦✳
❋ís✐❝❛ ❛❧é♠ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❡♠ ❚❡♦r✐❛s ❝♦♠
❉✐♠❡♥sõ❡s ❊①tr❛s
❘❡s✉♠♦
❖ ♦❜❥❡t✐✈♦ ❞❡st❛ ❞✐ss❡rt❛çã♦ é ❡st✉❞❛r t❡♦r✐❛s q✉❡ ✉t✐❧✐③❛♠ ❞✐♠❡♥sõ❡s ❡①tr❛s ♣❛r❛ ❡①♣❧✐❝❛r ♦s ♣r♦❜❧❡♠❛s q✉❡ s✉r❣❡♠ ♥♦ ▼♦❞❡❧♦ P❛❞rã♦ q✉❛♥❞♦ ❛ ❡♥❡r❣✐❛ ❛t✐♥❣❡ ✈❛❧♦r❡s ♠✉✐t♦ ❛❧t♦s ❝❤❡❣❛♥❞♦ à ♦r❞❡♠T eV✳ ❚r❛❜❛❧❤❛♠♦s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ❝♦♠ ♠♦❞❡❧♦s ❝♦♠
♠❛✐s ❞❡ ✹ ❞✐♠❡♥sõ❡s✱ ♦♥❞❡ ❛s ❞✐♠❡♥sõ❡s ❡①tr❛s sã♦ ❡s♣❛❝✐❛✐s ❡ ❝♦♠♣❛❝t✐✜❝❛❞❛s ❝♦♠ ♦ ♣r♦❝❡❞✐♠❡♥t♦S1/Z2✳
❙❛❜❡♠♦s q✉❡ ♦ ▼♦❞❡❧♦ P❛❞rã♦ é ❝♦♥s✐st❡♥t❡ ❝♦♠ t♦❞♦s ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s q✉❡ ♠❡❞✐♠♦s ❛té ❤♦❥❡✱ ♠❛s ❡①✐st❡♠ ♠✉✐t❛s r❛③õ❡s ♣❛r❛ ❡s♣❡r❛r♠♦s ♥♦✈❛ ❢ís✐❝❛ ♥❛ ❡s❝❛❧❛
T eV✳ ■♥✐❝✐❛♠♦s ♦ tr❛❜❛❧❤♦ ❛♣r❡s❡♥t❛♥❞♦ ♦s ❛s♣❡❝t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦ ▼♦❞❡❧♦
P❛❞rã♦✳ ❙❡❣✉✐♠♦s ❡s♣❡❝✐✜❝❛♥❞♦ ❛❧❣✉♥s ♣r♦❜❧❡♠❛s q✉❡ s✉r❣❡♠ ♥♦ ▼♦❞❡❧♦ P❛❞rã♦ ♥♦ ❧✐♠✐t❡ ♣❛r❛ ❛❧t❛s ❡♥❡r❣✐❛s q✉❡ r❡s✉❧t❛r❛♠ ♥❛ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛ ❝r✐❛çã♦ ❞❡ ❚❡♦r✐❛s ❆❧é♠ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦✳ ❊①♣❧✐❝✐t❛♠♦s ❛❧❣✉♥s ❞❡ s❡✉s ♣r♦❜❧❡♠❛s✱ ♠❛s ❡♥tr❛♠♦s ❡♠ ❞❡t❛❧❤❡s ♥♦ ❡st✉❞♦ ❞❡ ❞♦✐s ♣r✐♥❝✐♣❛✐s✿ ♦ Pr♦❜❧❡♠❛ ❞❛ ❍✐❡r❛rq✉✐❛ ❡ ♦ Pr♦❜❧❡♠❛ ❞❡ ▼❛ss❛ ❞♦s ❋ér♠✐♦♥s✳
❊♠ s❡❣✉✐❞❛✱ ❞❡✜♥✐♠♦s ♦s três t✐♣♦s ❞❡ t❡♦r✐❛s q✉❡ ✉t✐❧✐③❛♠ ❞✐♠❡♥sõ❡s ❡①tr❛s ♣❛r❛ s♦❧✉❝✐♦♥❛r ♦ Pr♦❜❧❡♠❛ ❞❛ ❍✐❡r❛rq✉✐❛ ❡ ❛s ❛♣r❡s❡♥t❛♠♦s ♥❛ ♦r❞❡♠ ❡♠ q✉❡ ❢♦r❛♠ ✐❞❡❛❧✐③❛❞❛s✳ ❆s ❞✉❛s ♣r✐♠❡✐r❛s✱ ❞❡♥♦♠✐♥❛❞❛s ✏▲❛r❣❡ ❊①tr❛ ❉✐♠❡♥s✐♦♥s✑ ✭▲❊❉✮ ❡ ✏❯♥✐✈❡rs❛❧ ❊①tr❛ ❉✐♠❡♥s✐♦♥s✑ ✭❯❊❉✮ ✉t✐❧✐③❛♠ ✉♠❛ ♠étr✐❝❛ ♣❧❛♥❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ t♦t❛❧ ❡ sã♦ ❞✐❢❡r❡♥t❡s ♥❛ ❞❡✜♥✐çã♦ ❞❛ ♣r♦♣❛❣❛çã♦ ❞♦s ❝❛♠♣♦s ❡♠ ❞❡t❡r♠✐♥❛❞❛s ❞✐✲ ♠❡♥sõ❡s✳ ❆ ✏❲❛r♣❡❞ ❊①tr❛ ❉✐♠❡♥s✐♦♥s✑ ✭❲❊❉✮ ✉t✐❧✐③❛ ✉♠❛ ✉♠❛ ♠étr✐❝❛ ❝✉r✈❛ ❞♦ ❡s♣❛ç♦✲t❡♠♣♦ ✺✲❞✐♠❡♥s✐♦♥❛❧ ❡ s♦❧✉❝✐♦♥❛ ♦ Pr♦❜❧❡♠❛ ❞❛ ❍✐❡r❛rq✉✐❛ ❞❡ ♠❛♥❡✐r❛ ❞✐❢❡r❡♥❝✐❛❞❛✳
P❤②s✐❝s ❇❡②♦♥❞ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ✐♥ ❚❤❡♦r✐❡s ✇✐t❤
❊①tr❛ ❉✐♠❡♥s✐♦♥s
❆❜str❛❝t
❚❤❡ ❣♦❛❧ ♦❢ t❤✐s ❞✐ss❡rt❛t✐♦♥ ✐s t♦ st✉❞② t❤❡♦r✐❡s t❤❛t ✉s❡ ❡①tr❛ ❞✐♠❡♥s✐♦♥s t♦ s♦❧✈❡ t❤❡ ♣r♦❜❧❡♠s t❤❛t ❛♣♣❡❛r ✐♥ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❛t ❡♥❡r❣✐❡s ♦❢ t❤❡ ♦r❞❡r ✶ ❚❡❱✳ ❙♣❡❝✐✜❝❛❧❧②✱ ✇❡ ✇♦r❦❡❞ ✇✐t❤ ♠♦❞❡❧s ✇✐t❤ ♠♦r❡ t❤❛♥ ✹ ❞✐♠❡♥s✐♦♥s✱ ✇❤❡r❡ t❤❡ s♣❛t✐❛❧ ❞✐♠❡♥s✐♦♥s ❛r❡ ❝♦♠♣❛❝t✐✜❡❞ ✐♥ ❛♥S1/Z2 ♦r❜✐❢♦❧❞✳
❚❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❛❣r❡❡s t♦ ❛ ❣r❡❛t ❞❡❣r❡❡ ✇✐t❤ t❤❡ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ✇❡ ❤❛✈❡ t♦❞❛② ❜✉t t❤❡r❡ ❛r❡ s❡✈❡r❛❧ r❡❛s♦♥s t♦ ❡①♣❡❝t ♥❡✇ ♣❤②s✐❝s ❛t t❤❡ ❚❡❱ s❝❛❧❡✳ ❲❡ st❛rt ♣r❡s❡♥t✐♥❣ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ❛s♣❡❝ts ♦❢ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧✳ ❲❡ t❤❡♥ s♣❡❝✐❢② s♦♠❡ ♦❢ t❤❡ ♣r♦❜❧❡♠s t❤❛t ❛♣♣❡❛r ❛t ❤✐❣❤ ❡♥❡r❣✐❡s ✭❤✐❣❤❡r t❤❛♥ t❤❡ ✇❡❛❦ s❝❛❧❡✮ ❛♥❞ t❤❛t ❛r❡ t❤❡ ♠♦t✐✈❛t✐♦♥ t♦ ❝♦♥s✐❞❡r t❤❡♦r✐❡s ❜❡②♦♥❞ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧✳ ❲❡ ❢♦❝✉s ♦♥ t✇♦ s✉❝❤ ♣r♦❜❧❡♠s✿ t❤❡ ❤✐❡r❛r❝❤② ♣r♦❜❧❡♠ ❛♥❞ t❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ ❢❡r♠✐♦♥ ♠❛ss❡s✳ ❲❡ ♣r❡s❡♥t t❤r❡❡ t②♣❡s ♦❢ t❤❡♦r✐❡s ✉s✐♥❣ ❡①tr❛ ❞✐♠❡♥s✐♦♥s t♦ ❛❞❞r❡ss t❤❡ ❤✐❡r❛r❝❤② ♣r♦❜❧❡♠✳ ❚❤❡ ✜rst t✇♦✱ ▲❛r❣❡ ❊①tr❛ ❉✐♠❡♥s✐♦♥s ✭▲❊❉✮ ❛♥❞ ❯♥✐✈❡rs❛❧ ❊①tr❛ ❉✐✲ ♠❡♥s✐♦♥s ✭❯❊❉✮ ✉s❡ ❛ ✢❛t ♠❡tr✐❝ ❛♥❞ ♦♥❧② ❞✐✛❡r ♦♥ t❤❡ ✜❡❧❞s t❤❛t ❛r❡ ❛❧❧♦✇❡❞ t♦ ♣r♦♣❛❣❛t❡ ✐♥ t❤❡ ❡①tr❛ ❞✐♠❡♥s✐♦♥s✳ ❲❛r♣❡❞ ❊①tr❛ ❉✐♠❡♥s✐♦♥s ✭❲❊❉✮ ✉s❡ ❛ ❝✉r✈❡❞ ♠❡tr✐❝ t♦ s♦❧✈❡ t❤❡ ❤✐❡r❛r❝❤② ♣r♦❜❧❡♠ ✐♥ ❛ ✉♥✐q✉❡ ✇❛②✳
❙✉♠ár✐♦
❆❣r❛❞❡❝✐♠❡♥t♦s ✈
❘❡s✉♠♦ ✈✐✐
❆❜str❛❝t ✐①
✶ ■♥tr♦❞✉çã♦ ✶
✷ ❖ ▼♦❞❡❧♦ P❛❞rã♦ ✸
❙❯▼➪❘■❖ ①✐✐
✷✳✸ ■♥t❡r❛çõ❡s ❢♦rt❡s ❡ ❛ ◗❈❉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸✳✶ ■♥❝❧✉✐♥❞♦ ♦s ❍á❞r♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✶✳✶ ■♥tr♦❞✉③✐♥❞♦ ♦s q✉❛r❦s ♥❛ ❧❛❣r❛♥❣❡❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✸✳✶✳✷ ◗✉❛r❦s ♠❛ss✐✈♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✷✳✹ ❆ ❧❛❣r❛♥❣❡❛♥❛ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹✳✶ ❇ós♦♥s ❞❡ ❣❛✉❣❡ ✰ ❊s❝❛❧❛r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✹✳✷ ▲é♣t♦♥s ✰ ❨✉❦❛✇❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✹✳✸ ◗✉❛r❦s ✰ ❨✉❦❛✇❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✷✳✹✳✹ ❈♦♥❝❧✉✐♥❞♦✿ ❛s ♣❛rtí❝✉❧❛s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼
✸ ❆ ❋ís✐❝❛ ❆❧é♠ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ✸✶
❙❯▼➪❘■❖ ①✐✐✐
✹ ❉✐♠❡♥sõ❡s ❊①tr❛s ✹✾
✹✳✶ ❖ s✉r❣✐♠❡♥t♦ ❞❛ ✐❞é✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✵ ✹✳✷ ▲❛r❣❡ ❊①tr❛ ❉✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✸ ❯♥✐✈❡rs❛❧ ❊①tr❛ ❉✐♠❡♥s✐♦♥s ✭❯❊❉✮ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✹✳✸✳✶ Pr♦♣❛❣❛çã♦ ❞♦s ❜ós♦♥s ❞❡ ❣❛✉❣❡ ❡ ❢ér♠✐♦♥s ♥❛s ❞✐♠❡♥sõ❡s ❡①tr❛s ✻✵ ✹✳✸✳✷ ❖s ❝❛♠♣♦s ❢❡r♠✐ô♥✐❝♦s ❡ s✉❛ q✉✐r❛❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✹✳✸✳✸ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✹ ✹✳✹ ❲❛r♣❡❞ ❊①tr❛ ❉✐♠❡♥s✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✹✳✶ ❘❡s♦❧✈❡♥❞♦ ♦ Pr♦❜❧❡♠❛ ❞❛ ❍✐❡r❛rq✉✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✹✳✷ ❆ ♣r♦♣❛❣❛çã♦ ✺❉ ❞♦s ❝❛♠♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✵ ✹✳✹✳✸ ❆ r❡❞✉çã♦ ❞✐♠❡♥s✐♦♥❛❧ ❡ s✉❛s ❝♦♥s❡qüê♥❝✐❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✹✳✸✳✶ ❈❛♠♣♦s ❡s❝❛❧❛r❡s ♥♦ ❜✉❧❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✹✳✹✳✸✳✷ ❇ós♦♥s ❞❡ ●❛✉❣❡ ♥♦ ❜✉❧❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✻ ✹✳✹✳✸✳✸ ❋ér♠✐♦♥s ♥♦ ❜✉❧❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✶ ✹✳✺ ❈♦♥❝❧✉sõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾
✺ ❱✐♦❧❛çã♦ ❞❡ ❙❛❜♦r❡s ❡♠ ❲❊❉ ♥♦ ▲❍❈ ✾✶
❙❯▼➪❘■❖ ①✐✈
▲✐st❛ ❞❡ ❋✐❣✉r❛s
✷✳✶ P♦t❡♥❝✐❛❧ ❡❢❡t✐✈♦ V(φ) ♣❛r❛ ❛ ❧❛❣r❛♥❣❡❛♥❛ ♠♦str❛❞❛ ❡♠ ✭✷✳✺✮ ❝♦♠ µ2 >0 ❡♠ ✭❛✮ ❡ µ2 <0 ❡♠ ✭❜✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✷✳✷ ❚❡r♠♦s ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s ❜ós♦♥s ❞❡ ❣❛✉❣❡ W±
µ✱ Aµ ❡ Zµ0. ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✷✳✸ ❚❡r♠♦s ❞❡ ✐♥t❡r❛çã♦ ❞♦ ❜ós♦♥ ❞❡ ❍✐❣❣s ❝♦♠ ♦s ❜ós♦♥s Wµ±✱Aµ ❡ Zµ0. ✳ ✳ ✳ ✷✻ ✷✳✹ ❚❡r♠♦s ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ❧é♣t♦♥s ❡ ♦s ❜ós♦♥s ❞❡ ❣❛✉❣❡ Wµ±✱ Aµ ❡ Zµ0. ✳ ✳ ✷✼ ✷✳✺ ❚❡r♠♦s ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s q✉❛r❦s ❡ ♦s ❜ós♦♥s ❞❡ ❣❛✉❣❡ W±
µ✱Aµ ❡ Zµ0. ✳ ✷✼ ✸✳✶ ❉✐❛❣r❛♠❛s ❞❡ ❋❡②♥♠❛♥ q✉❡ ❝♦♥tr✐❜✉❡♠ ♣❛r❛ ♦ ❡s♣❛❧❤❛♠❡♥t♦ WLWL →
WLWL✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✹
✸✳✷ ❆s ♦♥❞❛s ♣❛r❝✐❛✐s ❞❡✈❡♠ ❡st❛r ❞❡♥tr♦ ❝ír❝✉❧♦ ✉♥✐tár✐♦ ♣❛r❛ q✉❡ ❛ ❝♦♥❞✐çã♦ s❡❥❛ s❛t✐s❢❡✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✸✳✸ ❈♦rr❡çã♦ r❛❞✐♦❛t✐✈❛ ♣❛r❛ ❛ ♠❛ss❛ ❞♦ ❍✐❣❣s ❝♦♥s✐❞❡r❛♥❞♦ ❛ ❛♠♣❧✐t✉❞❡ ❞❡
✐♥t❡r❛çã♦ ❝♦♠ ♦ q✉❛r❦ t♦♣✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽ ✸✳✹ ❊sq✉❡♠❛ ❞♦ Pr♦❜❧❡♠❛ ❞❛ ❍✐❡r❛rq✉✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾
✹✳✶ ❆ t♦rr❡ ❞❡ ❑❛❧✉③❛✲❑❧❡✐♥ ❢♦r♠❛❞❛ ♣♦r ❝❛♠♣♦s ❡s❝❛❧❛r❡s✱ ❝♦♠♦ ♥♦s ♠♦str❛ ❛ ❡q✉❛çã♦ ✭✹✳✶✷✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾ ✹✳✷ ❊①❡♠♣❧♦s ❞❡ ❞✐❛❣r❛♠❛s ✭❛✮ ♣r♦✐❜✐❞♦ ❡ ✭❜✮ ♣❡r♠✐t✐❞♦ ♣❡❧❛ ❝♦♥s❡r✈❛çã♦ ❞♦
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①✈✐
✹✳✹ ❊sq✉❡♠❛ ❞❛ ✏❞❡❝♦♠♣♦s✐çã♦✑ ❞❡ ✉♠ ♠✉♥❞♦ ✺❉✿ ❡♠ ❝❛❞❛ ❧✉❣❛r ❞❛ ❝♦♦r❞❡✲ ♥❛❞❛ ❞❛ ❞✐♠❡♥sã♦ ❡①tr❛ t❡♠♦s ✉♠ ♠✉♥❞♦ ✹❉ ♠✐♥❦♦✇s❦✐❛♥♦ ❝♦♠♦ ❡st❡ q✉❡ ❡♥①❡r❣❛♠♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✺ ✹✳✺ ●rá✜❝♦ ❞♦ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❛ ❢✉♥çã♦ Xn(y)✳ P❛r❛ ♣❧♦t❛r♠♦s ❡st❡ ❣rá✜❝♦
✉t✐❧✐③❛♠♦s ✉♠❛ ♥♦✈❛ ✈❛r✐á✈❡❧ t ≡ eσ−kπR q✉❡ é ♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧✳ ◆❡st❡
❝❛s♦✱ t ❡✈♦❧✉✐ ❞❡ 0 ❛ 1 q✉❡ ❡q✉✐✈❛❧❡♠ r❡s♣❡❝t✐✈❛♠❡♥t❡ às ❜r❛♥❛s ❞❡ P❧❛♥❝❦ ❡ ❚❡❱✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✵ ✹✳✻ P❧♦t❡s ♣❛r❛ ❛s ❢✉♥çõ❡s ✭❛✮ F0R(y) ❡ ✭❜✮ F0L(y) ✭q✉❡ sã♦ ❛s ❢✉♥çõ❡s f0L,R
❝♦♠ ♦s ❢❛t♦r❡s ❞❛ ♠étr✐❝❛ ❡ ❞♦ ✈✐❡r❜❡✐♥ ❥á ✐♥❝❧✉s♦s✮ ❝♦♠ r❡s♣❡❝t✐✈♦s ✈❛❧♦r❡s ❞❛❞♦s ❛♦ ♣❛râ♠❡tr♦ cR ❡ cL✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✹✳✼ ●rá✜❝♦s ❞❡ f1L(t) ❡ f1R(t) ✭❞✉❜❧❡t♦s✮ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞♦ ♣❛râ♠❡tr♦ c✳ ✽✽
✹✳✽ ●rá✜❝♦s ❞❡ f1L(t) ❡ f1R(t) ✭s✐♥❣❧❡t♦s✮ ♣❛r❛ ❞✐❢❡r❡♥t❡s ✈❛❧♦r❡s ❞♦ ♣❛râ♠❡tr♦c✳ ✽✾
✺✳✶ ❉✐❛❣r❛♠❛ ❡①❡♠♣❧✐✜❝❛♥❞♦ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❡♥tr❡ ♦ ♣r✐♠❡✐r♦ ♠♦❞♦ ❡①❝✐t❛❞♦ ❞❡ ✉♠ ❣❧ú♦♥ ❝♦♠ ♦ ♠♦❞♦ ③❡r♦ ❞❡ ❞♦✐s q✉❛r❦s✱ ♦✉ s❡❥❛✱ ❞♦✐s q✉❛r❦s ❡①✐st❡♥t❡s ♥♦ ▼♦❞❡❧♦ ♣❛❞rã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✾ ✺✳✷ ❉✐❛❣r❛♠❛ ❞❡ ❋❡②♥♠❛♥ r❡♣r❡s❡♥t❛♥❞♦ ❛ ✐♥t❡r❛çã♦ qq¯ → qq¯ ✈✐❛ t♦❞♦s ♦s
♠♦❞♦s ❞❡ ❑❛❧✉③❛✲❑❧❡✐♥ ❞♦ ❣❧ú♦♥✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✵ ✺✳✸ ●rá✜❝♦ ❞❛ ✈❛r✐❛çã♦ ❞♦ ❛❝♦♣❧❛♠❡♥t♦ ❡♠ ❢✉♥çã♦ ❞♦ ♣❛râ♠❡tr♦c❞♦s ❢ér♠✐♦♥s✳
❉❡♣❡♥❞❡♥❞♦ ❞❛ ♠❛ss❛ ❢❡r♠✐ô♥✐❝❛ ♥♦ ❜✉❧❦ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❝♦♠ ♦ ♣r✐♠❡✐r♦ ♠♦❞♦ ❞♦ ❣❧ú♦♥ ♣♦❞❡ s❡r ♠❛✐♦r ♦✉ ♠❡♥♦r✳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✺✳✹ ●rá✜❝♦ ❞❛ s❡çã♦ ❞❡ ❝❤♦q✉❡ ❞✐❢❡r❡♥❝✐❛❧ ♣❡❧❛ ♠❛ss❛ ✐♥✈❛r✐❛♥t❡ ❞♦ ♣r♦❝❡ss♦ ❡♠
q✉❡stã♦ ♣❛r❛ ♦ ▼♦❞❡❧♦ P❛❞rã♦ ✭❝✉r✈❛ ✐♥❢❡r✐♦r✮ ❡ ♣❛r❛ ❛ ❚❡♦r✐❛ ❞❡ ❉✐♠❡♥✲ sõ❡s ❊①tr❛s ❈✉r✈❛s q✉❡ ❡stá s❡♥❞♦ ❝♦♥s✐❞❡r❛❞❛✳ ❋♦✐ ✉t✐❧✐③❛❞♦ MG≃2T eV✳ ✶✵✻ ✺✳✺ ❖ ❛❝♦♣❧❛♠❡♥t♦ q✉❡ ♥♦ ▼♦❞❡❧♦ P❛❞rã♦ só ❡①✐st✐❛ ❝♦♠ ❞♦✐s s❛❜♦r❡s ✐❣✉❛✐s
▲■❙❚❆ ❉❊ ❋■●❯❘❆❙ ①✈✐✐
✺✳✽ ●rá✜❝♦ ❞❛ s❡çã♦ ❞❡ ❝❤♦q✉❡ ❞✐❢❡r❡♥❝✐❛❧ ♣❡❧❛ ♠❛ss❛ ✐♥✈❛r✐❛♥t❡ ❞♦ ♣r♦❝❡ss♦ ❡♠ q✉❡stã♦ ✳ ❋♦r❛♠ ♦❜t✐❞❛s três ❝✉r✈❛s ❞✐❢❡r❡♥t❡s ♣❛r❛ ♦s s✐♥❛✐s ❝♦♥s✐❞❡r❛♥❞♦
▲✐st❛ ❞❡ ❚❛❜❡❧❛s
✷✳✶ ❈♦♥t❛❣❡♠ ❞♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❛ t❡♦r✐❛ ❛♥t❡s ❡ ❞❡♣♦✐s ❞❛ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡tr✐❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✷ ❈♦♥t❛❣❡♠ ❞♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❛ t❡♦r✐❛ ❛♥t❡s ❡ ❞❡♣♦✐s ❞❛
q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡tr✐❛ ❝♦♠ ♦ ❣❛✉❣❡ ❡s❝♦❧❤✐❞♦ ❡♠ ✭✷✳✶✹✮✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✸ ▼❛ss❛s ❞❛s ♣r✐♥❝✐♣❛✐s ♣❛rtí❝✉❧❛s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ♦❜t✐❞❛s ❡①♣❡r✐♠❡♥t❛❧✲
♠❡♥t❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✹ ◆ú♠❡r♦s q✉â♥t✐❝♦s ✭✐s♦s♣✐♥ ❡ ❤✐♣❡r❝❛r❣❛✮ ♣❛r❛ ♦s ❧é♣t♦♥s ❡ q✉❛r❦s ❞♦ ▼♦✲
❞❡❧♦ P❛❞rã♦✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵
✷✳✺ ◆ú♠❡r♦s q✉â♥t✐❝♦s ❞♦s ❢ér♠✐♦♥s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦✱ ❝♦♠ Q=T3+Y /2✳✳ ✳ ✷✽
✸✳✶ ▼❛ss❛s ♦❜s❡r✈❛❞❛s ❞♦s ❧é♣t♦♥s ❡ q✉❛r❦s ❡♠ M eV✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷
✹✳✶ P❛r❛ ❞❡t❡r♠✐♥❛❞♦s ✈❛❧♦r❡s ❞❡ m∗ ❡ n ♦❜t❡♠♦s ✉♠ ✈❛❧♦r ♣❛r❛ R✳ ◗✉❛♥❞♦ n = 1✱ R = m
2
planck m3
∗ ❀ ♣❛r❛ n = 2✱ R =
mplanck
m∗ ❀ ❡ ♣❛r❛ n = 3✱ R =
m2
planck m5
∗ 1/3
✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺
✺✳✶ ❈♦rt❡s ❛❞✐❝✐♦♥❛✐s ✉t✐❧✐③❛❞♦s ♣❛r❛ ❛ r❡❞✉çã♦ ❞♦ ❜❛❝❦❣r♦✉♥❞✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷ ✺✳✷ ❙❡çõ❡s ❞❡ ❝❤♦q✉❡ ♦❜t✐❞❛s ♣❛r❛ ♦s s✐♥❛✐s ❡ ❜❛❝❦❣r♦✉♥❞s ❛ ♣❛rt✐r ❞❡ ♣r♦❝❡ss♦s
❝♦♠ MG= 1T eV ❡URtq = 1✳ ❖ s✐♥❛❧ ❡stá r❡♣r❡s❡♥t❛❞♦ ❡♠ ✈❡r♠❡❧❤♦✱ ❡ ♦s ❜❛❝❦❣r♦✉♥❞s ♣♦ss✉❡♠ s✉❛ ❞❡t❡r♠✐♥❛❞❛ ❡q✉❛çã♦ ❡♥tr❡ ♣❛rê♥t❡s❡s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✷ ✺✳✸ ❙❡çõ❡s ❞❡ ❝❤♦q✉❡ ❞♦s s✐♥❛✐s ❡ ❜❛❝❦❣r♦✉♥❞ ♦❜t✐❞♦s ♣❛r❛ MG = 2T eV ❡
URtq = 1✳ ◆♦✈❛♠❡♥t❡ ♦ s✐♥❛❧ ❡stá ❡♥❢❛t✐③❛❞♦ ❡♠ ✈❡r♠❡❧❤♦✱ ❡ ♦s ❜❛❝❦❣r♦✉♥❞s
▲■❙❚❆ ❉❊ ❚❆❇❊▲❆❙ ①✐①
❈❛♣ít✉❧♦ ✶
■♥tr♦❞✉çã♦
❆ ❋ís✐❝❛ ❞❡ P❛rtí❝✉❧❛s s❡ ❡♥❝♦♥tr❛ ❡♠ ✉♠ ♠♦♠❡♥t♦ ❝rít✐❝♦ ♥❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ ❣r❛♥❞❡s ♠✉❞❛♥ç❛s✳ ❖ ▼♦❞❡❧♦ P❛❞rã♦ é ✉♠❛ t❡♦r✐❛ ❡①tr❡♠❛♠❡♥t❡ ❜❡♠ s✉❝❡❞✐❞❛ q✉❛♥❞♦ ❝♦♠✲ ♣❛r❛❞♦ ❛♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s q✉❡ t❡♠♦s ❤♦❥❡✳ ❊❧❡ t❡♠ s✐❞♦ ❝♦♥✜r♠❛❞♦ ❛♦ ❧♦♥❣♦ ❞♦ t❡♠♣♦ ❡ ❝♦♥s❡❣✉✐✉ ♣❛ss❛r ♣♦r ❞✐✈❡rs❛s ♣r♦✈❛s✱ ❞❡s❞❡ ❛ ❞❡s❝♦❜❡rt❛ ❞❛s ❝♦rr❡♥t❡s ♥❡✉tr❛s ♥♦s ❛♥♦s ✼✵✱ ❛ ❞❡s❝♦❜❡rt❛ ❞♦s ❜ós♦♥s ❞❡ ❣❛✉❣❡W ❡ Z q✉❡ ✐♥t❡r♠❡❞✐❛♠ ❛s
✐♥t❡r❛çõ❡s ❡❧❡tr♦❢r❛❝❛s ♥♦s ❛♥♦s ✽✵✱ ❛té ♦s t❡st❡s ❞❡ ❣r❛♥❞❡ ♣r❡❝✐sã♦ ✭< 1%✮ ❝♦♥✲
s✐❞❡r❛♥❞♦ ✈❛❧♦r❡s ❞❡ ❡♥❡r❣✐❛ tã♦ ❛❧t♦s q✉❛♥t♦ ♣✉❞❡♠♦s ♠❡❞✐r t❛♥t♦ ♥♦ ❚❡✈❛tr♦♥ q✉❛♥t♦ ♥♦ ❈❊❘◆✳ ❈♦♥t✉❞♦✱ ❡①✐st❡♠ ♠✉✐t♦s ✐♥❞í❝✐♦s q✉❡ s✉❣❡r❡♠ ❛ ❡①✐stê♥❝✐❛ ❞❡ ❢ís✐❝❛ ❛❧é♠ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦✳ ❖ ♣r✐♠❡✐r♦ ✐♥❞í❝✐♦ q✉❡ ♣♦❞❡♠♦s ❝✐t❛r é ♦ ♣r♦❜❧❡♠❛ ❞❛ ❤✐❡r❛rq✉✐❛ ❞❡ ❡s❝❛❧❛s✳ ◆♦ ▼♦❞❡❧♦ P❛❞rã♦✱ ❛ ❡s❝❛❧❛ ❡❧❡tr♦❢r❛❝❛ é ✐♥stá✈❡❧ s♦❜ ❝♦r✲ r❡çõ❡s r❛❞✐❛t✐✈❛s✳ ❊st❛ ✐♥st❛❜✐❧✐❞❛❞❡ s❡ ♠❛♥✐❢❡st❛✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❢❛t♦ ❞❛ ♠❛ss❛ ❞♦ ❜ós♦♥ ❞❡ ❍✐❣❣s r❡❝❡❜❡r ❝♦rr❡çõ❡s r❛❞✐❛t✐✈❛s q✉❡ ❞✐✈❡r❣❡♠ q✉❛❞r❛t✐❝❛♠❡♥t❡ ❝♦♠ ❛ ❡♥❡r❣✐❛✳ P♦rt❛♥t♦✱ s❡ ♦ ▼♦❞❡❧♦ P❛❞rã♦ ❢♦ss❡ ✈á❧✐❞♦ ❛té ❛ ❡s❝❛❧❛ ❞❡ P❧❛♥❝❦✱ ♦ ✈❛❧♦r ♥❛t✉r❛❧ ❞❛ ♠❛ss❛ ❞♦ ❍✐❣❣s s❡r✐❛ Mplanck = 1019GeV✳ ❖ ♣r♦❜❧❡♠❛ é q✉❡ ♦ ❜ós♦♥ ❞❡ ❍✐❣❣s é ❛ ♣❛rtí❝✉❧❛ q✉❡ ✉♥✐t❛r✐③❛ ❛ t❡♦r✐❛✱ ♦ q✉❡ ❢❛③ ♥❡❝❡ssár✐♦ q✉❡ s✉❛ ♠❛ss❛ s❡❥❛ ♠❡♥♦r q✉❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 1 T eV✳ ❆ss✐♠✱ s❡ ♦ ✈❛❧♦r ♥❛t✉r❛❧ ❞❛ ♠❛ss❛ ❞♦
❍✐❣❣s é Mplanck✱ ♦ ✈❛❧♦r ♥❛t✉r❛❧ ❞❛ ❡s❝❛❧❛ ❡❧❡tr♦❢r❛❝❛ t❛♠❜é♠ ❞❡✈❡r✐❛ s❡r ❡st❡ ✈❛❧♦r ❡ ♥ã♦ 1017 ✈❡③❡s ♠❡♥♦r✳ ❆❧é♠ ❞❡st❡ ♣r♦❜❧❡♠❛ ❡①✐st❡♠ ♠✉✐t♦s ♦✉tr♦s q✉❡ ❝♦♠❡ç❛♠
❛ ❛♣❛r❡❝❡r q✉❛♥❞♦ ❡♥❡r❣✐❛s r❛③♦❛✈❡❧♠❡♥t❡ ❛❧t❛s sã♦ ❛t✐♥❣✐❞❛s✳
❉❡ ❢♦r♠❛ ♥♦tá✈❡❧✱ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❤✐❡r❛rq✉✐❛ s✉❣❡r❡ ❢♦rt❡♠❡♥t❡ ❛ ♣r❡s❡♥ç❛ ❞❡ ♥♦✈❛ ❢ís✐❝❛ ♥❛ ❡s❝❛❧❛ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 1T eV✳ ❖ ▲❛r❣❡ ❍❛❞r♦♥ ❈♦❧❧✐❞❡r ✭▲❍❈✮ é ✉♠
❈❛♣ít✉❧♦ ✶✳ ■♥tr♦❞✉çã♦ ✷
❡①♣❧♦r❛r ❝♦♠ ❣r❛♥❞❡ ❛❧❝❛♥❝❡ ❡st❛ ❡s❝❛❧❛ ❞❡ ❡♥❡r❣✐❛✳ ❊stá ♣❧❛♥❡❥❛❞♦ ♣❛r❛ ❝♦♠❡ç❛r ❛ r♦❞❛r ❡♠ ✷✵✵✼✱ ♦❜t❡♥❞♦ ♦s ♣r✐♠❡✐r♦s ❞❛❞♦s ❢ís✐❝♦s ❡♠ ✷✵✵✽ ❡ ♣♦ss✉✐ ♣♦t❡♥❝✐❛❧ ♣❛r❛ r❡✈♦❧✉❝✐♦♥❛r ❛ ❢ís✐❝❛ ❞❡ ♣❛rtí❝✉❧❛s ❡♠ ♣♦✉❝♦s ❛♥♦s✳
❈♦♠♦ ❝♦♥s❡qüê♥❝✐❛✱ t❡♦r✐❛s q✉❡ s♦❧✉❝✐♦♥❛♠ ♦s ♣r♦❜❧❡♠❛s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❡ r❡✲ q✉❡r❡♠ ♥♦✈❛ ❢ís✐❝❛ ♥❛ ❡s❝❛❧❛ ❚❡❱ ✈❡♠ s✐❞♦ ♣r♦♣♦st❛s ❛♦ ❧♦♥❣♦ ❞♦s ú❧t✐♠♦s ❛♥♦s✳ ❯♠ ❡①❡♠♣❧♦ ❞✐st♦ sã♦ ❛s t❡♦r✐❛s q✉❡ ✉t✐❧✐③❛♠ ❞✐♠❡♥sõ❡s ❡①tr❛s ❡s♣❛❝✐❛✐s ❝♦♠♣❛❝t❛s ♣❛r❛ r❡s♦❧✈❡r ♦ ♣r♦❜❧❡♠❛ ❞❛ ❤✐❡r❛rq✉✐❛✳ ❊st❡s t✐♣♦s ❞❡ t❡♦r✐❛s s❡ ❞✐✈✐❞❡♠ ❡♠ três ♣r✐♥❝✐♣❛✐s ✈❡rt❡♥t❡s✿ ❛s ❚❡♦r✐❛s ❝♦♠ ❉✐♠❡♥sõ❡s ❊①tr❛s ●r❛♥❞❡s ✭♦✉ ✏▲❛r❣❡ ❊①tr❛ ❉✐♠❡♥s✐♦♥s✑✱ ▲❊❉✮✱ ❛s ❚❡♦r✐❛s ❝♦♠ ❉✐♠❡♥sõ❡s ❊①tr❛s ❯♥✐✈❡rs❛✐s ✭♦✉ ✏❯♥✐✈❡rs❛❧ ❊①tr❛ ❉✐♠❡♥s✐♦♥s✑✱ ❯❊❉✮ ❡ ❛s ❚❡♦r✐❛s ❝♦♠ ❉✐♠❡♥sõ❡s ❊①tr❛s ❈✉r✈❛s ✭♦✉ ✏❲❛r♣❡❞ ❊①tr❛ ❉✐♠❡♥s✐♦♥s✑✱ ❲❊❉✮✳ ❆s ❞✉❛s ♣r✐♠❡✐r❛s ♣♦ss✉❡♠ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ t♦t❛❧ ❝♦♠ ♠étr✐❝❛ ♣❧❛♥❛✱ ❡ ❛ ú❧t✐♠❛ ✉♠ ❡s♣❛ç♦✲t❡♠♣♦ ❝♦♠ ♠étr✐❝❛ ❝✉r✈❛❞❛✳
◆❡st❡ tr❛❜❛❧❤♦ t✐✈❡♠♦s ❛ ♣r❡♦❝✉♣❛çã♦ ✐♥✐❝✐❛❧ ❞❡ ❡♥t❡♥❞❡r♠♦s ♦ ❢✉♥❝✐♦♥❛♠❡♥t♦ ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❡ ❞❡ s❡✉s ♣r♦❜❧❡♠❛s ♣❛r❛ q✉❡ ♣❛ssáss❡♠♦s ❛♦ ❡st✉❞♦ ❞❡st❛s ♥♦✈❛s t❡♦r✐❛s✳ ❆ss✐♠✱ ♦ ❝❛♣ít✉❧♦ ✷ s❡ ❝♦♥❝❡♥tr❛ ❡♠ ❡♥❢❛t✐③❛r ♦s ❛s♣❡❝t♦s ♠❛✐s ✐♠♣♦rt❛♥t❡s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦✳ ◆❡st❡ ❝❛♣ít✉❧♦ ❡①♣❧✐❝✐t❛♠♦s t♦❞❛s ❛s ♣❛rtí❝✉❧❛s ♣r❡✈✐st❛s✱ ❛ss✐♠ ❝♦♠♦ s✉❛s ✐♥t❡r❛çõ❡s✳
◆♦ ❝❛♣ít✉❧♦ ✸ ❡♥tr❛r❡♠♦s ♥♦s ♣r♦❜❧❡♠❛s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦✳ ◆❡❧❡ ❡①♣❧✐❝❛♠♦s ❞❡t❛✲ ❧❤❛❞❛♠❡♥t❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ❤✐❡r❛rq✉✐❛ ♠❡♥❝✐♦♥❛❞♦ ❛ ♣♦✉❝♦✱ ❡ ♦ ♣r♦❜❧❡♠❛ ❞❛ ♠❛ss❛ ❞♦s ❢ér♠✐♦♥s✱ ❛❧é♠ ❞❡ ❝✐t❛r♠♦s ❛❧❣✉♥s ♦✉tr♦s ♣❛r❛ q✉❡ ♦ ❧❡✐t♦r ❡st❡❥❛ ❝✐❡♥t❡ ❞❡ s✉❛s ❡①✐stê♥❝✐❛s✳
❆s ❚❡♦r✐❛s ❝♦♠ ❉✐♠❡♥sõ❡s ❊①tr❛s sã♦ ✐♥tr♦❞✉③✐❞❛s ♥♦ ❝❛♣ít✉❧♦ ✹ ♣❛r❛ s♦❧✉❝✐♦♥❛r ❛❧❣✉♥s ❞♦s ♣r♦❜❧❡♠❛s ❞♦ ▼♦❞❡❧♦ P❛❞rã♦ ❞❡s❝r✐t♦s ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✳ ◆❡st❡ ❝❛♣í✲ t✉❧♦ ❝❛❞❛ ✈❡rt❡♥t❡ ❞❛ t❡♦r✐❛ ❡ s✉❛ ❢♦r♠❛ ❞❡ s♦❧✉❝✐♦♥❛r ♦ Pr♦❜❧❡♠❛ ❞❛ ❍✐❡r❛rq✉✐❛ é ❡①♣❧✐❝❛❞❛ ❝♦♠ ❞❡t❛❧❤❡s✱ ❡ ❛ ❚❡♦r✐❛ ❞❡ ❉✐♠❡♥sõ❡s ❊①tr❛s ❈✉r✈❛s é ❡s❝♦❧❤✐❞❛ ♣❛r❛ s❡r ❛ ❜❛s❡ ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦✳
❈❛♣ít✉❧♦ ✷
❖ ▼♦❞❡❧♦ P❛❞rã♦
✧❚❤❡ ♠♦r❡ s✉❝❝❡ss t❤❡ q✉❛♥t✉♠ t❤❡♦r② ❤❛s✱ t❤❡ s✐❧❧✐❡r ✐t ❧♦♦❦s✧ ✭❆❧❜❡rt ❊✐♥st❡✐♥✮
❖ ▼♦❞❡❧♦ P❛❞rã♦ ❞❛ ❢ís✐❝❛ ❞❡ ♣❛rtí❝✉❧❛s é ✉♠❛ t❡♦r✐❛ q✉❡ ❞❡s❝r❡✈❡ ❛s três ❢♦rç❛s ❢✉♥✲ ❞❛♠❡♥t❛✐s ✭❢♦rt❡✱ ❢r❛❝❛✱ ❡❧❡tr♦♠❛❣♥ét✐❝❛✮ ❡ s✉❛s ✐♥t❡r❛çõ❡s✳ ➱ ✉♠❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❞❡ ❝❛♠♣♦s ❞❡s❡♥✈♦❧✈✐❞❛ ♥❛ ❞é❝❛❞❛ ❞❡ ✼✵✱ ❡ t❡♥t❛ ❡①♣❧✐❝❛r ❛ ❞✐♥â♠✐❝❛ ❞♦ ✉♥✐✈❡rs♦ ❡♠ q✉❡ ✈✐✈❡♠♦s ❛tr❛✈és ❞❛ ♠❛tér✐❛ ❡ ❞❛s ❢♦rç❛s q✉❡ ❛❣❡♠ s♦❜r❡ ❡❧❛✳
◆♦ ▼♦❞❡❧♦ P❛❞rã♦ ❛ ♠❛tér✐❛ ❢✉♥❞❛♠❡♥t❛❧ ❢❡r♠✐ô♥✐❝❛ sã♦ ♦s q✉❛r❦s ❡ ♦s ❧é♣t♦♥s✱ q✉❡ ♣♦ss✉❡♠ s♣✐♥ ✶✴✷ ❡ s❡❣✉❡♠ ❛ ❡st❛tíst✐❝❛ ❞❡ ❋❡r♠✐✲❉✐r❛❝✳ ❖s q✉❛r❦s ❛✐♥❞❛ s♦❢r❡♠ ✐♥t❡r❛çõ❡s ❢♦rt❡s ❞❡✈✐❞♦ à s✉❛ ❝♦r✱ ♠❛s ♦s ❧é♣t♦♥s s♦❢r❡♠ ❛♣❡♥❛s ✐♥t❡r❛çõ❡s ❡❧❡tr♦❢r❛❝❛s✳ ❯♠ q✉❛❞r♦ ❣❡r❛❧ q✉❡ ♠♦str❛ ❛ ♠❛tér✐❛ ❢❡r♠✐ô♥✐❝❛ ♣♦❞❡ s❡r ✈✐st♦ ❡♠ ✭✷✳✹✮ ❡ ❡s♣❡❝✐✜❝❛ s❡✉s r❡s♣❡❝t✐✈♦s ♥ú♠❡r♦s q✉â♥t✐❝♦s✳
✷✳✶ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s
❆s s✐♠❡tr✐❛s s❡♠♣r❡ ♦❝✉♣❛r❛♠ ❧✉❣❛r❡s ❞❡ ❞❡st❛q✉❡ ♥❛ ❢ís✐❝❛✳ ❈♦♠ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ❢ís✐❝❛ ❞❡ ♣❛rtí❝✉❧❛s ♥ã♦ ❢♦✐ ❞✐❢❡r❡♥t❡✳
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✹
♠♦str❛ q✉❡ ❧❡✐s ❞❡ ❝♦♥s❡r✈❛çã♦ sã♦ ❝♦♥s❡qüê♥❝✐❛s ❞❛s s✐♠❡tr✐❛s ❞❡ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ♠♦❞❡❧♦✳
❆ ❊❧❡tr♦❞✐♥â♠✐❝❛ ◗✉â♥t✐❝❛✱ ♦✉ ◗❊❉ ✭✏◗✉❛♥t✉♠ ❊❧❡tr♦❞②♥❛♠✐❝s✑✮ é ✉♠ ❡①❡♠♣❧♦ ❞✐st♦✳ ◆❡❧❛✱ ♦ ♣r✐♥❝í♣✐♦ ❞❡ ✐♥✈❛r✐â♥❝✐❛ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❧♦❝❛✐s ❞❡ ❣❛✉❣❡ ✭❞♦ ❣r✉♣♦
U(1)✮ ✐♠♣❧✐❝❛ ♥❛ ❡①✐stê♥❝✐❛ ❞♦s ❝❛♠♣♦s ❞❡ ❣❛✉❣❡ ❝♦♠ ❞❡t❡r♠✐♥❛❞❛s ♣r♦♣r✐❡❞❛❞❡s
q✉❡ ❝♦♥❤❡❝❡♠♦s✱ ❝♦♠♦ ♣♦❞❡r❡♠♦s ✈❡r ❛ s❡❣✉✐r✶✳
✷✳✶✳✶ ❖ ♣r✐♥❝í♣✐♦ ❞❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ ❣❛✉❣❡
❙❡ ❝♦♥s✐❞❡r❛r♠♦s ✉♠❛ t❡♦r✐❛ ❝♦♠ ❢ér♠✐♦♥s ❝❛rr❡❣❛❞♦s✱ ❛ ❡q✉❛çã♦ ❞❡ ♠♦✈✐♠❡♥t♦ ♣❛r❛ ♦ ❝❛♠♣♦ ❧✐✈r❡ ❞❡✈❡ s❡r ❛ ❡q✉❛çã♦ ❞❡ ❉✐r❛❝✱ q✉❡ ♣♦❞❡ s❡r ♦❜t✐❞❛ ❛tr❛✈és ❞❡ s✉❛ ❧❛❣r❛♥❣❡❛♥❛✿
L0 = ¯ψ(x)(i6∂−m)ψ(x) ✭✷✳✶✮
❊st❛ ❧❛❣r❛♥❣❡❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜r❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❢❛s❡ ❣❧♦❜❛❧ψ →eiαψ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ♥❛ ❝♦♥s❡r✈❛çã♦ ❞❛ ❝♦rr❡♥t❡ ❞❡ ❉✐r❛❝✿
∂µjµ= 0
♦♥❞❡ jµ(x) = ¯ψ(x)γµψ(x) é ❛ ❝♦rr❡♥t❡ ❞❡ ❉✐r❛❝✳
◆♦ ❡♥t❛♥t♦✱ ❛ ❧❛❣r❛♥❣❡❛♥❛ ✭✷✳✶✮ ♥ã♦ é ✐♥✈❛r✐❛♥t❡ s♦❜r❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ ❢❛s❡ ❧♦❝❛❧ ψ → eiα(x)ψ ❞❡✈✐❞♦ ❛♦s t❡r♠♦s q✉❡ ❡♥✈♦❧✈❡♠ ❛s ❞❡r✐✈❛❞❛s✳ P❛r❛ ❣❡r❛r♠♦s
❡st❛ ✐♥✈❛r✐â♥❝✐❛✱ ♦s t❡r♠♦s ❞❡r✐✈❛t✐✈♦s ❞❛ ❧❛❣r❛♥❣❡❛♥❛ ❞❡✈❡♠ s❡r s✉❜st✐t✉í❞♦s ♣❡❧❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡✱ ❞❡✜♥✐❞❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
∂µ→Dµ≡∂µ+iqAµ
♦♥❞❡ q é ❛ ❝❛r❣❛ ❞❛ ♣❛rtí❝✉❧❛✱ ❡ Aµ é ♦ ♣♦t❡♥❝✐❛❧ ✈❡t♦r✳ ❆ ❧❛❣r❛♥❣❡❛♥❛ ✭✷✳✶✮ ✜❝❛✿
L0 = ¯ψ(x)(i6D−m)ψ(x) = ¯ψ(x)[iγµ(∂µ+iqAµ)−m]ψ(x) ❡ é ❛❣♦r❛ ✐♥✈❛r✐❛♥t❡ s♦❜r❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❧♦❝❛✐s
ψ(x) → ψ′(x) = eiα(x)ψ(x)
Aµ(x) → Aµ′ =Aµ(x)− 1
q∂
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✺
◆♦t❡ q✉❡ ♦ t❡r♠♦
Lint=−qψγ¯ µAµψ
♣♦❞❡ s❡r ✐❞❡♥t✐✜❝❛❞♦ ❝♦♠♦ ♦ t❡r♠♦ ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s ❝❛♠♣♦s ❞♦ ❢ót♦♥ ❡ ❞♦ ❧é♣t♦♥✳
❈♦♠♦ ❝♦♥s❡qüê♥❝✐❛ ♣♦❞❡♠♦s ❡♥①❡r❣❛r ❞♦✐s ❢❛t♦s ❞❡ ❡①tr❡♠❛ ✐♠♣♦rtâ♥❝✐❛✿
✶✳ ❆♦ ✐♠♣♦r♠♦s ❛ ✐♥✈❛r✐â♥❝✐❛ ❧♦❝❛❧ ♥❛ ❧❛❣r❛♥❣❡❛♥❛✱ ♦❜s❡r✈❛♠♦s ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ♥♦✈♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧ ♥❛ t❡♦r✐❛✿ ♦ ❝❛♠♣♦ ❞❡ ❣❛✉❣❡✳
✷✳ ❊st❛ s✐♠❡tr✐❛ ❡s♣❡❝✐✜❝❛ ❛ ♥❛t✉r❡③❛ ❞♦s t❡r♠♦s ❞❡ ✐♥t❡r❛çã♦ ❡♥tr❡ ♦s ❝❛♠♣♦s✳
❆ ✐❞é✐❛ ❡♥tã♦ ❞❡ ❙❛❧❛♠ ❡ ❲❛r❞ ❬✹✶❪ ❡r❛ ❣❡♥❡r❛❧✐③❛r ❡st❡ ♣r✐♥❝í♣✐♦ ♣❛r❛ ♦✉tr❛s ✐♥t❡✲ r❛çõ❡s✱ ♦✉ s❡❥❛✱ ❣❡r❛r ♦s t❡r♠♦s ❞❡ ✐♥t❡r❛çõ❡s ❢♦rt❡s✱ ❢r❛❝❛s ❡ ❡❧❡tr♦❢r❛❝❛s ❛tr❛✈és ❞❡ ❞❡t❡r♠✐♥❛❞❛s s✐♠❡tr✐❛s ✐♠♣♦st❛s ♥♦ ♠♦❞❡❧♦✳
➱ ♥❛t✉r❛❧ ♣❡♥s❛r♠♦s q✉❡ ❛ ❧❛❣r❛♥❣❡❛♥❛ t♦t❛❧ ❞❛ t❡♦r✐❛ t❡♥❤❛ ✉♠ t❡r♠♦ ❝✐♥ét✐❝♦✱ ♣r♦♣♦r❝✐♦♥❛❧ ❛ ❞❡r✐✈❛❞❛ ❞♦ ❝❛♠♣♦ ❡ ✉♠ t❡r♠♦ ❞❡ ♠❛ss❛ ❞♦ t✐♣♦AµAµ✳ ▼❛s ♥❡st❡ ❝❛s♦✱ ✉♠ t❡r♠♦ ❞❡ ♠❛ss❛ ❞❡st❡ t✐♣♦ ♥ã♦ é ✐♥✈❛r✐❛♥t❡ s♦❜ ❛s tr❛♥s❢♦r♠❛çõ❡s ❧♦❝❛✐s ♠♦str❛❞❛s ❡♠ ✭✷✳✷✮✱ ♦ q✉❡ ♥♦s ♠♦str❛ q✉❡ ❛ ♠❛ss❛ ❞♦ ❢ót♦♥ ❞❡✈❡ s❡r ♥✉❧❛✳ P♦rt❛♥t♦✱ ❞❛❞♦ q✉❡ ♦ t❡♥s♦r ❡❧❡tr♦♠❛❣♥ét✐❝♦ Fµν é ✐♥✈❛r✐❛♥t❡ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ ✭✷✳✷✮✱ ♦ t❡r♠♦ ❝✐♥ét✐❝♦ ♣♦❞❡ s❡r ❞❛❞♦ ♣♦r✿
LA=− 1 4FµνF
µν ✭✷✳✸✮
♦♥❞❡Fµν ≡∂µAν−∂νAµ✳ ❈♦♠ ✐st♦✱ ❛ s♦♠❛ ❞❡ ✭✷✳✶✮ ❝♦♠ ✭✷✳✸✮ ❞❡s❝r❡✈❡ ❛ ❊❧❡tr♦❞✐✲ ♥â♠✐❝❛ ◗✉â♥t✐❝❛✳
❆ ✐❞é✐❛ s❡❣✉✐♥t❡✱ ♣r♦♣♦st❛ ✐♥✐❝✐❛❧♠❡♥t❡ ♣♦r ❨❛♥❣ ❡ ▼✐❧❧s✱ ❢♦✐ ❡st❡♥❞❡r ❡st❡ t✐♣♦ ❞❡ t❡♦r✐❛ ♣❛r❛ ❣r✉♣♦s ♥ã♦✲❛❜❡❧✐❛♥♦s✳ ❙✉♣♦♥❤❛ q✉❡ ❛ ❧❛❣r❛♥❣❡❛♥❛ é ❛❣♦r❛ ❞❛❞❛ ♣❡❧❛ s♦♠❛ ❞❛s ❡q✉❛çõ❡s ✭✷✳✶✮ ❡ ✭✷✳✸✮✿
Lcampos= ¯ψ(x)(i6D−m)ψ(x)− 1 4FµνF
µν.
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✻
❡ ✐♥tr♦❞✉③✐r ♦s ❝❛♠♣♦s Ψ(x)
Ψ =
Ψ1
Ψ2
✳✳✳
Ψn
q✉❡ ❡stã♦ ♥❛ r❡♣r❡s❡♥t❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ ❡ s❡ tr❛♥s❢♦r♠❛♠ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿
Ψ(x)→Ψ′(x)≡e−iTaαa(x)Ψ(x) =U(x)Ψ(x)
♦♥❞❡ Ta é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❝♦♥✈❡♥✐❡♥t❡ ❞♦s ❣❡r❛❞♦r❡sta✳ ❆ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡ ♥❡st❡ ❝❛s♦ é ❞❛❞❛ ♣♦r✿
Dµ ≡∂µ−igTaAaµ
♦ q✉❡ ✐♠♣❧✐❝❛ ♥❛ ✐♥✈❛r✐â♥❝✐❛ ❞❡ Lcampos s♦❜r❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ ❧♦❝❛✐s✳ ◆❡st❡ ❝❛s♦✿
TaAa µ→U
TaAa
µ+ 1
g∂µ
U−1
♦✉ ❡①♣❛♥❞✐♥❞♦U ❡♠ s✉❛ ❢♦r♠❛ ✐♥✜♥✐t❡s✐♠❛❧ ✭U ≃1−iTaαa(x)✮✿
Aaµ′ =Aaµ− 1 g∂µα
a+C
abcαbAcµ
❈♦♠♣❛r❛♥❞♦ ❝♦♠ ❛ ❡①♣r❡ssã♦ ♣❛r❛ ❛ ◗❊❉ ❝♦rr❡s♣♦♥❞❡♥t❡ ✭❡q✉❛çã♦ ✭✷✳✷✮✮✱ ✈❡♠♦s ❡①♣❧✐❝✐t❛♠❡♥t❡ q✉❡ ♦ t❡r❝❡✐r♦ t❡r♠♦ é ♥♦✈♦✱ ❝♦♥s❡qüê♥❝✐❛ ❞✐r❡t❛ ❞❛ s✐♠❡tr✐❛ ♥ã♦✲ ❛❜❡❧✐❛♥❛✳ ❈♦♠ ✐st♦✱ ♦ t❡♥s♦r Fµν
Fµνa ≡∂µAaν −∂νAaµ+gCabcAbµAcν
s❡ tr❛♥s❢♦r♠❛ ❝♦♠♦Fa′
µν →Fµνa +CabcαbFµνc ✳
❈♦♥s❡qü❡♥t❡♠❡♥t❡✱ ♦ t❡r♠♦ ❝✐♥ét✐❝♦ ❞❛ ❧❛❣r❛♥❣❡❛♥❛ ♣❛r❛ ♦s ❜ós♦♥s ❞❡ ❣❛✉❣❡ ✭❝♦r✲ r❡s♣♦♥❞❡♥t❡s à ✭✷✳✸✮ ❞❛ ◗❊❉✮ ♣♦❞❡ s❡r ❡s❝r✐t♦ ❝♦♠♦✿
LA=− 1 4F
a
µνFa µν ✭✷✳✹✮
♠❛s ♦ t❡r♠♦ ❞❡ ♠❛ss❛ ✭Aa
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✼
✷✳✶✳✷ ❆ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❡ s✐♠❡tr✐❛
❱✐♠♦s q✉❡ s✐♠❡tr✐❛s ✐♠♣❧✐❝❛♠ ❡♠ ❧❡✐s ❞❡ ❝♦♥s❡r✈❛çã♦✳ ◆❡st❡s ❝❛s♦s✱ ❛ ❧❛❣r❛♥❣❡❛♥❛ ❡ ♦ ✈á❝✉♦ ✭♠❡♥♦r ❡st❛❞♦ ❞❡ ❡♥❡r❣✐❛✮ ❞❛ t❡♦r✐❛ sã♦ ✐♥✈❛r✐❛♥t❡s✱ ♠❛s ♣♦❞❡ ❡①✐st✐r ✉♠❛ s✐t✉❛çã♦ ❡♠ q✉❡ ❛ ❧❛❣r❛♥❣❡❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜ ✉♠❛ s✐♠❡tr✐❛ q✉❡ ♥ã♦ ❞❡✐①❛ ♦ ✈á❝✉♦ s❡r ✐♥✈❛r✐❛♥t❡✳ ❙❡ ✐st♦ ♦❝♦rr❡ ❞✐③❡♠♦s q✉❡ ❛ s✐♠❡tr✐❛ ❢♦✐ q✉❡❜r❛❞❛✱ ♦✉ ❡s❝♦♥❞✐❞❛✱ ❞❡ ❢♦r♠❛ q✉❡ ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠ ♦✉tr♦ ✈á❝✉♦✱ ♦♥❞❡ ❛♠❜♦s sã♦ ✐♥✈❛r✐❛♥t❡s✳
❱❛♠♦s r❡s♦❧✈❡r ✉♠ ❡①❡♠♣❧♦ s✐♠♣❧❡s✱ ♣❛r❛ ❞❡✐①❛r ❝❧❛r♦ ❝♦♠♦ ❢✉♥❝✐♦♥❛✳ ❈♦♥s✐❞❡r❡ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r r❡❛❧ φ✱ ❝✉❥❛ ❧❛❣r❛♥❣❡❛♥❛ é ❞❛❞❛ ♣♦r✿
Lφ = 1
2(∂µφ)(∂ µφ)
−V(φ) ✭✷✳✺✮
♦♥❞❡ ♦ ♣♦t❡♥❝✐❛❧ ❡❢❡t✐✈♦ éV(φ) = 1 2µ
2φ2+ 1 4λφ
4 ❝♦♠ λ >0✳
❊st❛ ❧❛❣r❛♥❣❡❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜r❡ ❛ tr❛♥s❢♦r♠❛çã♦ ❞❡ r❡✢❡①ã♦
φ→ −φ, ✭✷✳✻✮
♥♦ ❡♥t❛♥t♦✱ ♣♦❞❡♠♦s ❛✜r♠❛r ♦ ♠❡s♠♦ s♦❜r❡ ♦ ✈á❝✉♦❄ ❆♦ ❝❛❧❝✉❧❛r♠♦s ❡st❡ ✈á❝✉♦✱ ♥♦s ❞❡♣❛r❛♠♦s ❝♦♠ ❞✉❛s s✐t✉❛çõ❡s ❞✐st✐♥t❛s✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ s✐♥❛❧ ❞♦ ❝♦❡✜❝✐❡♥t❡µ2✿
✶✳ ❙❡ µ2 > 0✱ ♦ ♣♦t❡♥❝✐❛❧ é ❛q✉❡❧❡ ♠♦str❛❞♦ ♥❛ ✜❣✉r❛ ✭✷✳✶❛✮✳ ◆❡st❡ ❝❛s♦ ✈❡♠♦s
❝❧❛r❛♠❡♥t❡ q✉❡ ♦ ✈á❝✉♦ é✿
hφi0 ≡ h0|φ|0i= 0 (µ2 >0)
q✉❡ é ✐♥✈❛r✐❛♥t❡ s♦❜r❡ ❛ tr❛♥s❢♦r♠❛çã♦ ♠♦str❛❞❛ ❡♠ ✭✷✳✻✮✳
✷✳ ❙❡ µ2 < 0✱ ♦ ♠í♥✐♠♦ ❞♦ ♣♦t❡♥❝✐❛❧ é ❛q✉❡❧❡ ✈✐st♦ ♥❛ ✜❣✉r❛ ✭✷✳✶❜✮✱ ❡ ♣♦❞❡ s❡r
❝❛❧❝✉❧❛❞♦✿
hφi0 =±
r
−µ2
λ ≡ ±v (µ
2 <0). ✭✷✳✼✮
◆♦t❡ q✉❡ ♥❡st❡ ❝❛s♦ ♦ ✈á❝✉♦ é ❞❡❣❡♥❡r❛❞♦✱ ❡ ♣♦rt❛♥t♦ s✉❛ ✐♥✈❛r✐â♥❝✐❛ s♦❜r❡ ❛ tr❛♥s❢♦r♠❛çã♦ ✭✷✳✻✮ ❞❡✈❡ ❞❡♣❡♥❞❡r ❞❛ ❡s❝♦❧❤❛ ❡♥tr❡ +v ♦✉−v✳
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✽
V( )f
f
V( )f
f
(a) (b)
❋✐❣✉r❛ ✷✳✶✿ P♦t❡♥❝✐❛❧ ❡❢❡t✐✈♦V(φ) ♣❛r❛ ❛ ❧❛❣r❛♥❣❡❛♥❛ ♠♦str❛❞❛ ❡♠ ✭✷✳✺✮ ❝♦♠ µ2 >0 ❡♠ ✭❛✮ ❡µ2 <0 ❡♠ ✭❜✮✳
❱❛♠♦s ❞❡✜♥✐r ❝❛♠♣♦ξ✱ ♣❛r❛ q✉❡ ♦ ❡st❛❞♦ ❞❡ ✈á❝✉♦ ❡st❡❥❛ ♥♦ ③❡r♦✿
ξ(x)≡φ(x)− hφi0 =φ(x)−v
♦♥❞❡ hξi0 = 0✳
❆❣♦r❛✱ ❛ ❧❛❣r❛♥❣❡❛♥❛ ✭✷✳✶✮ ♣♦❞❡ s❡r r❡✲❡s❝r✐t❛ ❡♠ t❡r♠♦s ❞❡st❡ ♥♦✈♦ ❝❛♠♣♦✿
Lξ = 1
2(∂µφξ)(∂ µξ)
−λv2ξ2−λvξ3 −1
4λξ
4. ✭✷✳✽✮
q✉❡ é ❛ ❧❛❣r❛♥❣❡❛♥❛ ❞❡ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r ❧✐✈r❡ ❞❡ ♠❛ss❛mξ=
p
−2µ2✷✳
P♦rt❛♥t♦✱ ❛ s✐♠❡tr✐❛ ❡st❛✈❛ ❡s❝♦♥❞✐❞❛✱ ❡ ♣✉❞❡♠♦s ❡♥❝♦♥trá✲❧❛ r❡❞❡✜♥✐♥❞♦ ♦ ❝❛♠♣♦ ❢ís✐❝♦ ❞❛ ❧❛❣r❛♥❣❡❛♥❛✳ ❊st❡ é ✉♠ tí♣✐❝♦ ❡①❡♠♣❧♦ ♣❛r❛ ❡①♣❧✐❝❛r ♦ ❝♦♥❝❡✐t♦ ❞❛ s✐♠❡tr✐❛ ❡s❝♦♥❞✐❞❛✱ ❞❡♥tr♦ ❞❛ ❝❤❛♠❛❞❛ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❡ s✐♠❡tr✐❛✱ ♦♥❞❡ ❛ ❧❛❣r❛♥❣❡❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜r❡ ❞❡t❡r♠✐♥❛❞❛ ♦♣❡r❛çã♦✱ ♥♦ ❡♥t❛♥t♦ ♦ ✈á❝✉♦ ♥ã♦ é✳
✷✳✶✳✷✳✶ ❖s ❜ós♦♥s ❞❡ ●♦❧❞st♦♥❡
❈♦♥s✐❞❡r❡ ❛❣♦r❛ ✉♠ s✐st❡♠❛ ❢♦r♠❛❞♦ ♣♦r ❝❛♠♣♦s ❡s❝❛❧❛r❡s ❝♦♠♣❧❡①♦s✳ ❆ ❧❛❣r❛♥❣❡❛♥❛ ♣♦❞❡ s❡r ❡s❝r✐t❛ ❝♦♠♦✿
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✾
❊st❛ ❧❛❣r❛♥❣❡❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜r❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❢❛s❡ ❣❧♦❜❛✐s ❞♦ ❣r✉♣♦ ❞❡ s✐♠❡tr✐❛U(1)✿
φ(x)→φ′(x) = eiαφ(x) ✭✷✳✶✵✮
♦♥❞❡ α é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ①✳
❉❡✜♥✐♥❞♦ρ=φ†φ✱ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r ♦ ♣♦t❡♥❝✐❛❧ ❡❢❡t✐✈♦ ❞❡ ✭✷✳✾✮ ❝♦♠♦✿
V(ρ) =µ2ρ+λρ2
❡ s❡♣❛r❛r ♦ r❡s✉❧t❛❞♦ ❡♠ ❞♦✐s ❝❛s♦s✱ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ✜③❡♠♦s ❛♥t❡r✐♦r♠❡♥t❡✿
✶✳ ❙❡ µ2 > 0✱ ♦ ♠í♥✐♠♦ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡✈❡ ❡st❛r ❡♠ ρ = φ = 0, ❡ ♦ ✈á❝✉♦ é
s✐♠étr✐❝♦ s♦❜r❡ ❡st❛s tr❛♥s❢♦r♠❛çõ❡s ❣❧♦❜❛✐s✳
✷✳ ❙❡ µ2 <0✱ ♦ ♠í♥✐♠♦ ❞❡✈❡ ❡st❛r ♥♦ ♣❧❛♥♦ ❝♦♠♣❧❡①♦✱ ❡♠ ✉♠ ❝ír❝✉❧♦ ❞❡ r❛✐♦
|φ|=
r
−µ2
2λ ≡ v
√
2.
P♦❞❡♠♦s ❡s❝♦❧❤❡r ✉♠ ❞♦s ♣♦♥t♦s ❞❡ ✈á❝✉♦ ♣♦ssí✈❡✐s ❡ ❡①♣❛♥❞✐r ❛ ❧❛❣r❛♥❣❡❛♥❛ ❡♠ s❡✉ ❡♥t♦r♥♦✱ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ✜③❡♠♦s ❡♠ ✭✷✳✽✮✳ ❊s❝♦❧❤❡♥❞♦ ❝♦♠♦ ♦ ✈á❝✉♦✱ ♦ ♣♦♥t♦ ❞❡ ♠í♥✐♠♦ ❞♦ ♣♦t❡♥❝✐❛❧ ♥♦ ❡✐①♦ r❡❛❧ ✭Re(φ) = √v
2✮✱ ♣♦❞❡♠♦s r❡❞❡✜♥✐r✿
φ(x) = √1
2[v+ξ(x) +iX(x)]
❡ r❡❡s❝r❡✈❡r ❛ ❧❛❣r❛♥❣❡❛♥❛
L= 1 2(∂µξ)
2+ 1
2(∂µX)
2
−λv2ξ2−λvξ(ξ2+X2)− 1 4λ(ξ
2+
X2)2+cte. ✭✷✳✶✶✮
❱❡❥❛ q✉❡ ♣♦❞❡♠♦s ✐♥t❡r♣r❡t❛r ✭✷✳✶✶✮ ❝♦♠♦ ✉♠❛ ❧❛❣r❛♥❣❡❛♥❛ ❞❡ ✉♠❛ t❡♦r✐❛ q✉â♥t✐❝❛ ❞❡ ❞♦✐s ❝❛♠♣♦s✿ ξ ❡ X✳ ❖ ❝❛♠♣♦ X ♥ã♦ ❞❡✈❡ t❡r ♠❛ss❛✱ ♣♦✐s ♥ã♦ ❡①✐st❡ ♥❡♥❤✉♠
t❡r♠♦ ♣r♦♣♦r❝✐♦♥❛❧ ❛ s♦♠❡♥t❡X2✳ ❏á ❝♦♠ r❡❧❛çã♦ à ♠❛ss❛ ❞❡ξ✱ ❡st❛ ❞❡✈❡ s❡r ❞❛❞❛
♣♦r✿
mξ =
√
2λv2 =p−2µ2 (µ2 <0)
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✶✵
❆ss✐♠✱ ❛ ❧❛❣r❛♥❣❡❛♥❛ ❞❡ ✉♠❛ t❡♦r✐❛ ❝♦♠ ✉♠ ❝❛♠♣♦ ❡s❝❛❧❛r ❝♦♠♣❧❡①♦ ♣ô❞❡ s❡r r❡✲❡s❝r✐t❛ r❡❞❡✜♥✐♥❞♦ ❡st❡ ❝❛♠♣♦ ❡♠ ❢✉♥çã♦ ❞❡ ♦✉tr♦s ❞♦✐s✱ ♦ q✉❡ ❛ss❡❣✉r♦✉ ❛ ✐♥✈❛✲ r✐â♥❝✐❛ s♦❜r❡ ❛s tr❛♥s❢♦r♠❛çõ❡s ✭✷✳✶✵✮✳ ❈♦♠ ✐st♦✱φ♣♦❞❡rá s❡r ♦❜s❡r✈❛❞♦ ✜s✐❝❛♠❡♥t❡
❛tr❛✈és ❞♦s ❝❛♠♣♦sξ ❡X✱ q✉❡ ❞❡✈❡♠ ♣♦ss✉✐r ❛♦ t♦❞♦ ♦ ♠❡s♠♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡
❧✐❜❡r❞❛❞❡ ❞♦ ♣ró♣r✐♦φ✳
❖ t❡♦r❡♠❛ ❞❡ ●♦❧❞st♦♥❡ r❡s✉♠❡ ❡st❛ ✐❞é✐❛ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✿ ✏❙❡ ✉♠❛ s✐♠❡tr✐❛ ❣❧♦❜❛❧ ❝♦♥tí♥✉❛ é ❡s♣♦♥t❛♥❡❛♠❡♥t❡ q✉❡❜r❛❞❛✱ ♣❛r❛ ❝❛❞❛ ❣❡r❛❞♦r ❞♦ ❣r✉♣♦ q✉❡❜r❛❞♦✱ ❞❡✈❡ ❛♣❛r❡❝❡r ♥❛ t❡♦r✐❛ ✉♠❛ ♣❛rtí❝✉❧❛ s❡♠ ♠❛ss❛✳✑
❊st❛s ♣❛rtí❝✉❧❛s s❡♠ ♠❛ss❛ ❛ q✉❡ ♦ t❡♦r❡♠❛ s❡ r❡❢❡r❡ ✭r❡❧❛❝✐♦♥❛❞❛s ❝♦♠ ♦ ❝❛♠♣♦
X(x)♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛✮ sã♦ ❝❤❛♠❛❞❛s ❞❡ ❜ós♦♥s ❞❡ ●♦❧❞st♦♥❡✳
✷✳✶✳✸ ❖ ♠❡❝❛♥✐s♠♦ ❞❡ ❍✐❣❣s
❖ t❡♦r❡♠❛ ❞❡ ●♦❧❞st♦♥❡ ✐♠♣❧✐❝❛ ♥❛ ❡①✐stê♥❝✐❛ ❞❡ ♣❛rtí❝✉❧❛s ❡s❝❛❧❛r❡s ♥ã♦ ♠❛ss✐✈❛s✱ ❡ s✉r❣❡ ❛ ♣❛rt✐r ❞❡ ✉♠❛ t❡♦r✐❛ ✐♥✈❛r✐❛♥t❡ s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ ❣❧♦❜❛✐s✳ ❖ ♠❡❝❛♥✐s♠♦ ❞❡ ❍✐❣❣s s❡ ❜❛s❡✐❛ ♥❡st❛ ✐❞é✐❛ ❞♦s ❜ós♦♥s ❞❡ ●♦❧❞st♦♥❡✱ ❡ ❢♦✐ ❡❧❛❜♦r❛❞❛ ♣❛r❛ t❡♦r✐❛s ✐♥✈❛r✐❛♥t❡s s♦❜ tr❛♥s❢♦r♠❛çõ❡s ❞❡ ❣❛✉❣❡ ❧♦❝❛✐s✳ ◆❡st❡✱ ♦s ❝❛♠♣♦s ❞❡ ❣❛✉❣❡ s❡♠ ♠❛ss❛ ❞❡✈❡♠ s❡r ❝♦♠❜✐♥❛❞♦s ❛♦s ❜ós♦♥s ❞❡ ●♦❧❞st♦♥❡✱ ❞❛♥❞♦ ♦r✐❣❡♠ ❛ ❝❛♠♣♦s ❞❡ ❣❛✉❣❡ ♠❛ss✐✈♦s✳ ❱❛♠♦s r❡s♦❧✈❡r ✉♠ ❡①❡♠♣❧♦ ♣❛r❛ ❡♥t❡♥❞❡r♠♦s ❝♦♠♦ ♦ ♠❡❝❛♥✐s♠♦ ♦❝♦rr❡✳
❱❛♠♦s ❝♦♥s✐❞❡r❛r ✉♠❛ t❡♦r✐❛ ❝♦♠ ❡s❝❛❧❛r❡s ❝♦♠♣❧❡①♦s ✐♥✈❛r✐❛♥t❡ s♦❜r❡ tr❛♥s❢♦r✲ ♠❛çõ❡s ❧♦❝❛✐s ❞♦ ❣r✉♣♦ ❯✭✶✮✱ ❝♦♠♦ ♥❛ s❡çã♦ ✭✷✳✶✳✶✮✳ ◆❡st❡ ❝❛s♦✱ ❛ ❧❛❣r❛♥❣❡❛♥❛ é ❞❛❞❛ ♣♦r✿
L= (Dµφ)†(Dµφ)−µ2φ†φ−λ(φ†φ)2− 1 4FµνF
µν ✭✷✳✶✷✮
♦♥❞❡ λ >0✱ φ é ✉♠ ❝❛♠♣♦ ❝♦♠♣❧❡①♦ ❡✿
Dµ = ∂µ+iqAµ
Fµν = ∂µAν −∂νAµ.
❆ ❧❛❣r❛♥❣❡❛♥❛ é ✐♥✈❛r✐❛♥t❡ s♦❜r❡ tr❛♥s❢♦r♠❛çõ❡s ❞❡ r♦t❛çã♦✿
φ(x) → φ′(x) =eiqα(x)φ(x)
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✶✶
P♦rt❛♥t♦✱ ❞❡♣❡♥❞❡♥❞♦ ❞♦ ✈❛❧♦r ❞❡ µ2 ♦❜t❡r❡♠♦s ✉♠❛ ❞❛s s✐t✉❛çõ❡s ❞❡s❝r✐t❛s ♥♦s
❡①❡♠♣❧♦s ❛♥t❡r✐♦r❡s✿
✶✳ ❙❡ µ2 >0✱ ♦ ♠í♥✐♠♦ ❞♦ ♣♦t❡♥❝✐❛❧ é ú♥✐❝♦ ❡ ❡stá ❡♠ φ = φ† = 0✳ ◆❡st❡ ❝❛s♦✱
❛ s✐♠❡tr✐❛ ❞❛ ❧❛❣r❛♥❣❡❛♥❛ é ❛ ♠❡s♠❛ s✐♠❡tr✐❛ ❞♦ ❡st❛❞♦ ❞❡ ♠❡♥♦r ❡♥❡r❣✐❛✱ ❡ ❛ t❡♦r✐❛ é ❢♦r♠❛❞❛ ♣♦r ❜ós♦♥s ❞❡ ❣❛✉❣❡ ♥ã♦ ♠❛ss✐✈♦s ❡ ❝❛♠♣♦s ❡s❝❛❧❛r❡s ❞❡ ♠❛ss❛ µ✳
✷✳ ❙❡ µ2 <0✱ ♦ ✈á❝✉♦ é ❞❡❣❡♥❡r❛❞♦✱ ❡ ♦ ♠í♥✐♠♦ ❞♦ ♣♦t❡♥❝✐❛❧ ❞❡✈❡ ❡st❛r ❡♠✿
|φ|2 =−µ
2
2λ ≡ v2
2 .
❊s❝♦❧❤❡♥❞♦ ♦ s❡❣✉✐♥t❡ ✈á❝✉♦
hφi0 = √v 2
✭❝♦♠ ✈ r❡❛❧ ❡ ♣♦s✐t✐✈♦✮ ❡ ❡①♣❛♥❞✐♥❞♦ ❡♠ ❝♦♦r❞❡♥❛❞❛s ♣♦❧❛r❡s✱
φ(x) = √1
2[v+η(x)]e
iξ(x)/v =
= √1
2[v+η(x) +iξ(x) +...].
❙✉❜st✐t✉✐♥❞♦ ♥❛ ❧❛❣r❛♥❣❡❛♥❛✱ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ✜③❡♠♦s ❡♠ ✭✷✳✶✶✮✱ t❡♠♦s✿
L = 1
2(∂µη)(∂ µη)
− 122)η2+1
2(∂µξ)(∂ µξ)
− 14FµνFµν +
+ q
2v2
2 AµA
µ+qvA
µ∂µξ+int. ✭✷✳✶✸✮
❉❡st❛ ❢♦r♠❛✱ ❛tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞♦s t❡r♠♦s ❞❡st❛ ❧❛❣r❛♥❣❡❛♥❛ ♣♦❞❡♠♦s r❡✲✐♥t❡r♣r❡t❛r ❡st❛ t❡♦r✐❛ ❝♦♠♦ ✉♠❛ t❡♦r✐❛ ❞❡ três ❝❛♠♣♦s✿ ✉♠ ❡s❝❛❧❛r ♠❛ss✐✈♦✱ ❞❡ ♠❛ss❛ mη =
p
−2µ2❀ ✉♠ ❡s❝❛❧❛r s❡♠ ♠❛ss❛✱ m
ξ = 0❀ ❡ ✉♠ ❜ós♦♥ ❞❡ ❣❛✉❣❡ ❝♦♠ ♠❛ss❛ mA=qv✳ ◆♦ ❡♥t❛♥t♦✱ ♦ ú❧t✐♠♦ t❡r♠♦ ❞❡ ✭✷✳✶✸✮ é ✉♠ t❛♥t♦ ✐♥❝♦♥✈❡♥✐❡♥t❡✱ ❥á q✉❡ ♠✐st✉r❛ ❛ ♣❛rtí❝✉❧❛ r❡❧❛❝✐♦♥❛❞❛ ❛♦ ❝❛♠♣♦ξ ❝♦♠ ♦ ❜ós♦♥ r❡❧❛❝✐♦♥❛❞♦ à Aµ✳
✷✳✶✳ ❚❡♦r✐❛s ❞❡ ❣❛✉❣❡ ❡ s✉❛s s✐♠❡tr✐❛s ✶✷ ▲❛❣r❛♥❣❡❛♥❛ ■♥✐❝✐❛❧ ✭✷✳✶✷✮ ▲❛❣r❛♥❣❡❛♥❛ ❋✐♥❛❧ ✭✷✳✶✸✮
❝❛♠♣♦ ❡s❝❛❧❛rφ✿ ✷ ❝❛♠♣♦ ❡s❝❛❧❛r η ✿ ✶
❝❛♠♣♦ ✈❡t♦r✐❛❧Aµ s❡♠ ♠❛ss❛✿ ✷ ❝❛♠♣♦ ❡s❝❛❧❛r ξ✿ ✶ ❈❛♠♣♦ ✈❡t♦r✐❛❧Aµ ♠❛ss✐✈♦✿ ✸
❚❖❚❆▲✿ ✹ ❚❖❚❆▲✿ ✺
❚❛❜❡❧❛ ✷✳✶✿ ❈♦♥t❛❣❡♠ ❞♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❛ t❡♦r✐❛ ❛♥t❡s ❡ ❞❡♣♦✐s ❞❛ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡tr✐❛✳
◆❛ r❡❛❧✐❞❛❞❡✱ ❡st❛ ❞✐❢❡r❡♥ç❛ é ❢r✉t♦ ❞❡ ✉♠❛ ✐❧✉sã♦ ❡ ♣♦❞❡ s❡r ❝♦rr✐❣✐❞❛ ♣♦r ✉♠❛ ❡s❝♦❧❤❛ ❝♦♥✈❡♥✐❡♥t❡ ❞♦ ❣❛✉❣❡✿
α(x) =− 1
qvξ(x) ✭✷✳✶✹✮
❈♦♠ ✐st♦✱ ✭✷✳✶✸✮ ✜❝❛✿
L= 1
2(∂µη)(∂
µη) +µ2η2+1
2q
2v2A
µAµ− 1 4FµνF
µν ✭✷✳✶✺✮
❡ ♦ ❜ós♦♥ ❞❡ ●♦❧❞st♦♥❡ ❞❡st❛ t❡♦r✐❛ s♦♠❡✳ ❉✐③❡♠♦s q✉❡ ❡❧❡ ❢♦✐ ❛❜s♦r✈✐❞♦ ♣❡❧♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧Aµ♣❛r❛ q✉❡ ❡❧❡ s❡ t♦r♥❡ ♠❛ss✐✈♦✳ ❈♦♥t❛♥❞♦ ❛❣♦r❛ ♦s ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❛ t❡♦r✐❛✱ t❡♠♦s✿
▲❛❣r❛♥❣❡❛♥❛ ■♥✐❝✐❛❧ ✭✷✳✶✷✮ ▲❛❣r❛♥❣❡❛♥❛ ❋✐♥❛❧ ✭✷✳✶✺✮ ❝❛♠♣♦ ❡s❝❛❧❛r φ✿ ✷ ❝❛♠♣♦ ❡s❝❛❧❛r η ✿ ✶
❝❛♠♣♦ ✈❡t♦r✐❛❧Aµ s❡♠ ♠❛ss❛✿ ✷ ❝❛♠♣♦ ❡s❝❛❧❛r ξ✿ ✵ ❈❛♠♣♦ ✈❡t♦r✐❛❧Aµ ♠❛ss✐✈♦✿ ✸
❚❖❚❆▲✿ ✹ ❚❖❚❆▲✿ ✹
❚❛❜❡❧❛ ✷✳✷✿ ❈♦♥t❛❣❡♠ ❞♦ ♥ú♠❡r♦ ❞❡ ❣r❛✉s ❞❡ ❧✐❜❡r❞❛❞❡ ❞❛ t❡♦r✐❛ ❛♥t❡s ❡ ❞❡♣♦✐s ❞❛ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡tr✐❛ ❝♦♠ ♦ ❣❛✉❣❡ ❡s❝♦❧❤✐❞♦ ❡♠ ✭✷✳✶✹✮✳
❖ ❝❛♠♣♦ ❢ís✐❝♦ ❡s❝❛❧❛r ♠❛ss✐✈♦ηé ❝❤❛♠❛❞♦ ❞❡ ❝❛♠♣♦ ❞❡ ❍✐❣❣s✱ ❡ ♦ ❣❛✉❣❡ ❡s❝♦❧❤✐❞♦
❡♠ ✭✷✳✶✹✮ é ❞❡♥♦♠✐♥❛❞♦ ❣❛✉❣❡ ✉♥✐tár✐♦✳
❊st❛ ♠❡s♠❛ ❛♥á❧✐s❡ ♣♦❞❡ s❡r ❡st❡♥❞✐❞❛ ♣❛r❛ ❝❛♠♣♦s ♥ã♦✲❛❜❡❧✐❛♥♦s✱ ❝♦♠♦ ✜③❡♠♦s ♥❛ s❡çã♦ ✭✷✳✶✳✶✮✳ ❉❛ ♠❡s♠❛ ❢♦r♠❛✱ ♦s ❝❛♠♣♦s ❞❡ ❣❛✉❣❡ ❞❡✈❡rã♦ ❛❞q✉✐r✐r ♠❛ss❛ q✉❡❜r❛♥❞♦ ❛ s✐♠❡tr✐❛ ❞♦ ✈á❝✉♦ ❝♦♠ ❛ ❛❥✉❞❛ ❞♦s ❝❛♠♣♦s ❡s❝❛❧❛r❡s✳ ❆❧❣✉♠❛s ♣❛rt❡s ❞♦s ❝❛♠♣♦s ❡s❝❛❧❛r❡s ✈ã♦ ❞❛r ❧✉❣❛r à ❝♦♠♣♦♥❡♥t❡ ❧♦♥❣✐t✉❞✐♥❛❧ ❞♦ ❝❛♠♣♦ ✈❡t♦r✐❛❧
Ba
✷✳✷✳ ❖ ▼♦❞❡❧♦ ❊❧❡tr♦❢r❛❝♦ ✶✸
❡①✐st❡♥t❡s✱ é ♣♦ssí✈❡❧ q✉❡ ♣❛rt❡ ❞❡❧❡s ❛❞q✉✐r❛ ♠❛ss❛✱ ❡ ✐st♦ ❞❡✈❡ ♦❝♦rr❡r ❛tr❛✈és ❞❡st❡ ♠❡s♠♦ ♣r♦❝❡❞✐♠❡♥t♦ ❞❡ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡tr✐❛✳
✷✳✷ ❖ ▼♦❞❡❧♦ ❊❧❡tr♦❢r❛❝♦
❖ ▼♦❞❡❧♦ ❊❧❡tr♦❢r❛❝♦ ❞❡ ♣❛rtí❝✉❧❛s ❢♦✐ s❡♥❞♦ ♠♦♥t❛❞♦ ❝♦♠♦ ✉♠ q✉❡❜r❛✲❝❛❜❡ç❛s✱ ❡ ✉♠❛ ❞❛s ♣r✐♠❡✐r❛s ♣❡ç❛s q✉❡ ♦ ❧❡✈❛r❛♠ ❛ ♠♦♥t❛❣❡♠ ✜♥❛❧ ❢♦✐ ❡♥❝❛✐①❛❞❛ ♣♦r ●❧❛s❤♦✇ ❡♠ ✶✾✻✶ ❬✷✹❪✳ ❆ s✉❛ ✐❞é✐❛ ❡r❛ ✉t✐❧✐③❛r ♦ ❣r✉♣♦ ❞❡ ❣❛✉❣❡ SU(2)×U(1) ♣❛r❛ ✐♥❝❧✉✐r
❛s ✐♥t❡r❛çõ❡s ❢r❛❝❛s ❡ ❡❧❡tr♦♠❛❣♥ét✐❝❛s✱ ♦♥❞❡
Q=T3+Y /2 ✭✷✳✶✻✮
r❡❧❛❝✐♦♥❛♥❞♦ ❛ ❝❛r❣❛ ❡❧étr✐❝❛ ❝♦♠ ♦ ✐s♦s♣✐♥ ❡ ❛ ❤✐♣❡r❝❛r❣❛ ❧❡♣tô♥✐❝❛✳ ❊st❛ ♥♦✈❛ t❡♦r✐❛ ❞❡✈❡ ♣♦ss✉✐r q✉❛tr♦ ❜ós♦♥s ❞❡ ❣❛✉❣❡✿ ✉♠ tr✐♣❧❡t♦ ❛ss♦❝✐❛❞♦ ❝♦♠ ♦s ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦ SU(2) ❡ ✉♠ s✐♥❣❧❡t♦ ❛ss♦❝✐❛❞♦ ❛♦ ❣r✉♣♦ U(1)✳ ❖s ❜ós♦♥s ❝❛rr❡❣❛❞♦s
✭W±✮ ❞❡✈❡♠ s❡r ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❡♥tr❡ ❞♦✐s ❜ós♦♥s ❞♦ tr✐♣❧❡t♦✱ ❡ ♦s ♥❡✉tr♦s ✭Z0 ❡
♦ ❢ót♦♥✮ ✉♠❛ ❝♦♠❜✐♥❛çã♦ ❞♦s ♦✉tr♦s ❞♦✐s ❜ós♦♥s r❡st❛♥t❡s✳ ❊♠ ✶✾✻✼✱ ❲❡✐♥❜❡r❣ ❡ ❙❛❧❛♠ ❬✹✹✱ ✹✵❪ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ♣r♦♣✉s❡r❛♠ q✉❡ ❛s ♠❛ss❛s ❞❡W± ❡Z0 ♣♦❞❡r✐❛♠
s❡r ♦❜t✐❞❛s ❛ ♣❛rt✐r ❞♦ ♠❡❝❛♥✐s♠♦ ❞❡ ❍✐❣❣s ❡ ❞❛ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡tr✐❛✳ ❋✐♥❛❧♠❡♥t❡ ❡♠ ✶✾✼✶✱ ✬t ❍♦♦❢t ❬✹✷✱ ✹✸❪ ❝♦❧♦❝♦✉ ❛ ú❧t✐♠❛ ♣❡ç❛ ♠♦str❛♥❞♦ q✉❡ ❛ t❡♦r✐❛ ❡r❛ r❡♥♦r♠❛❧✐③á✈❡❧✳
✷✳✷✳✶ ❆ ▲❛❣r❛♥❣❡❛♥❛ ♣❛r❛ ❛s ✐♥t❡r❛çõ❡s ❡❧❡tr♦❢r❛❝❛s
❖ ❣r✉♣♦ ❞❡ ❣❛✉❣❡ éSU(2)L×U(1)✱ r❡q✉❡r❡♥❞♦ q✉❛tr♦ ❜ós♦♥s ❞❡ ❣❛✉❣❡ ♣❛r❛ ❛ t❡♦r✐❛✿ ✉♠ tr✐♣❧❡t♦ (b1
µ, b2µ, b3µ) ✈✐♥❞♦ ❞❡ SU(2)L ❡ ✉♠ s✐♥❣❧❡t♦ aµ ❞❡ U(1)✳ ❆ ❧❛❣r❛♥❣❡❛♥❛ ♣❛r❛ ❡st❛ t❡♦r✐❛ ♣♦❞❡ s❡r ❞❛❞❛ ♣♦r✿
L =Lescalar+Lf ermion+Lgauge+Lesc−f er
❝♦♠ Lescalar s❡♥❞♦ ❛ ❧❛❣r❛♥❣❡❛♥❛ ❞♦s ❝❛♠♣♦s ❡s❝❛❧❛r❡s ❞❛ t❡♦r✐❛✱ Lf ermion ❛ ❧❛✲ ❣r❛♥❣❡❛♥❛ ❞♦s ❝❛♠♣♦s ❢❡r♠✐ô♥✐❝♦s✱ Lgauge ❛ ❧❛❣r❛♥❣❡❛♥❛ ❞♦s ❝❛♠♣♦s ❞❡ ❣❛✉❣❡ ❡
✷✳✷✳ ❖ ▼♦❞❡❧♦ ❊❧❡tr♦❢r❛❝♦ ✶✹
❆ ♣❛rt❡ ❞❡ ❣❛✉❣❡ ♣♦❞❡ s❡r ❝♦❧♦❝❛❞❛ ❝♦♠♦ ❡♠ ✭✷✳✸✮ ❡ ✭✷✳✹✮✿
Lgauge=− 1 4F j µνF µν j − 1 4fµνf
µν ✭✷✳✶✼✮
♦♥❞❡ ♦s t❡♥s♦r❡s Fj
µν ✭♥ã♦✲❛❜❡❧✐❛♥♦✮ ❡ fµν ✭❛❜❡❧✐❛♥♦✮ sã♦ ❞❛❞♦s ♣♦r✿
fµν = ∂µaν −∂νaµ ✭✷✳✶✽✮
Fj
µν = ∂µbjν −∂νbµj −gEjklbkµblν ✭✷✳✶✾✮
❆ ♣❛rt❡ ❞♦s ❢ér♠✐♦♥s ❞❛ ❧❛❣r❛♥❣❡❛♥❛ é ❝♦❧♦❝❛❞❛ ❝♦♠♦✿
Lf ermion= ¯Riγµ
∂µ+
ig′
2 aµY
R+ ¯Liγµ
∂µ+
ig′
2 aµY +
ig
2τ jbj
µ
L ✭✷✳✷✵✮
♦♥❞❡ ♦s ❝❛♠♣♦s ❞❡ ♠ã♦ ❡sq✉❡r❞❛ ❡ ♠ã♦ ❞✐r❡✐t❛ sã♦ ❞❡✜♥✐❞♦s ❝♦♠♦✿
R ≡ eR =
1
2(1 +γ5)e ✭✷✳✷✶✮
L ≡ ν
e !
L = 1
2(1−γ5)
ν e
!
✭✷✳✷✷✮
❊st❛♠♦s r❡♣r❡s❡♥t❛♥❞♦ ♦s ❧é♣t♦♥s ❛♣❡♥❛s ❝♦♠ ♦s ♥❡✉tr✐♥♦s ❡ ♦s ❡❧étr♦♥s✳ ❆s ♦✉tr❛s ❢❛♠í❧✐❛s ❞❡✈❡♠ s❡❣✉✐r ❡st❛ ♠❡s♠❛ ❡str✉t✉r❛✱ ❡ s❡rã♦ ❛❞✐❝✐♦♥❛❞❛s ❛♣❡♥❛s q✉❛♥❞♦ r❡q✉✐s✐t❛❞❛s✳ ❆s ❝♦♥st❛♥t❡s ❞❡ ❛❝♦♣❧❛♠❡♥t♦ sã♦ g ❡ g2′ ❛ss♦❝✐❛❞❛s r❡s♣❡❝t✐✈❛♠❡♥t❡
❛ ❤✐♣❡r❝❛r❣❛ ❞♦s ❣r✉♣♦s SU(2)L ❡ U(1)✳ ❖❜s❡r✈❡ q✉❡ ❡♠ ✭✷✳✷✵✮ ♦s ❝❛♠♣♦s ❞❡ ♠ã♦ ❡sq✉❡r❞❛ ❡ ❞❡ ♠ã♦ ❞✐r❡✐t❛ s❡ ❛❝♦♣❧❛♠ ❞✐❢❡r❡♥t❡♠❡♥t❡✳ ❚❡♠♦s ✉♠ t❡r♠♦ ❡①tr❛ q✉❡ é s❡♥sí✈❡❧ s♦♠❡♥t❡ ❛♦s t❡r♠♦s ❞❡ ♠ã♦ ❡sq✉❡r❞❛ q✉❡ s❡rá r❡s♣♦♥sá✈❡❧ ♣♦r ❡st❛ ❞✐❢❡r❡♥ç❛ ♥❡st❡s ❛❝♦♣❧❛♠❡♥t♦s ❡♥tr❡ ❝❛♠♣♦s ❞❡ ♠ã♦ ❡sq✉❡r❞❛ ❡ ❞✐r❡✐t❛✳
❖s ❡s❝❛❧❛r❡s ❞❡✈❡♠ s❡r ✉♠ ❞✉❜❧❡t♦ ❝♦♠♣❧❡①♦ ❞♦ ❣r✉♣♦SU(2)
φ ≡ φ
+
φ0
!
= √1 2
φ1+iφ2
φ3+iφ4
!
✭✷✳✷✸✮
❡ r❡♣r❡s❡♥t❛❞♦s ♣❡❧❛ ❧❛❣r❛♥❣❡❛♥❛ ❛ s❡❣✉✐r✿
✷✳✷✳ ❖ ▼♦❞❡❧♦ ❊❧❡tr♦❢r❛❝♦ ✶✺
❝♦♠ ❛ ❞❡r✐✈❛❞❛ ❝♦✈❛r✐❛♥t❡
Dµ=∂µ+
ig′
2 aµY +
ig
2τ jbj
µ ✭✷✳✷✺✮
❡ ♦ ♣♦t❡♥❝✐❛❧
V(φ†φ) =µ2φ†φ+λ(φ†φ)2 (λ >0). ✭✷✳✷✻✮
P❛r❛ ✜♥❛❧✐③❛r t❡♠♦s ♦ ❛❝♦♣❧❛♠❡♥t♦ ❞♦s ❢ér♠✐♦♥s ❝♦♠ ♦s ❡s❝❛❧❛r❡s ❞❡✜♥✐❞♦ ❝♦♠♦ ♦ ❛❝♦♣❧❛♠❡♥t♦ ❞❡ ❨✉❦❛✇❛✿
Lf er−esc =−Gℓ
¯
R(φ†L) + ¯(Lφ)R
✭✷✳✷✼✮
❡♠ q✉❡ Gℓ é ✉♠❛ ❝♦♥st❛♥t❡ ❧❡♣tô♥✐❝❛ ✐♥❞❡♣❡♥❞❡♥t❡ ❞♦s ❛❝♦♣❧❛♠❡♥t♦s g ❡ g
′
2✳
✷✳✷✳✷ ❆ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡tr✐❛
P❛r❛ ❣❡r❛r ❛s ♠❛ss❛s✱ ♣r❡❝✐s❛♠♦s ❞♦ ♠❡❝❛♥✐s♠♦ ❞❛ q✉❡❜r❛ ❡s♣♦♥tâ♥❡❛ ❞❛ s✐♠❡✲ tr✐❛✱ ❝♦♠♦ ❡①♣❧✐❝❛❞♦ ♥❛ s❡çã♦ ✭✷✳✶✳✷✮✳ Pr✐♠❡✐r❛♠❡♥t❡ ✈❛♠♦s ♣❡♥s❛r ♥♦s ❡s❝❛❧❛r❡s✳ ◗✉❡❜r❛♠♦s ❡s♣♦♥t❛♥❡❛♠❡♥t❡ ❛ s✐♠❡tr✐❛ ❞❡ ❣❛✉❣❡ ❡s❝♦❧❤❡♥❞♦ µ2 < 0 ❡♠ ✭✷✳✷✻✮✳
❊s❝♦❧❤❡♠♦s t❛♠❜é♠ ♦ ❡st❛❞♦ ❞❡ ✈á❝✉♦ s❡♥❞♦✿
hφi0 ≡ h0|φ|0i= √v
2 0 1
!
❝♦♠
v =
r
−µ2
λ . ✭✷✳✷✽✮
❆✐♥❞❛ t❡♠♦s q✉❡ ♦s ❣❡r❛❞♦r❡s ❞♦ ❣r✉♣♦ SU(2) ❞❡✈❡♠ s❡r ❞❛❞♦s ♣❡❧❛s ♠❛tr✐③❡s✿
τ1 =
1 2
0 1 1 0
!
; τ2 =
1 2
0 −i
i 0
!
; τ3 =
1 2
1 0
0 −1
!
