Dynamical chiral symmetry breaking: the fermionic gap equation with dynamical gluon mass and confinement

✡✡✡ ✪✪ ✪ ✱✱

✱✱ ✑✑✑ ✟✟ ❡

❡❡ ❅

❅_❅ ❧

❧ ❧ ◗

◗◗ ❍ ❍P P P ❳❳ ❳ ❤❤ ❤ ❤

✭ ✭ ✭

✭✏✟✏

■❋❚

▼❆❙❚❊❘ ❉■❙❙❊❘❚❆❚■❖◆ ■❋❚✕❉✳✵✵✽✴✷✵✶✸

❉❨◆❆▼■❈❆▲ ❈❍■❘❆▲ ❙❨▼▼❊❚❘❨ ❇❘❊❆❑■◆●✿ ❚❍❊

❋❊❘▼■❖◆■❈ ●❆P ❊◗❯❆❚■❖◆ ❲■❚❍ ❉❨◆❆▼■❈❆▲ ●▲❯❖◆

▼❆❙❙ ❆◆❉ ❈❖◆❋■◆❊▼❊◆❚

❘♦❞♦❧❢♦ ▼❛r✐♦ ❈❛♣❞❡✈✐❧❧❛ ❘♦❧❞❛♥

❆❞✈✐s♦r

Pr♦❢✳ ❉r✳ ❆❞r✐❛♥♦ ❆♥t♦♥✐♦ ◆❛t❛❧❡

✧◆♦ q✉✐❡r♦ ❧❧❡❣❛r ❛ s❡r ❋ís✐❝♦ ❚❡ór✐❝♦✱ ♥✐ t❛♠♣♦❝♦ ❋ís✐❝♦ ❊①♣❡r✐♠❡♥t❛❧✳ ◗✉✐❡r♦ q✉❡ ♠✐ ❢♦r♠❛❝✐ó♥ ♠❡ ❝♦♥❞✉③❝❛ ❛ s❡r ❛❧❣♦ ♠ás✳ ◗✉✐❡r♦ ❧❧❡❣❛r ❛ s❡r✳✳✳ ❢ís✐❝♦✧✳

❯❞❡❆✷✵✶✵

✧❖ ❢ís✐❝♦ ♥ã♦ só ❝r✐❛ ♦s ♠♦❞❡❧♦s ♦✉ r❡❛❧✐③❛ ♦s ❡①♣❡r✐♠❡♥t♦s✳ ❖ ❢ís✐❝♦✱ ❞❡♣♦✐s ❞❡ ❝r✐❛r ♦s ♠♦❞❡❧♦s✱ ♣r♦♣õ❡ ❡①♣❡r✐♠❡♥t♦s✳ ❖✉ ❞❡♣♦✐s ❞❡ r❡❛❧✐③❛r ♦s ❡①♣❡r✐♠❡♥t♦s✱ ♣r♦♣õ❡ ♠♦❞❡❧♦s✧✳

✐

❆❝❦♥♦✇❧❡❞❣♠❡♥ts

❚♦ ●♦❞✱ ❢♦r ❤✐s ❝❛r❡ ❛♥❞ ♣r♦t❡❝t✐♦♥ t❤r♦✉❣❤♦✉t ♠② ❧✐❢❡✳

❚♦ ♠② ❢❛♠✐❧②✱ ♠② ❢r✐❡♥❞s ❛♥❞ ♣❛rt♥❡rs✱ ❛❧❧ ♦❢ t❤❡♠ ✇❤♦ ❤❡❧♣ ♠❡ ✐♥ ♠② ♣❡rs♦♥❛❧ ❢♦r♠❛t✐♦♥✱ ❡s♣❡❝✐❛❧❧② t♦ ♠② ✇✐❢❡✱ t❤❛t ♣r❡tt② ❣✐r❧✱ t❤❡ ❧♦✈❡ ♦❢ ♠② ❧✐❢❡✳

❚♦ ♣r♦❢❡ss♦r ❆❞r✐❛♥♦✱ ✇❤♦ ♠♦r❡ t❤❛♥ ❛ ❣✉✐❞❡ ❤❛s ❜❡❡♥ ❛ ❢r✐❡♥❞ t❤r♦✉❣❤ t❤✐s ♠❛st❡r✳ ❋♦r ❤✐s ❛❞✈✐❝❡s✱ ♣❛t✐❡♥❝❡ ❛♥❞ ✉♥❝♦♥❞✐t✐♦♥❛❧ s✉♣♣♦rt✳

✐✐

❆❜str❛❝t

❙♦♠❡ ❛s♣❡❝ts ♦❢ ❝❤✐r❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❢♦r q✉❛r❦s ✐♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ r❡♣r❡s❡♥t❛t✐♦♥ ❛r❡ ❞✐s❝✉ss❡❞ ✐♥ t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ t❤❡ ❙❝❤✇✐♥❣❡r✲❉②s♦♥ ❡q✉❛t✐♦♥s✳ ❲❡ st✉❞② t❤❡ ❢❡r♠✐♦♥✐❝ ❣❛♣ ❡q✉❛t✐♦♥ ✐♥❝❧✉❞✐♥❣ ❡✛❡❝ts ♦❢ ❞②♥❛♠✐❝❛❧ ❣❧✉♦♥ ♠❛ss✳ ❙t✉❞②✐♥❣ t❤❡ ❜✐❢✉r❝❛t✐♦♥ ❡q✉❛t✐♦♥ ♦❢ t❤✐s ❣❛♣ ❡q✉❛t✐♦♥ ✇❡ ✈❡r✐❢② t❤❛t t❤❡ ✐♥t❡r❛❝t✐♦♥ ✐s ♥♦t str♦♥❣ ❡♥♦✉❣❤ t♦ ❣❡♥❡r❛t❡ ❛ s❛t✐s❢❛❝t♦r② ❞②♥❛♠✐❝❛❧ q✉❛r❦ ♠❛ss✳ ❲❡ ❛❧s♦ ❞✐s❝✉ss ❤♦✇ t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❛ ❝♦♥✜♥✐♥❣ ♣r♦♣❛❣❛t♦r ♠❛② ❝❤❛♥❣❡ t❤✐s s❝❡♥❛r✐♦ ❛s r❡❝❡♥t❧② ♣♦✐♥t❡❞ ♦✉t ❜② ❈♦r♥✇❛❧❧ ❬✶❪✱ s♦ ✇❡ st✉❞② ❛ ✧❝♦♠♣❧❡t❡✧ ❣❛♣ ❡q✉❛t✐♦♥ ❝♦♠♣♦s❡❞ ❜② t❤❡ ♦♥❡✲❞r❡ss❡❞✲❣❧✉♦♥ ❡①❝❤❛♥❣❡ t❡r♠ ❛♥❞ ❛ ❝♦♥✜♥✐♥❣ t❡r♠✿ M(p2_{) =}

Mc(p2) +M1g(p2)✳ ❲❡ ✜♥❞ ❛s②♠♣t♦t✐❝ s♦❧✉t✐♦♥s ❢♦r t❤✐s ❣❛♣ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❝❛s❡s ♦❢ ✧❝♦♥st❛♥t

❝♦✉♣❧✐♥❣✧ ❛♥❞ ✧r✉♥♥✐♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t✧✳ ❚❤✐s ❧❛st ❝❛s❡ ✐s ❛♥ ✐♠♣r♦✈❡♠❡♥t ♦❢ t❤❡ ❝♦♥st❛♥t ❝♦✉♣❧✐♥❣ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❉♦✛✱ ▼❛❝❤❛❞♦ ❛♥❞ ◆❛t❛❧❡ ❬✷❪✳

❑❡② ✇♦r❞s✿✿ ❈❤✐r❛❧ ❙②♠♠❡tr② ❇r❡❛❦✐♥❣❀ ❉②♥❛♠✐❝❛❧ ◗✉❛r❦ ▼❛ss❀ ❙❝❤✇✐♥❣❡r✲❉②s♦♥ ❊q✉❛t✐♦♥s❀ ❈♦♥✜♥✐♥❣ Pr♦♣❛❣❛t♦r✳

✐✐✐

❘❡s✉♠♦

❆❧❣✉♥s ❛s♣❡❝t♦s ❞❛ q✉❡❜r❛ ❞❡ s✐♠❡tr✐❛ q✉✐r❛❧ ♣❛r❛ q✉❛r❦s ♥❛ r❡♣r❡s❡♥t❛çã♦ ❢✉♥❞❛♠❡♥t❛❧ sã♦ ❞✐s❝✉t✐❞♦s ♥♦ ❝♦♥t❡①t♦ ❞❛s ❡q✉❛çõ❡s ❞❡ ❙❝❤✇✐♥❣❡r✲❉②s♦♥✳ ❊st✉❞❛♠♦s ❛ ❡q✉❛çã♦ ❞❡ ❣❛♣ ❢❡r♠✐♦♥✐❝❛ ✐♥❝❧✉✐♥❞♦ ♦ ❡❢❡✐t♦ ❞❡ ✉♠❛ ♠❛ss❛ ❞✐♥â♠✐❝❛ ♣❛r❛ ♦s ❣❧✉♦♥s✳ ❆♦ ❡st✉❞❛r ❡st❛ ❡q✉❛çã♦ ❞❡ ❣❛♣ ✈❡r✐✜❝❛♠♦s q✉❡ ❛ ✐♥t❡r❛çã♦ ♥ã♦ é ❢♦rt❡ ♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❡r❛r ✉♠❛ ♠❛ss❛ ❞✐♥â♠✐❝❛ ❞♦s q✉❛r❦s ❝♦♠♣❛tí✈❡❧ ❝♦♠ ♦s ❞❛❞♦s ❡①♣❡r✐♠❡♥t❛✐s✳ ❚❛♠❜é♠ ❞✐s❝✉t✐♠♦s ❝♦♠♦ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ♣r♦♣❛❣❛❞♦r ❝♦♥✜♥❛♥t❡ ♣♦❞❡ ♠✉❞❛r ❡st❡ ❝❡♥ár✐♦✱ ❡①❛t❛♠❡♥t❡ ❝♦♠♦ ❢♦✐ ♣r♦♣♦st♦ ♣♦r ❈♦r♥✇❛❧❧ ❬✶❪ r❡❝❡♥t❡♠❡♥t❡✱ ❞❡st❛ ❢♦r♠❛ ❡st✉❞❛♠♦s ✉♠❛ ❡q✉❛çã♦ ❞❡ ❣❛♣ ✧❝♦♠♣❧❡t❛✧✱ ❝♦♠♣♦st❛ ♣❡❧❛ tr♦❝❛ ❞❡ ✉♠ ❣❧✉♦♥ ♠❛ss✐✈♦ ❡ ♣♦r ✉♠ t❡r♠♦ ❝♦♥✜♥❛♥t❡✿ M(p2_{) = M}

c(p2) +M1g(p2)✳ ❊♥❝♦♥tr❛♠♦s

s♦❧✉çõ❡s ❛ss✐♥tót✐❝❛ ❞❡st❛ ❡q✉❛çã♦ ❞❡ ❣❛♣ ♥♦s ❝❛s♦s ❞❡ ❝♦♥st❛♥t❡ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ✧❝♦♥st❛♥t❡✧ ❡ ✧❝♦rr❡❞♦r❛✧✳ ❊st❡ ú❧t✐♠♦ ❝❛s♦ ❝♦rr❡s♣♦♥❞❡ ❛ ✉♠ ❛♣r✐♠♦r❛♠❡♥t♦ ❞♦ ❝á❧❝✉❧♦ ❝♦♠ ❝♦♥st❛♥t❡ ❞❡ ❛❝♦♣❧❛♠❡♥t♦ ✧❝♦♥st❛♥t❡✧ ❢❡✐t♦ ♣♦r ❉♦✛✱ ▼❛❝❤❛❞♦ ❡ ◆❛t❛❧❡ ❬✷❪✳

P❛❧❛✈r❛s ❝❤❛✈❡✿ ◗✉❡❜r❛ ❞❡ s✐♠❡tr✐❛ q✉✐r❛❧❀ ▼❛ss❛ ❞✐♥â♠✐❝❛ ❞❡ q✉❛r❦s❀ ❊q✉❛çõ❡s ❞❡ ❙❝❤✇✐♥❣❡r✲❉②s♦♥❀ Pr♦♣❛❣❛❞♦r ❝♦♥✜♥❛♥t❡✳

❈♦♥t❡♥ts

✶ ■♥tr♦❞✉❝t✐♦♥ ✶

✷ ◗✉❛♥t✉♠ ❈❤r♦♠♦❞②♥❛♠✐❝s ✹

✷✳✶ ❍✐st♦r✐❝❛❧ ❘❡♠❛r❦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹

✷✳✷ ●❡♥❡r❛❧ Pr♦♣❡rt✐❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼

✷✳✸ ❙❝❤✇✐♥❣❡r ❉②s♦♥ ❊q✉❛t✐♦♥s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

✷✳✹ ◗❊❉ ❣❛♣ ❊q✉❛t✐♦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹

✸ ❈❤✐r❛❧ ❙②♠♠❡tr② ✶✼

✸✳✶ ❈❤✐r❛❧ ❙②♠♠❡tr② ❇r❡❛❦✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼

✸✳✷ ◆❛♠❜✉✲❏♦♥❛✲▲❛s✐♥✐♦ ▼♦❞❡❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✸✳✸ ❉②♥❛♠✐❝❛❧ ●❧✉♦♥ ▼❛ss ❬✹✵❪ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

✹ ❉②♥❛♠✐❝❛❧ ◗✉❛r❦ ▼❛ss ✷✷

✹✳✶ ❖♥❡✲❉r❡ss❡❞✲●❧✉♦♥ ❊①❝❤❛♥❣❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✹✳✶✳✶ ❈♦♥st❛♥t ❈♦✉♣❧✐♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

✹✳✶✳✷ ❘✉♥♥✐♥❣ ❈♦✉♣❧✐♥❣ ❈♦♥st❛♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺

✹✳✷ ❖♥❡✲❉r❡ss❡❞✲●❧✉♦♥ ❊①❝❤❛♥❣❡ ✰ ❈♦♥✜♥❡♠❡♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✹✳✷✳✶ ❈♦♥st❛♥t ❈♦✉♣❧✐♥❣ ❛♥❞ ❆s②♠♣t♦t✐❝ ❇❡❤❛✈✐♦r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾

❈❖◆❚❊◆❚❙ ✈

✹✳✷✳✷ ❘✉♥♥✐♥❣ ❈♦✉♣❧✐♥❣ ❈♦♥st❛♥t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✺ ❙✉♠♠❛r② ❛♥❞ ❈♦♥❝❧✉s✐♦♥s ✸✼

❆ ❙♦♠❡ ❝❛❧❝✉❧❛t✐♦♥s ✸✾

❆✳✶ Pr♦♦❢ ♦❢ ❡q✉❛t✐♦♥ ✭✷✳✷✹✮✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾

❆✳✷ Pr♦♦❢ ♦❢ ❡q✉❛t✐♦♥ ✭✹✳✶✸✮✿ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵

❈❤❛♣t❡r ✶

■♥tr♦❞✉❝t✐♦♥

❙②♠♠❡tr✐❡s ❛r❡ s♦♠❡ ♦❢ t❤❡ ♠♦st ✐♠♣♦rt❛♥t ♣r♦♣❡rt✐❡s t❤❛t ✇❡ ✉s❡ t♦ ❞❡s❝r✐❜❡ ♥❛t✉r❡✳ ❙②♠♠❡tr✐❡s ❛r❡ r❡❧❛t❡❞ t♦ ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s ❛♥❞ t❤✐s ✐s ✇❤② ✇❡ r❡❛❧❧② ❧✐❦❡ ✇❤❡♥ ❛ ♣❤②s✐❝❛❧ s②st❡♠ ✐s ✐♥✈❛r✐❛♥t ✉♥❞❡r s♣❛t✐❛❧ tr❛♥s❧❛t✐♦♥s✱ r♦t❛t✐♦♥s ♦r t✐♠❡ tr❛♥s❧❛t✐♦♥✱ ❜❡❝❛✉s❡ t❤✐s ✐♠♣❧② ❝♦♥s❡r✈❛t✐♦♥s ♦❢ t❤❡ ❧✐♥❡❛r ♠♦♠❡♥t✉♠✱ ❛♥❣✉❧❛r ♠♦♠❡♥t✉♠ ❛♥❞ ❡♥❡r❣② ✭r❡s♣❡❝t✐✈❡❧②✮✱ ❛♥❞ t❤✐s ❝♦♥s❡r✈❛t✐♦♥ ❧❛✇s ❛r❡ ♣♦✇❡r❢✉❧ t♦♦❧s ✉s❡❞ t❤r♦✉❣❤ ❛❧❧ t❤❡ ♣❤②s✐❝s✱ ❢r♦♠ ❝❧❛ss✐❝❛❧ t♦ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s✳ ❉❡s♣✐t❡ t❤❡ s②♠♠❡tr✐❡s ✐♠♣♦rt❛♥❝❡✱ ♦♥❡ ❛s♣❡❝t t❤❛t ✐s ♠♦r❡ ✐♠♣♦rt❛♥t ✐s ✇❤❡♥ t❤❡ s②♠♠❡tr✐❡s ❛r❡ ❜r♦❦❡♥✳ ❋♦r ❡①❛♠♣❧❡✱ ✐t ✐s ✐♥ t❡r♠s ♦❢ ❛ s♣♦♥t❛♥❡♦✉s❧② s②♠♠❡tr② ❜r❡❛❦✐♥❣ ✭❙❙❇✮ t❤❛t ▲❛♥❞❛✉ ❞❡s❝r✐❜❡❞ t❤❡ ♠❛❣♥❡t✐③❛t✐♦♥ ♦❢ ❢❡rr♦♠❛❣♥❡t✐❝ s②st❡♠s ❬✸❪✱ ❛♥❞ t❤✐s ✐s ❛♥ ✐♠♣♦rt❛♥t ♠♦❞❡❧ ♦❢ ♣❤❛s❡ tr❛♥s✐t✐♦♥s ✇❤✐❝❤ ❤❛s ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ♦t❤❡r ✜❡❧❞s ❧✐❦❡ ♣❛rt✐❝❧❡ ♣❤②s✐❝s ❧❡❛❞✐♥❣ t♦ t❤❡ ❍✐❣❣s ♠❡❝❤❛♥✐s♠ ❬✹❪✳ ❚❤✐s ♠♦❞❡❧ ♦❢ s♣♦♥t❛♥❡♦✉s❧② s②♠♠❡tr② ❜r❡❛❦✐♥❣ ✐s ❛ s✉❝❝❡ss ✐♥ t❤❡ ❡❧❡❝tr♦✇❡❛❦ ♠♦❞❡❧✱ ✇❤❡r❡ ✐t ✇❛s ✐♥tr♦❞✉❝❡❞ ✐♥ ♦r❞❡r t♦ ❞❡s❝r✐❜❡ t❤❡ ❣❛✉❣❡ ❜♦s♦♥ ❛♥❞ ❢❡r♠✐♦♥s ♠❛ss ❣❡♥❡r❛t✐♦♥✳ ❚❤✐s ♣r♦❝❡ss✱ ✇❤❡r❡ ✇❡ st❛rt ✇✐t❤ ❛ ▲❛❣r❛♥❣✐❛♥ ✇✐t❤ ♠❛ss❧❡ss ❢❡r♠✐♦♥s ❛♥❞ ✐♥ t❤❡ ❡♥❞ t❤❡② ♦❜t❛✐♥ ♠❛ss❡s ✐s ❝❛❧❧❡❞ ❈❤✐r❛❧ ❙②♠♠❡tr② ❇r❡❛❦✐♥❣ ✭❈❙❇✮✳ ❚❤❡ ❈❙❇ ❝❛♥ ♦❝❝✉rs ✐♥ t✇♦ ✇❛②s❀ s♣♦♥t❛♥❡♦✉s❧② ✭❛s ✐♥ t❤❡ ▲❛♥❞❛✉ ♠♦❞❡❧✮✱ ❛♥❞ ❞②♥❛♠✐❝❛❧❧② ✭❛s ✇❡ ✇✐❧❧ ❞✐s❝✉ss ❤❡r❡✮✳

❉②♥❛♠✐❝❛❧ ❝❤✐r❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ✭❉❈❙❇✮ ✐s ✉♥❞❡rst♦♦❞ ❛s t❤❡ ♠❡❝❤❛♥✐s♠ ✇❤❡r❡ t❤❡ ♠❛ss❡s ❛r❡ ❝r❡❛t❡❞ ❜② s❡❧❢✲✐♥t❡r❛❝t✐♦♥ ✇✐t❤♦✉t ✐♥tr♦❞✉❝t✐♦♥ ♦❢ s❝❛❧❛r ✜❡❧❞s✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ♣r♦t♦♥ ✐s ❝♦♠♣♦s❡❞ ❜② t❤r❡❡ ❧✐❣❤t q✉❛r❦s✱ ❡❛❝❤ ✇✐t❤ ❛ ❝✉rr❡♥t ♠❛ss ♦❢ ❛❜♦✉t ✺▼❡❱✱ ❜✉t t❤❡

❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✷

♣r♦t♦♥ ❤❛s ❛ ♠❛ss ❛❜♦✉t ✶●❡❱✳ ■❢ ✇❡ s✉♠ t❤❡ ♠❛ss ♦❢ t❤r❡❡ ❧✐❣❤t q✉❛r❦s✱ t❤❡ ♣r♦t♦♥ ✇♦✉❧❞ ❤❛✈❡ ❛ ♠❛ss ♦❢ ❛❜♦✉t ✶✺▼❡❱✱ s♦ t❤❡ ❜✐❣ q✉❡st✐♦♥ ✐s✿ ✧❋r♦♠ ✇❤❡r❡ ❝♦♠❡ t❤❡ ♣r♦t♦♥ ♠❛ss❄✧✱ ❛♥❞ ❉❈❙❇ ✐s t❤❡ ❛♥s✇❡r t♦ t❤✐s q✉❡st✐♦♥✳

❉❈❙❇ ❤❛s ❜❡❡♥ st✉❞✐❡❞ ❛❧s♦ ✐♥ ◗❊❉ ❛♥❞ ♣✉r❡ ◗❈❉ ♦r ✇✐t❤ t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❛ ◆❏▲ t②♣❡ ♦❢ ✐♥t❡r❛❝t✐♦♥ ❬✺✱ ✻✱ ✼✱ ✽❪✳ ❚❤❡s❡ ❛♥❛❧②s✐s ❛r❡ ♠♦❞✐✜❡❞ ❜② t❤❡ ✐♥tr♦❞✉❝t✐♦♥ ♦❢ ❛ ❞②♥❛♠✐❝❛❧ ❣❧✉♦♥s ♠❛ss ✭mg✮ ✇❤✐❝❤ ✐s ❛ ♥❡❝❡ss❛r② ✐♥❣r❡❞✐❡♥t ❢♦r ❛ ❝♦♠♣❧❡t❡ ❛♥❛❧②s✐s ♦❢ ❉❈❙❇ ❜❡❝❛✉s❡ ✐t ❡①✐st❡♥❝❡

❤❛s ❜❡❡♥ ❝♦♥✜r♠❡❞ ✈✐❛ ❛♥❛❧②t✐❝❛❧ ❛♥❞ ♥✉♠❡r✐❝❛❧ ❝❛❧❝✉❧❛t✐♦♥s ❬✾✱ ✶✵✱ ✶✶❪✳

Pr❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ✇♦r❦

❚❤❡ ♣r❡s❡♥t ✇♦r❦ ✐s ♦r❣❛♥✐③❡❞ ✐♥ t❤❡ ♥❡①t ✇❛②✿ ■♥ s❡❝t✐♦♥ t✇♦ ✇❡ ❜r✐❡✢② r❡✈✐❡✇ s♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ ◗✉❛♥t✉♠ ❈❤r♦♠♦❞②♥❛♠✐❝ ✭◗❈❉✮❀ ✇❡ st❛rt ✇✐t❤ ❛ ❤✐st♦r✐❝❛❧ r❡♠❛r❦ ✇❤❡r❡ ✐t ✐s ❞❡✜♥❡❞ ❛❧❧ t❤❡ ❧❛♥❣✉❛❣❡ ✉s❡❞ ✇❤❡♥ s♣❡❛❦✐♥❣ ❛❜♦✉t ◗❈❉ ❛♥❞ s♦♠❡ r❡❢❡r❡♥❝❡s ❢♦r ❛ ❞❡❡♣❡r st✉❞②✳ ❚❤❡ ❧✐♥❡ ❢♦❧❧♦✇❡❞ ✐♥ t❤❡ ❤✐st♦r✐❝❛❧ r❡♠❛r❦ ✐s✿

❙tr♦♥❣ ✐♥t❡r❛❝t✐♦♥ → ❨✉❦❛✇❛ ♠♦❞❡❧ → ❨❛♥❣✲▼✐❧❧s t❤❡♦r②

❊①♣❡r✐♠❡♥t❛❧ ❢❛❝ts✿ ❈♦❧♦r✱ ❋❧❛✈♦r → ◗❈❉

P❤❡♥♦♠❡♥♦❧♦❣②✿ ⎧

⎪ ⎨

⎪ ⎩

N onrelativistic interacting potential

Bag model

⎫

⎪ ⎬

⎪ ⎭

→ ❉❈❙❇

■♥ s❡❝t✐♦♥ t✇♦✱ ✇❡ ❛❧s♦ ♣r❡s❡♥t t❤❡ ◗❈❉ ▲❛❣r❛♥❣✐❛♥✱ t❤❡ ❋❡②♥♠❛♥ r✉❧❡s✱ t❤❡ ❙❝❤✇✐♥❣❡r✲ ❉②s♦♥ ❡q✉❛t✐♦♥s✱ ❛♥❞ ✇✐t❤ t❤❡s❡ t♦♦❧s✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❣❛♣ ❡q✉❛t✐♦♥✱ ❡q✉❛t✐♦♥ ✇❤✐❝❤ ❧❡❛❞s ✉s t♦ st✉❞② t❤❡ ♣❤❡♥♦♠❡♥♦♥ ♦❢ ❉❈❙❇✳ ■♥ s❡❝t✐♦♥ t❤r❡❡ ✇❡ ♣r❡s❡♥t ❢❡✇ ❞❡t❛✐❧s ❛❜♦✉t ❝❤✐r❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣✱ ❥✉st ✐♥ ♦r❞❡r t♦ ♣♦✐♥t ♦✉t s♦♠❡ ❞❡✜♥✐t✐♦♥s✱ ❝♦♠♠❡♥t ❛❜♦✉t t❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ ❈❙❇✱ ❛♥❞ r❡✈✐❡✇ ♦❢ ♦♥❡ ♦❢ t❤❡ ✜rst ♠♦❞❡❧s ✐♥ t❤❛t ❞✐r❡❝t✐♦♥✱ t❤❡ ◆❛♠❜✉✲❏♦♥❛✲▲❛s✐♥✐♦ ♠♦❞❡❧ ❛♥❞ ✜♥❛❧❧② ✇❡ ♣r❡s❡♥t ❛♥ ✐♠♣♦rt❛♥t r❡s✉❧t ♦❢ ❞②♥❛♠✐❝❛❧ ♠❛ss ❣❡♥❡r❛t✐♦♥ ✇❤✐❝❤ ✐s t❤❡ ✧●❧✉♦♥ ♠❛ss ❣❡♥❡r❛t✐♦♥✧✳

❈❍❆P❚❊❘ ✶✳ ■◆❚❘❖❉❯❈❚■❖◆ ✸

s♦♠❡ ❛♣♣r♦①✐♠❛t✐♦♥s✱ ✇❡ ♦❜t❛✐♥ r❡s✉❧ts ❛❜♦✉t t❤❡ ❞②♥❛♠✐❝❛❧ q✉❛r❦ ♠❛ss✳ ❋♦r s♦♠❡ r❡❛s♦♥s✱ ♣r❡s❡♥t❡❞ ✐♥ t❤✐s ✇♦r❦✱ ✇❡ s❡❡ t❤❛t t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❛ ❞②♥❛♠✐❝❛❧ ❣❧✉♦♥ ♠❛ss ✐♥t♦ t❤❡ ◗❈❉ ❣❛♣ ❡q✉❛t✐♦♥ ❞♦❡s ♥♦t ❤❛✈❡ str❡♥❣t❤ ❡♥♦✉❣❤ t♦ ❣❡♥❡r❛t❡ ❛ s❛t✐s❢❛❝t♦r② ❞②♥❛♠✐❝❛❧ q✉❛r❦ ♠❛ss ❬✶✷✱ ✶✸❪ ❛♥❞ ✇❡ ❝❛♥ s❡❡ t❤✐s ✐♥ t❤❡ ❝❛s❡s ♦❢ ❝♦♥st❛♥t ❝♦✉♣❧✐♥❣ ❛♥❞ ✇✐t❤ ❛ r✉♥♥✐♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t✳ ❚❤✐s s❝❡♥❛r✐♦ ✐s ♠♦❞✐✜❡❞ ✇✐t❤ t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❛ ♥❡✇ ✐♥❣r❡❞✐❡♥t ✐♥t♦ t❤❡ ❣❛♣ ❡q✉❛t✐♦♥ ❛s ♣r♦♣♦s❡❞ ❜② ❈♦r♥✇❛❧❧ ❬✶❪✱❛♥❞ t❤✐s ✐♥❣r❡❞✐❡♥t ✐s ❝♦♥✜♥❡♠❡♥t ✐♥ t❤❡ ❢♦r♠ ♦❢ ❛ ❝♦♥✜♥✐♥❣ ❡✛❡❝t✐✈❡ ♣r♦♣❛❣❛t♦r ✭❈❊P✮✱ s♦ ✇❡ st✉❞② ❛ ✧❝♦♠♣❧❡t❡✧ ❣❛♣ ❡q✉❛t✐♦♥ M(p2_{) =M}

c(p2) +M1g(p2) ✇❤✐❝❤ ♠✐♠✐❝s t❤❡

♣❤❡♥♦♠❡♥♦❧♦❣✐❝❛❧ ❢♦r♠✉❧❛ VF(r) =KFr−4αs/3r ❢♦r t❤❡ ✐♥t❡r❛❝t✐♦♥ ♣♦t❡♥t✐❛❧ ❜❡t✇❡❡♥ q✉❛r❦s✳

❈❤❛♣t❡r ✷

◗✉❛♥t✉♠ ❈❤r♦♠♦❞②♥❛♠✐❝s

✷✳✶ ❍✐st♦r✐❝❛❧ ❘❡♠❛r❦

❙✐♥❝❡ ✐ts ♣r♦♣♦s❛❧ ✐♥ t❤❡ ✸✵✬s✱ t❤❡ str♦♥❣ ✐♥t❡r❛❝t✐♦♥ ✐s ♦♥❡ ♦❢ t❤❡ ❝❤❛❧❧❡♥❣❡s ♦❢ t❤❡ t❤❡♦r❡t✐❝❛❧ ♣❤②s✐❝s ❢♦r ✉♥❞❡rst❛♥❞✐♥❣ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ❜✉✐❧❞✐♥❣ ❜❧♦❝❦s ♦❢ ♠❛tt❡r ❛♥❞ t❤❡✐r ✐♥t❡r❛❝t✐♦♥s✳ ■t ✐s ♣r♦♣♦s❡❞ ❛s t❤❡ ✐♥t❡r❛❝t✐♦♥ r❡s♣♦♥s✐❜❧❡ ❢♦r t❤❡ st❛❜✐❧✐t② ♦❢ t❤❡ ❛t♦♠✐❝ ♥✉❝❧❡✐✱ ✇❤✐❝❤ ♦❢ ❝♦✉rs❡✱ ♠✉st ❜❡ ✈❡r② ✧str♦♥❣✧ ✐♥ ♦r❞❡r t♦ s✉r♣❛ss t❤❡ ✧str♦♥❣✧ ❡❧❡❝tr♦♠❛❣♥❡t✐❝ r❡♣✉❧s✐♦♥ ✭❞✉❡ t♦ t❤❡ s❤♦rt ❞✐st❛♥❝❡✱ ∽ 10f m✮❜❡t✇❡❡♥ t❤❡ ♣r♦t♦♥s ♣r❡s❡♥t ✐♥ t❤❡ ❛t♦♠✐❝ ♥✉❝❧❡✐✳ ❖♥❡ ♦❢ t❤❡ ✜rst

♠♦❞❡❧s ✐♥ ♦r❞❡r t♦ ❞❡s❝r✐❜❡ t❤✐s ✐♥t❡r❛❝t✐♦♥ ✇❛s ♣r♦♣♦s❡❞ ❜② ❨✉❦❛✇❛ ❬✶✹❪ ✇❤♦ t❤♦✉❣❤t ✐♥ t✇♦ ✐♥t❡r❛❝t✐♥❣ ❢❡r♠✐♦♥s ✭◆✉❝❧❡♦♥s❀ Pr♦t♦♥ ❛♥❞✴♦r ◆❡✉tr♦♥s✮✇❤✐❝❤ ❡①❝❤❛♥❣❡ ❛ ✧✈✐rt✉❛❧✧ ♣❛rt✐❝❧❡ ✐♥ ❛♥❛❧♦❣② ✇✐t❤ ◗✉❛♥t✉♠ ❊❧❡❝tr♦❞②♥❛♠✐❝s ✭◗❊❉✮✳ ❙✐♥❝❡ t❤✐s ✐♥t❡r❛❝t✐♦♥ ♠✉st ❜❡ ♦❢ s❤♦rt r❛♥❣❡ ✭✐s ✧❧✐♠✐t❡❞✧ t♦ t❤❡ s✐③❡ ♦❢ t❤❡ ❛t♦♠✐❝ ♥✉❝❧❡✐✮✱ ❛♥❞ ❜② t❤❡ ❍❡✐ss❡♥❜❡r❣ ✉♥❝❡rt❛✐♥t② ♣r✐♥❝✐♣❧❡✱ ❨✉❦❛✇❛ ♣r♦♣♦s❡❞ t❤❛t t❤✐s q✉❛♥t❛ ♠✉st ❜❡ ♠❛ss✐✈❡ ❛♥❞ ♣r❡❞✐❝t❡❞ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♣❛rt✐❝❧❡ ✇✐t❤ ❛ ♠❛ss ♦❢ ❛❜♦✉t ✶✷✵▼❡❱ ✭♣❛rt✐❝❧❡ ✐❞❡♥t✐✜❡❞ ❧❛t❡r ✇✐t❤ t❤❡ π ♠❡s♦♥✮✳

❍❡✐ss❡♥❜❡r❣ ♣r♦♣♦s❡❞ t❤❛t ✐♥ t❤❡ ❨✉❦❛✇❛ ♠♦❞❡❧ t❤❡r❡ ✐s ❛ s②♠♠❡tr② ❬✶✺❪✱ ✇❤✐❝❤ ❲✐❣♥❡r ❝❛❧❧❡❞ ✐s♦s♣✐♥ ❬✶✻❪✳ ■s♦s♣✐♥ r❡✢❡❝t t❤❡ ❢❛❝t t❤❛t t❤❡ str♦♥❣ ✐♥t❡r❛❝t✐♦♥ ✐s t❤❡ s❛♠❡ ❜❡t✇❡❡♥ t✇♦ ♣r♦t♦♥s ♦r t✇♦ ♥❡✉tr♦♥s✱ ♦r ❜❡t✇❡❡♥ ❛ ♣r♦t♦♥ ❛♥❞ ❛ ♥❡✉tr♦♥✱ ♥❛♠❡❧②✱ str♦♥❣ ✐♥t❡r❛❝t✐♦♥ ✐s ✐♥✈❛r✐❛♥t ❜② ❛ r♦t❛t✐♦♥ ✐♥ t❤❡ ✧✐s♦s♣✐♥ s♣❛❝❡✧ ✭♦r ❜② ❛ ❡①❝❤❛♥❣❡ ♦❢ ♣r♦t♦♥s ❜② ♥❡✉tr♦♥s✮✱ ❜✉t t❤✐s s②♠♠❡tr② ✐s s❛✐❞ ❛♣♣r♦①✐♠❛t❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❞✐✛❡r❡♥❝❡ ♦❢ ♠❛ss ❜❡t✇❡❡♥ ♥❡✉tr♦♥s ❛♥❞

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✺

N

N

N

N

n, p

p, n π±

p, n

n, p π0

❋✐❣✉r❡ ✷✳✶✿ ❋❡②♥♠❛♥ ❞✐❛❣r❛♠s ♦❢ ◆✉❝❧❡♦♥ ✐♥t❡r❛❝t✐♦♥ ✐♥ t❤❡ ❨✉❦❛✇❛ ♠♦❞❡❧✳

♣r♦t♦♥s✳ ■s♦s♣✐♥ ✐s ❛ss♦❝✐❛t❡❞ t♦ ❛ s②♠♠❡tr② ❣r♦✉♣✱ t❤❡ ❙❯✭✷✮ ❣r♦✉♣✱ ❛♥❞ ✐t ✇❛s ❜❡❝❛✉s❡ ♦❢ t❤✐s t❤❛t ❨❛♥❣ ❛♥❞ ▼✐❧❧s ❬✶✼❪ ♣r♦♣♦s❡❞ ❛ ✜❡❧❞ t❤❡♦r② ❢♦r ❙❯✭✷✮ ✐♥ ♦r❞❡r t♦ st✉❞② str♦♥❣ ✐♥t❡r❛❝t✐♦♥ ✭❝❛❧❧❡❞ ❧❛t❡r ❨❛♥❣✲▼✐❧❧s t❤❡♦r② ❛♥❞ ❡①t❡♥❞❡❞ t♦ ❙❯✭◆✮✮✳

■♥ t❤❡ ❨❛♥❣✲▼✐❧❧s ♠♦❞❡❧✱ ✇❡ ❤❛✈❡ ❛ ▲❛❣r❛♥❣✐❛♥ ✇❤✐❝❤ ✐s ❛♥❛❧♦❣♦✉s t♦ t❤❡ ◗❊❉ ▲❛❣r❛♥❣✐❛♥✱ ❜✉t ❣❡♥❡r❛❧✐③❡❞ t♦ ❙❯✭✷✮ ●r♦✉♣✿

L=₋1 4F

µν a F

a

µν + ¯ψN(γµDµ−im)ψN ✭✷✳✶✮

❲✐t❤ Fa

µν = ∂µπaν −∂νπaµ +gǫabcπµbπcν ❜❡✐♥❣ t❤❡ str❡♥❣t❤ t❡♥s♦r ❢♦r ❙❯✭✷✮✱ ❣ t❤❡ ❝♦✉♣❧✐♥❣

❝♦♥st❛♥t✱ N = n, p ♠❡❛♥s ✧♥✉❝❧❡♦♥✧✱ ❛♥❞ Dµ = ∂µ−igτaπµa t❤❡ ❝♦✈❛r✐❛♥t ❞❡r✐✈❛t✐✈❡ ✇❤❡r❡ τ

❛r❡ t❤❡ P❛✉❧✐✬s ♠❛tr✐❝❡s ✭❛✱ ❜✱ ❝ ❂ ✶✱ ✷✱ ✸✮✳ ❚❤❡ ✜❡❧❞ π r❡♣r❡s❡♥t t❤❡ ♣✐♦♥ ✐♥ t❤❡ s❡♥s❡ t❤❛t

π0_{= π}3_✱π+ _=π1_+iπ2_✱π−_=π1_−iπ2_✳

❚❤✐s ▲❛❣r❛♥❣✐❛♥ ✐s ❣❛✉❣❡ ✐♥✈❛r✐❛♥t ✐❢ t❤❡ π q✉❛♥t❛s ❛r❡ ♠❛ss❧❡ss✱ ❛♥❞ ❜❡❝❛✉s❡ ♦❢ t❤✐s✱ t❤✐s

♠♦❞❡❧ ✇❛s ♥♦t ❣❡♥❡r❛❧❧② ❛❝❝❡♣t❡❞ ❛t t❤❛t t✐♠❡ ❜② t❤❡ s❝✐❡♥t✐✜❝ ❝♦♠♠✉♥✐t②✱ ❛t t❤❡ ♣♦✐♥t ♦❢ ❛❧♠♦st ❜❡✐♥❣ ❢♦r❣♦tt❡♥ ❬✶✽❪✳

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✻

♠♦❞❡❧ t❤❡r❡ ❛r❡ t❤r❡❡ t②♣❡s ♦❢ q✉❛r❦s ✭✉✱ ❞✱ s✮ ❛ss♦❝✐❛t❡❞ t♦ ❛ ✧✢❛✈♦✉r✧ ❙❯✭✸✮ s②♠♠❡tr② ❣r♦✉♣✳ ❚❤r❡❡ q✉❛r❦s ❝♦♥st✐t✉t❡ ❛ ❇❛r✐♦♥ ✭♣r♦t♦♥✱ ♥❡✉tr♦♥✱ Λ✱ ❡t❝✳✮ ❛♥❞ t❤❡ ▼❡s♦♥s ✭π✱ ρ✱ ❡t❝✳✮

❛r❡ ❝♦♥st✐t✉t❡❞ ❜② ❛ q✉❛r❦✲❛♥t✐q✉❛r❦ ♣❛✐r✳ ❇❡s✐❞❡s ✢❛✈♦r✱ ❛♥♦t❤❡r q✉❛♥t✉♠ ♥✉♠❜❡r ❢♦r t❤❡ q✉❛r❦s ✇❛s ♣♦st✉❧❛t❡❞✱ t❤❡ ✧❝♦❧♦r✧ ✭❣r❡❡♥✱ ❜❧✉❡✱ r❡❞✮✱ ✜rst ♣r♦♣♦s❡❞ ❜② ❖✳❲✳ ●r❡❡♥❜❡r❣ ❬✷✵❪ ✐♥ ♦r❞❡r t♦ s♦❧✈❡ t❤❡ st❛t✐st✐❝ ♣r♦❜❧❡♠ ✐♥ ❤❛❞r♦♥s ❛♥❞ ✐t ✇❛s ❝♦♥✜r♠❡❞ ❧❛t❡r t❤❡♦r❡t✐❝❛❧❧② ❛♥❞ ❜② ❡①♣❡r✐♠❡♥t❛❧ ❢❛❝ts ✭❈❛♥❝❡❧❧❛t✐♦♥ ♦❢ ❛♥♦♠❛❧✐❡s✱ ❉❡❝❛② ♦❢ π0 _{✐♥t♦ t✇♦ ♣❤♦t♦♥s✱ t❤❡ ❜r❛♥❝❤✐♥❣}

r❛t✐♦ R = σ(e+_e−

→ hadrons)/σ(e+_e−

→ µ+_µ−_{)/✱ ❡t❝✳ ❬✷✶❪✮✳ ❚❤❡ q✉❛r❦✲q✉❛r❦ ✐♥t❡r❛❝t✐♦♥}

✇❛s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❝♦❧♦r ❝❤❛r❣❡ ❛♥❞ ✉s✐♥❣ t❤❡ ❨❛♥❣✲▼✐❧❧s t❤❡♦r② ❢♦r ✧❝♦❧♦r✧ ❙❯✭✸✮ s②♠♠❡tr② ❣r♦✉♣✱ ✇❛s ❜♦r♥ t❤❡ ◗✉❛♥t✉♠ ❈❤r♦♠♦❞②♥❛♠✐❝s ✭◗❈❉✮ ❛s t❤❡ t❤❡♦r② t♦ ✉♥❞❡rst❛♥❞ t❤❡ str♦♥❣ ✐♥t❡r❛❝t✐♦♥❬✷✷❪✳ ■♥◗❈❉ t❤❡ ❣❛✉❣❡ ❜♦s♦♥s ❛r❡ ❝❛❧❧❡❞ ✧❣❧✉♦♥s✧ ✇❤✐❝❤ ❝♦✉♣❧❡ ✇✐t❤ t❤❡ q✉❛r❦s ❞✉❡ t♦ t❤❡ ❝♦❧♦r ❝❤❛r❣❡✱ ❛s ✇❡❧❧ ❛s ❝♦✉♣❧❡ t♦ t❤❡♠s❡❧✈❡s✳

◗❈❉ ❤❛❞ ❛ ❣r❡❛t s✉❝❝❡ss ✐♥t❤❡ ❡①♣❧❛♥❛t✐♦♥♦❢ t❤❡ str♦♥❣ ✐♥t❡r❛❝t✐♦♥❛t ❤✐❣❤ ❡♥❡r❣✐❡s✱ ❡s♣❡❝✐❛❧❧② ✐♥ t❤❡ ❡①♣❡r✐♠❡♥ts ♦❢ ❉❡❡♣ ■♥❡❧❛stt✐❝ ❙❝❛tt❡r✐♥❣ ❛♥❞ t❤❡ ♣❛rt♦♥ ♠♦❞❡❧✱ ✇❤❡r❡ t❤❡ ❝❛❧❝✉❧❛t❡❞ s❝❛tt❡r✐♥❣ ♣r♦❝❡ss❡s ❛r❡ ♠❡❛s✉r❡❞ ❡①♣❡r✐♠❡♥t❛❧❧② ✇✐t❤ ❤✐❣❤ ♣r❡❝✐s✐♦♥ ❬✷✸❪✳ ❇❡❝❛✉s❡ ♦❢ ♥♦ ❡✈✐❞❡♥❝❡ ♥♦r ❞❡t❡❝t✐♦♥ ♦❢ ❢r❡❡ q✉❛r❦s✱ ✐t ✇❛s ♣r♦♣♦s❡❞ ♦♥❡ ♦❢ t❤❡ ♠❛✐♥ ♣r♦♣❡rt✐❡s ♦❢ t❤❡ ◗❈❉✱ t❤❡ ✧❝♦♥✜♥❡♠❡♥t ♦❢ q✉❛r❦s ❛♥❞ ❣❧✉♦♥s✧ ✐✳❡✳ q✉❛r❦s ❛♥❞ ❣❧✉♦♥s ✐♥t❡r❛❝t str♦♥❣❧② ✐♥ ❛ ❧✐♠✐t❡❞ r❡❣✐♦♥ ♦❢ s♣❛❝❡ ❛♥❞ t❤❡ ❢♦r❝❡ t♦ s❡♣❛r❛t❡ t❤❡♠ ❜❡②♦♥❞ t❤❛t ❧✐♠✐t ♠✉st ❜❡ ✐♥✜♥✐t② ❬✷✹❪✳ ❚❤❡ ♦t❤❡r ♠❛✐♥◗❈❉ ♣r♦♣❡rt② ✐s ❛s②♠♣t♦t✐❝ ❢r❡❡❞♦♠✱ ❞❡✈❡❧♦♣❡❞ ❜② ❉✳ ●r♦ss✱ ❋✳ ❲✐❧❝③❡❦ ❛♥❞ ❍✳ P♦❧✐t③❡r ❬✷✺✱ ✷✻❪✱ ✐♥ ♦r❞❡r t♦ ❡①♣❧❛✐♥ t❤❡ ♣r♦❜❧❡♠ ♦❢ t❤❡ s❝❛❧✐♥❣ ✐♥ t❤❡ ❞❡❡♣ ✐♥❡❧❛stt✐❝ s❝❛tt❡r✐♥❣✳ ❆s②♠♣t♦t✐❝ ❢r❡❡❞♦♠ t❡❧❧s ✉s t❤❛t t❤❡ str♦♥❣ ✐♥t❡r❛❝t✐♦♥✐s ✧✇❡❛❦✧ ✐♥t❤❡ ❧✐♠✐t ♦❢ s❤♦rt ❞✐st❛♥❝❡s ✭♦r ❤✐❣❤ ♠♦♠❡♥t❛✮✱ ❛♥❞ ✐s str♦♥❣❡r ✇❤❡♥ t❤❡ ❞✐st❛♥❝❡ ✐s ✐♥❝r❡❛s❡❞✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ t❤❡ ✧str♦♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t✧ ✐♥❝r❡❛s❡s ✇✐t❤ t❤❡ ❞✐st❛♥❝❡✳ ◗❈❉ ❞❡s❝r✐❜❡ t❤❡ ❛s②♠♣t♦t✐❝ ❢r❡❡❞♦♠✱ ❜✉t ✐t ❞♦❡s ♥♦t ❤❛✈❡ ✉♣ t♦ ♥♦✇ ❛♥ ❛♥❛❧②t✐❝❛❧ ❞❡s❝r✐♣t✐♦♥ ♦❢ ❝♦♥✜♥❡♠❡♥t✳

❙✐♥❝❡ t❤❡ ✼✵✬s t❤❡ s♣❡❝tr✉♠ ♦❢ ♠❡s♦♥s ✇❛s ❡①t❡♥s✐✈❡❧② st✉❞✐❡❞✱ s♦ ♠❛♥② ❞✐✛❡r❡♥t ♠♦❞❡❧s ♦❢ ♥♦♥r❡❧❛t✐✈✐st✐❝ ✐♥t❡r❛❝t✐♥❣ q✉❛r❦s ❜♦✉♥❞❡❞ ❜② ❛ r❛❞✐❛❧ ♣♦t❡♥t✐❛❧ V(r)✇❡r❡ ♣r♦♣♦s❡❞✳ ❚❤❡ ♠♦st

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✼

V(r) =₋4 3

αs

r +KFr ✭✷✳✷✮

❲❤❡r❡✱ KF ✐s t❤❡ str✐♥❣ t❡♥s✐♦♥ ❛♥❞ αs ✐s t❤❡ str♦♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t✳

❚❤❡ ✜rst t❡r♠ ✐♥ ❈♦r♥❡❧❧✬s ♣♦t❡♥t✐❛❧ ✐s t❤❡ ❝♦✉❧♦♠❜✲❧✐❦❡ t❡r♠✱ ✇❤✐❝❤ ❡①♣r❡ss❡s t❤❡ ❛♥❛❧♦❣② ❜❡t✇❡❡♥ t❤❡ ◗❊❉ ❛♥❞ ◗❈❉ ❛♥❞ t❤❡ s❡❝♦♥❞ ✐s t❤❡ ❝♦♥✜♥✐♥❣ t❡r♠✱ ✇❤✐❝❤ ❞❡♣❡♥❞s ❧✐♥❡❛r❧② ✇✐t❤ t❤❡ ❞✐st❛♥❝❡✱ s♦ ✐t ✐s ❞♦♠✐♥❛♥t ❢♦r ❧♦♥❣ ❞✐st❛♥❝❡s✳

❆♥♦t❤❡r ♠♦❞❡❧ ✉s❡❞ t♦ st✉❞② t❤❡ ❤❛❞r♦♥ ❞②♥❛♠✐❝ ❛♥❞ t❤❡ ❉■❙ ✇❛s t❤❡ ❜❛❣ ♠♦❞❡❧ ✐♥ ✇❤✐❝❤ r❡❧❛t✐✈✐st✐❝ ♠❛ss❧❡ss q✉❛r❦s s✉rr♦✉♥❞❡❞ ❜② ❛ ❝♦♥✜♥✐♥❣ ✧❜❛❣✧ ✐♥t❡r❛❝t ✇❡❛❦❧② ❡①❝❡♣t ✐♥ t❤❡ ❧✐♠✐t✐♥❣ r❡❣✐♦♥ ❬✷✾❪✳ ❇♦t❤ ♠♦❞❡❧s✱ t❤❡ ♥♦♥r❡❧❛t✐✈✐st✐❝ ✐♥t❡r❛❝t✐♥❣ ♣♦t❡♥t✐❛❧ ❛♥❞ t❤❡ ❝♦♥✜♥✐♥❣ ❜❛❣ ❞✐❞ s❤♦✇ ❛ str✐❦✐♥❣ s✉❝❝❡ss ❬✷✵✱ ✷✾❪✱ t❤❡r❡❢♦r❡ t❤❡② ♠✉st ❤❛✈❡ s♦♠❡ ❡❧❡♠❡♥ts ♦❢ tr✉t❤✱ ❞❡s♣✐t❡ t❤❡② ❛r❡ ❝♦♥tr❛❞✐❝t♦r✐❡s❀ ✐♥ ♦♥❡ ♣❧❛❝❡ ✇❡ ❤❛✈❡ ❤❡❛✈② ♥♦♥r❡❧❛t✐✈✐st✐❝ ❝♦♥st✐t✉❡♥t q✉❛r❦s✱ ♦❢ ♠❛ss ▼✱ ✇✐t❤ ❧✐tt❧❡ ❜✐♥❞✐♥❣✱ ❛♥❞ ❢♦r t❤❡ ♦t❤❡r ✇❡ ❤❛✈❡ ✈❡r② ❧✐❣❤t r❡❧❛t✐✈✐st✐❝ q✉❛r❦✱ ✇✐t❤ ❝✉rr❡♥t ♠❛ss ♠✱ ♦❜❡②✐♥❣ t❤❡ ❝❤✐r❛❧ s②♠♠❡tr② ❝♦♥str❛✐♥s ♦❢ ❝✉rr❡♥t ❛❧❣❡❜r❛ ✭✇❤❡r❡ M _≫ m✮✳

❈❤✐r❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ✐s ❛ s❝❤❡♠❡ t❤❛t ✉♥✐❢② t❤❛t ❦✐♥❞ ♦❢ ❝♦♥tr❛❞✐❝t✐♦♥s✱ ❛s ♣♦✐♥t❡❞ ♦✉t ❜② ❇✳ ▼❛❝❑❡❧❧❛r ❡t✳❛❧ ❬✸✵❪ ❛♥❞ ❡♠❡r❣❡s ❛s ❛ ❡①♣❧❛♥❛t✐♦♥ ♦❢ ♦t❤❡r ❦✐♥❞ ♦❢ ♣❤❡♥♦♠❡♥❛ ❛s ✇❡ ✇✐❧❧ ❞✐s❝✉ss ✐♥ t❤❡ ♥❡①t s❡❝t✐♦♥s✳

✷✳✷ ●❡♥❡r❛❧ Pr♦♣❡rt✐❡s

◗❈❉ ✐s ❛ ❣❛✉❣❡ t❤❡♦r② ❢♦r ❙❯✭✸✮ s②♠♠❡tr② ❣r♦✉♣✱ ✇❤✐❝❤ ✐s ✉s❡❞ t♦ ❞❡s❝r✐❜❡ t❤❡ ❢✉♥❞❛♠❡♥t❛❧ ✐♥t❡r❛❝t✐♦♥ ❜❡t✇❡❡♥ q✉❛r❦s ❛♥❞ t❤❡ ✐♥t❡r♠❡❞✐❛r② ❜♦s♦♥s✳ ❚❤❡ ◗❈❉ ▲❛❣r❛♥❣✐❛♥ ✐s✿

LQCD =−

1 4G

µν a G

a µν +

Nf

i,j

¯

ψi(γµDµ−im)_ijψj ✭✷✳✸✮

❲❤❡r❡Ga

µν =∂µAaν−∂νAaµ+gsǫabcAbµAcν✱ ❜❡✐♥❣ t❤❡ str❡♥❣t❤ t❡♥s♦r ❢♦r ❙❯✭✸✮✱ gs t❤❡ str♦♥❣

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✽

♠❛tr✐❝❡s ❛♥❞ ❛✱ ❜✱ ❝ ❂ ✶✱ ✳✳✳✱ ✽✮✳ ❚❤❡ ✜❡❧❞ Ar❡♣r❡s❡♥t t❤❡ ❣❧✉♦♥ ❛♥❞ Nf ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✢❛✈♦rs

♣r❡s❡♥t ✐♥ t❤❡ t❤❡♦r②✳

■t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ❞❡✜♥❡ ❛ ❣❧✉♦♥ ♣r♦♣❛❣❛t♦r ❢r♦♠ t❤✐s ▲❛❣r❛♥❣✐❛♥ ✇✐t❤♦✉t ♠❛❦✐♥❣ ❛ ❝❤♦✐❝❡ ♦❢ ❣❛✉❣❡✳ ❚❤✐s ❧❡❛❞ ✉s t♦ ♠❛❦❡ ✉s❡ ♦❢ t❤❡ ❋❛❞❡❡✈✲P♦♣♣♦✈ ♣r♦❝❡❞✉r❡ t♦ q✉❛♥t✐③❡ t❤❡ ❣❛✉❣❡ ✜❡❧❞s✱ ❛♥❞ ✇✐t❤ t❤✐s ✇❡ ♦❜t❛✐♥ ❛ ❣❛✉❣❡ ✜①✐♥❣ t❡r♠ ❛♥❞ ❛ ❣❤♦st ❝♦♥tr✐❜✉t✐♦♥ ❬✸✶❪✿

Lα=−

1 2α(∂µA

µ₎2 _❛♥❞

Lghost=∂µηa†

D_abµηb2

✭✷✳✹✮

❲✐t❤ t❤❡s❡ ❢♦r♠✉❧❛s ❡q✉❛t✐♦♥ ✭✷✳✸✮ ✐s tr❛♥s❢♦r♠❡❞ ✐♥t♦ t❤❡ ❢♦❧❧♦✇✐♥❣✿

Lef fQCD =−

1 4G µν a G a µν− 1 2α(∂µA

µ₎2

+

Nf

i,j

¯

ψi(γµDµ−im)_ijψj +∂µηa†

Dµ_abηb

✭✷✳✺✮

❚❤✐s ▲❛❣r❛♥❣✐❛♥ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ ♠♦♠❡♥t✉♠ s♣❛❝❡ ❋❡②♥♠❛♥ r✉❧❡s✿

i γp₋m+iǫ

−iδab

p2_+iǫ

gµν+ (α−1) pµpν

p2

µ _ν

p p

a b

❋✐❣✉r❡ ✷✳✷✿ ◗✉❛r❦ ❛♥❞ ❣❧✉♦♥ ❜❛r❡ ♣r♦♣❛❣❛t♦rs✳

µ, c µ, c

−igsTabc gsf

abc_qµ

a b a b q q q q

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✾

µ, a

ν, b

λ, c

−gsfabc[(p−q)λgµν+ (q−r)µgνλ+ (r−p)νgλµ]

p+q+r= 0

q

p r

❋✐❣✉r❡ ✷✳✹✿ ❚❤r❡❡ ✲ ❣❧✉♦♥s ✈❡rt❡①✳

µ, a

ν, b λ, c

ρ, d

−ig2 s

fabe_fcde_{(gµλgνρ−gµρgνλ)} face_fdbe_(gµρgνλ

−gµνgρλ)

fade_fbce_(gµνgλρ

−gµλgνρ)

❋✐❣✉r❡ ✷✳✺✿ ❋♦✉r ✲ ❣❧✉♦♥s ✈❡rt❡①✳

❚❤❡s❡ r✉❧❡s ❛❧❧♦✇ ✉s t♦ ♣❡r❢♦r♠ ♣❡rt✉r❜❛t✐✈❡ ❝❛❧❝✉❧❛t✐♦♥s ✐♥ t❤❡ ❤✐❣❤ ❡♥❡r❣② r❡❣✐♠❡ ♦❢ ◗❈❉ ✐♥ t❡r♠ ♦❢ ♣♦✇❡rs ♦❢ gs✳ ❚❤✐s s❝❤❡♠❡ ✐s ❥✉st✐✜❡❞ ❜② t❤❡ ❛s②♠♣t♦t✐❝ ❢r❡❡❞♦♠ ♣r♦♣❡rt②✿ ❚❤❡

r✉♥♥✐♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t ❜❡❝♦♠❡ ♠♦r❡ ❛♥❞ ♠♦r❡ s♠❛❧❧ ✐♥ t❤❡ ❤✐❣❤ ❡♥❡r❣② ❞♦♠❛✐♥✱ ❜✉t ✐♥ t❤❡ ❧♦✇ ♠♦♠❡♥t✉♠ ❞♦♠❛✐♥ ✐t ❜❡❝♦♠❡ ❤✐❣❤✱ s♦ ❤✐❣❤ t❤❛t t❤❡ ♣❡rt✉r❜❛t✐✈❡ ❡①♣❛♥s✐♦♥ ✐s ♥♦t ❛♥②♠♦r❡ ❥✉st✐✜❡❞ ❬✷✺❪✳

❙✐♥❝❡ ❞②♥❛♠✐❝❛❧ ❝❤✐r❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ✐s ❛ ❧♦✇ ❡♥❡r❣② ♣❤❡♥♦♠❡♥♦♥✱ ✇❡ ♥❡❡❞ ♦t❤❡r t♦♦❧s ✐♥ ♦r❞❡r t♦ st✉❞② ✐t✳ ❚❤❡ t✇♦ ♠❛✐♥ t♦♦❧s ✉s❡❞ t♦ st✉❞② t❤❡ ♥♦♥✲♣❡rt✉r❜❛t✐✈❡ r❡❣✐♠❡ ♦❢ ◗❈❉ ❛r❡ ▲❛tt✐❝❡ ◗❈❉ ❬✸✷❪ ❛♥❞ t❤❡ ❙❤✇✐♥❣❡r✲❉②s♦♥ ❡q✉❛t✐♦♥s ✭❙❉❊✮ ❬✸✸❪✳

✷✳✸ ❙❝❤✇✐♥❣❡r ❉②s♦♥ ❊q✉❛t✐♦♥s

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✶✵

❚❤❡ ❙❉❊ ❛r❡ ❛♥ ✐♥✜♥✐t❡ t♦✇❡r ♦❢ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥s t❤❛t r❡❧❛t❡ t❤❡ ♥✲♣♦✐♥t ❢✉♥❝t✐♦♥s ♦❢ t❤❡ t❤❡♦r② ❛♥❞ s✐♥❝❡ t❤❡② ❞♦ ♥♦t ❤❛✈❡ ❛♥ ❡①❛❝t s♦❧✉t✐♦♥ ✭❛s ❢❛r ❛s ✇❡ ❦♥♦✇✮✱ ✇❡ ❤❛✈❡ t♦ ❝♦♥s✐❞❡r s♦♠❡ ❛♣♣r♦①✐♠❛t✐♦♥s ♦r tr✉♥❝❛t✐♦♥s ✐♥ ♦r❞❡r t♦ s♦❧✈❡ t❤❡♠✳ ❍✐st♦r✐❝❛❧❧② t❤❡ ❙❉❊ ✇❡r❡ ❞❡❞✉❝❡❞ ❞✐❛❣r❛♠♠❛t✐❝❛❧❧②✱ ❜✉t ❛ s✐♠♣❧❡ ✇❛② t♦ ❞❡❞✉❝❡ t❤❡♠ ✐s ✐♥tr♦❞✉❝✐♥❣ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥❛❧ ❢♦r t❤❡ ♥✲♣♦✐♥t ❢✉♥❝t✐♦♥s ❬✸✹❪✿

Z[J, ξ,ξ¯] =

ˆ D

A,ψ, ψ¯

e−S[A,ψ,ψ¯ ;J,η,η¯] ✭✷✳✻✮

❲❤❡r❡ t❤❡ ♥♦t❛t✐♦♥ ✉s❡❞ ✐s✿

D

A,ψ, ψ¯

=DADψDψ¯ S[ϕi;ξi] =S[ϕi]−ϕiξi

ϕi ≡

A,ψ, ψ¯

ξi ≡(J, η,η¯)

ϕiξi=

ˆ dd_x

Jµ(x)Aµ(x) + ¯ψ(x)η(x) + ¯η(x)ψ(x)

■♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ✇❡ ✉s❡ t❤❡ ❛❝t✐♦♥✿

S[ϕi] =

ˆ ddx

₁

4(Fµν(x))

2₊ 1

2α(∂µAµ(x))

2_{+ ¯ψ(x) (γ.D+m)ψ(x) ✭✷✳✼✮}

❲❤❡r❡✿

Fµν(x) =∂µAν(x)−∂νAµ(x) ❛♥❞ Dµ =∂µ−ieAµ ✭✷✳✽✮

❋♦r ❛♥ ❛r❜✐tr❛r② ❢✉♥❝t✐♦♥ ❛♥❞ ❛ ❝❧♦s❡❞ ♣❛t❤ ✭❝✳♣✳✮ ´

c.p.dx d

dxf(x) = 0✱ ❛♥❞ s✐♥❝❡ ✐♥ t❤❡

❞❡✜♥✐t✐♦♥ ♦❢ Z[ϕi] ✐♥ t❤❡ ❊✉❝❧✐❞❡❛♥ ❙♣❛❝❡✱ ✇❡ ♣❡r❢♦r♠ t❤❡ ♣❛t❤ ✐♥t❡❣r❛❧ ♦✈❡r ❝❧♦s❡❞ ♣❛t❤s ♦❢

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✶✶

ˆ Dϕi

δ δϕi

e−S[ϕi]+ϕiξi _{= 0}

δS δϕi δ δξi

−ξi

Z[ξi] = 0 ✭✷✳✾✮

❚❤❡s❡ ❛r❡ t❤❡ ❙❝❤✇✐♥❣❡r✲❉②s♦♥ ❡q✉❛t✐♦♥s ✐♥ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✳

◆♦✇✱ ❝♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ ✭✷✳✾✮ ❢♦r ϕ(x) =A(x) ❛♥❞ ξ(x) =J(x)✿

δS δAµ(x)

δ δJµ

,−δ δη,

δ δη¯

−Jµ(x)

Z[J, η,η¯] = 0

❚❤❡ ❞❡r✐✈❛t✐✈❡ ♦✈❡r t❤❡ ❛❝t✐♦♥ st❛♥❞✿

δS δAµ(x)

=_−△−1

µν (α)Aν(x)−ieψ¯(x)γµψ(x)

❲❤❡r❡✿

−△−µν1(α) =δµν∂2−

1₋ 1

α

∂ν∂µ

■♥ t❡r♠s ♦❢ t❤❡ ❣❡♥❡r❛t✐♥❣ ❢✉♥❝t✐♦♥❛❧ ♦❢ t❤❡ ❢✉❧❧ ❝♦♥♥❡❝t❡❞ ●r❡❡♥✬s ❢✉♥❝t✐♦♥s W ❛♥❞ t❤❡

❝❧❛ss✐❝❛❧ ❛❝t✐♦♥ Γ✱ ✇❤✐❝❤ ❛r❡ ❞❡✜♥❡❞ ❛s✿

Z[ξi] = exp−w[ξi] and Γ

ϕcl i

=w[ξi] +ϕcliξi ✭✷✳✶✵✮

❲❡ ♦❜t❛✐♥ t❤❡ ❡q✉❛t✐♦♥ ∗_✿

−△−µν1δ(x−y)−ie

ˆ

dduddw T r[γµS(x, u) Γν(y;u, w)S(w, x)] =D−µν1(x, y) ✭✷✳✶✶✮

∗_{❍❡r❡ ✇❡ ❤❛✈❡ s❡t ξ}

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✶✷

❲❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ r❡❧❛t✐♦♥✿

− δ

2_w

δηA(x)δη¯B(z)

=

_δ2_Γ

δψB(z)δψ¯A(x)

−1

✭✷✳✶✷✮

❆♥❞ t❤❡ ❞❡✜♥✐t✐♦♥s✿

δ2_Γ

δAν(y)δAµ(x)⑤A=ψ= ¯ψ=0

=D−1

µν(x, y) =✭❋✉❧❧ ♣❤♦t♦♥ ♣r♦♣❛❣❛t♦r✮ ✭✷✳✶✸✮

δ2_Γ

δψ(y)δψ¯(x)⑤A=ψ= ¯ψ=0=S

−1_{(x, y) =✭❋✉❧❧ ❡❧❡❝tr♦♥ ♣r♦♣❛❣❛t♦r✮ ✭✷✳✶✹✮}

δ3_Γ

δAν(z)δψ(y)δψ¯(x)⑤A=ψ= ¯ψ=0

= Γν(z; x, y) =✭❋✉❧❧ ✈❡rt❡① ❢✉♥❝t✐♦♥✮ ✭✷✳✶✺✮

■♥ ♠♦♠❡♥t✉♠ s♣❛❝❡ t❤❡ ❡q✉❛t✐♦♥ ✭✷✳✶✶✮ ✐s ✇r✐tt❡♥ ❛s✿

Dµν(q)−1 =D(0)µν(q)

−1_−Π

µν(q)

❲❤❡r❡ ✇❡ ❤❛✈❡ ❞❡✜♥❡❞ t❤❡ ♣❤♦t♦♥ ♣♦❧❛r✐③❛t✐♦♥ t❡♥s♦r ✭❋✐❣✉r❡ ✷✳✻✮✿

Πµν(p) =−ie

ˆ _dd_k

(2π)d tr[γµS(k) Γν(k, p−k)S(p−k)]

❯s✐♥❣ t❤❡ ▲♦r❡♥t③ str✉❝t✉r❡ ❞❡✜♥❡❞ ❜② t❤❡ ❲❛r❞✲❚❛❦❛❤❛s❤✐ ✐❞❡♥t✐t✐❡s ✇❡ ❤❛✈❡✿

Dµν(q2) =

d(q2₎

q2

δµν −

qµqν

q2

+ α

q2

qµqν

q2 ✭✷✳✶✻✮

❲❤❡r❡ d(q2_{) = 1/[1 + Π(q}2_{)] r❡♣r❡s❡♥ts t❤❡ ♣❤♦t♦♥ ✇❛✈❡ ❢✉♥❝t✐♦♥ r❡♥♦r♠❛❧✐③❛t✐♦♥ ❛♥❞ α}

t❤❡ ❣❛✉❣❡✲✜①✐♥❣ ♣❛r❛♠❡t❡r✳

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✶✸

k

p₋k

p, µ p, ν

Πµν(p) =−i

❋✐❣✉r❡ ✷✳✻✿ P❤♦t♦♥ ♣♦❧❛r✐③❛t✐♦♥ t❡♥s♦r✳

❆ s✐♠✐❧❛r ♣r♦❝❡❞✉r❡ ❜✉t ✇✐t❤ t❤❡ s♣✐♥♦r ✜❡❧❞s ❧❡❛❞ ✉s t♦ ♦❜t❛✐♥ t❤❡ ❉❙❊❢♦r t❤❡ q✉❛r❦ ♣r♦♣❛❣❛t♦r✳ ❋♦r s♣✐♥♦rs✱ ❡q✉❛t✐♦♥ ✭✷✳✾✮ ✐s✿

δS δψ¯(x)

δ δJ,

−δ δη,

δ δη¯

−η(x)

Z[J, η,η¯] = 0

❲❤❡r❡✿

δ

δψ¯(x)S = (∂−ieA(x) +m)ψ(x)

❆❣❛✐♥ ✇❡ ✉s❡ t❤❡ ❝♦♥♥❡❝t❡❞ ●r❡❡♥ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ ❝❧❛ss✐❝❛❧ ❛❝t✐♦♥ ❛♥❞ ✉s❡ t❤❡ r❡❧❛t✐♦♥s ✉s❡❞ ❜❡❢♦r❡ ✭❡q✉❛t✐♦♥s ✭✷✳✶✵✮ t♦ ✭✷✳✶✺✮✮✿

(γ.∂+m)S(x, y) +ie ˆ

dd_w

dd_udd_{z D}

µν(x, z)γµS(x, u)Γν(z; u, w)

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✶✹

❖r✱ ✐♥ ♠♦♠❡♥t✉♠ s♣❛❝❡✿

S−1(p) =S₀−1(p) + Σ(p) ✭✷✳✶✼✮

❲❤❡r❡Σ(p)✐s t❤❡ q✉❛r❦ s❡❧❢✲❡♥❡r❣② ✭❋✐❣✉r❡ ✷✳✼✿

Σ(p) =ie

ˆ _dd_q

(2π)dγµDµν(p−q)S(q)Γν(p−q, q)

q p−q

p p

Σ(p) =−i

❋✐❣✉r❡ ✷✳✼✿ ❊❧❡❝tr♦♥ s❡❧❢✲❡♥❡r❣②✳

✷✳✹ ◗❊❉ ❣❛♣ ❊q✉❛t✐♦♥

❚❤❡ ❡q✉❛t✐♦♥ ✭✷✳✶✼✮ ❤❛s t❤❡ ❞✐❛❣r❛♠♠❛t✐❝ ❢♦r♠✿

= +

−1 ₋1

❋✐❣✉r❡ ✷✳✽✿ ❙❉❊ ❢♦r t❤❡ ❡❧❡❝tr♦♥ ♣r♦♣❛❣❛t♦r✳

❚❤✐s ❡q✉❛t✐♦♥ ✐s ♦✉r st❛rt✐♥❣ ♣♦✐♥t t♦ ❞❡❞✉❝❡ t❤❡ ❢♦r♠❛❧ ❡①♣r❡ss✐♦♥ ❢♦r t❤❡ ❣❛♣ ❡q✉❛t✐♦♥✳

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✶✺

S(p) = 1

γ.pA(p) +B(p) =

iZ(p)

γ.p+iM(p) ✭✷✳✶✽✮

❲✐t❤ t❤✐s✱ t❤❡ ❜❛r❡ ♣r♦♣❛❣❛t♦r ❛♥❞ t❤❡ s❡❧❢ ❡♥❡r❣② ✇❡ ♦❜t❛✐♥ t❤❡ r❡❧❛t✐♦♥s✿

M(p2₎

Z(p2₎ =m+

tr[

(p)]

tr[1] ✭✷✳✶✾✮

1

Z(p2₎ =1+

i tr[p

[p]]

p2_tr[1] ✭✷✳✷✵✮

■♥ t❤❡ ▲❛♥❞❛✉❣❛✉❣❡ ✭α = 0✮ t❤❡ ❣❧✉♦♥ ♣r♦♣❛❣❛t♦r ✐s ❝♦♥s✐❞❡r❛❜❧② s✐♠♣❧✐✜❡❞ ❛♥❞ ✐t ✐s ✇❡❧❧

❥✉st✐✜❡❞ t♦ ✉s❡ t❤❡ s♦ ❝❛❧❧❡❞ ✧r❛✐♥❜♦✇ ❛♣♣r♦①✐♠❛t✐♦♥✧ ❬✸✺❪✿

Γβ(q, p−q)≡ieγβ

Dαβ(p−q)≡

g

(p₋q)2

(p₋q)2

δαβ−

(p₋q)α(p−q)β

(p₋q)2

❆❢t❡r t❛❦✐♥❣ t❤❡ tr❛❝❡ t❤❡ ❡q✉❛t✐♦♥s ✭✷✳✶✾✮ ❛♥❞ ✭✷✳✷✵✮ ❜❡❝♦♠❡✿

M(p2₎

Z(p2₎ =m+ (d−1)e 2

ˆ _dd_q

(2π)d

¯

g2

(p₋q)2

(p₋q)2

M(q2₎

q2_+M2_(q2₎Z(q

2_{) ✭✷✳✷✶✮}

1

Z(p2₎ = 1 +ie 2

ˆ _dd_q

(2π)d

Z(q2_)¯g2

(p₋q)2

p2_[q2_+M2_(q2_)]

(d₋3)pq

(p₋q)2 +

2p(p₋q)q(p₋q) (p₋q)4

✭✷✳✷✷✮

❈❍❆P❚❊❘ ✷✳ ◗❯❆◆❚❯▼ ❈❍❘❖▼❖❉❨◆❆▼■❈❙ ✶✻

¯

g2_[(p−k)2_]≈¯g2_(p2_)θ

p2_−k2

+ ¯g2_(k2_)θ

k2_−p2

✭✷✳✷✸✮

❆❢t❡r t❤❡ ▲❆❑ ❛♣♣r♦①✐♠❛t✐♦♥ t❤❡ ❛♥❣✉❧❛r ✐♥t❡❣r❛t✐♦♥ ♦♥❧② ❛✛❡❝t t❤❡ ❧❛st ❜r❛❝❦❡t✳ ■t ❝❛♥ ❜❡ s❤♦✇♥ ✭s❡❡ ❆♣♣❡♥❞✐① ❆✮ t❤❛t t❤✐s ❛♥❣✉❧❛r ✐♥t❡❣r❛❧ ✐s ③❡r♦ ✐♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s✿

Id =

ˆ π 0

dθsind−2θ

(d₋3)pq(p₋q)2_{+ 2p}

(p−q)q(p−q)

(p₋q)4

= 0 ✭✷✳✷✹✮

❲✐t❤ t❤✐s✱ t❤❡ ❡q✉❛t✐♦♥s ✭✷✳✷✶✮ ❛♥❞ ✭✷✳✷✷✮ r❡❞✉❝❡ t♦✿

1

Z(q2₎ = 1

M(p2) =m0+e2

ˆ _d4_k

(2π)4

3d[(p₋k)2_]

(p₋k)2

M(k2₎

k2_+M2_(k2₎ ✭✷✳✷✺✮

❈❤❛♣t❡r ✸

❈❤✐r❛❧ ❙②♠♠❡tr②

▼❛ss❧❡ss ❢❡r♠✐♦♥s ❤❛✈❡ ❛ ❞❡✜♥❡❞ ❝❤✐r❛❧✐t②✳ ▲❛❣r❛♥❣✐❛♥s ✇✐t❤ ♠❛ss❧❡ss ❢❡r♠✐♦♥s ❛r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ✧❝❤✐r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✧✿

ψ(x)_→eiθγ5_ψ(x) ❲❤❡r❡θ ✐s ❛ ♣❛r❛♠❡t❡r ❛♥❞ γ5=iγ0γ1γ2γ3✳

❲❤❡♥ ❛ ♠❛ss t❡r♠ ❛♣♣❡❛rs ✐♥ t❤❡ ▲❛❣r❛♥❣✐❛♥ ✈✐❛ s♦♠❡ ♠❡❝❤❛♥✐s♠ ✭s♣♦♥t❛♥❡♦✉s ♦r ❞②♥❛♠✐❝❛❧✮✱ t❤✐s t❡r♠ ♠✐① t❤❡ t✇♦ ❝❤✐r❛❧ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ s♣✐♥♦r✱ s♦ t❤❛t t❤❡ ▲❛❣r❛♥❣✐❛♥ ✐s ♥♦t ❛♥② ♠♦r❡ ✐♥✈❛r✐❛♥t ✉♥❞❡r t❤❡ ❝❤✐r❛❧ tr❛♥s❢♦r♠❛t✐♦♥✳ ■♥ t❤✐s ❝❛s❡ ✇❡ s❛② t❤❛t ♦❝❝✉rr❡❞ ❛ ❈❤✐r❛❧ ❙②♠♠❡tr② ❇r❡❛❦✐♥❣✳

✸✳✶ ❈❤✐r❛❧ ❙②♠♠❡tr② ❇r❡❛❦✐♥❣

❆s ✇❡ s❛✐❞ ❜❡❢♦r❡✱ t❤❡ ♣r♦t♦♥ ♠❛ss ❛♥❞ t❤❡ ♠❡s♦♥ s♣❡❝tr♦s❝♦♣② s❤♦✇ ❡✈✐❞❡♥❝❡s ♦❢ s♦♠❡ ♠❡❝❤❛♥✐s♠ ♦❢ ♠❛ss ❣❡♥❡r❛t✐♦♥ ❛♥❞ s✐♥❝❡ ✐t ✐s ♣r❡s❡♥t ✐♥ t❤❡ ❤❛❞r♦♥s✱ ✐t ♠✉st ❜❡ r❡❧❛t❡❞ t♦ t❤❡ str♦♥❣ ✐♥t❡r❛❝t✐♦♥✳ ❉②♥❛♠✐❝❛❧ ❝❤✐r❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❝♦✉❧❞ ❡①♣❧❛✐♥ t❤❡ s✐❣♥✐✜❝❛♥t ❞✐✛❡r❡♥❝❡ ❜❡t✇❡❡♥ t❤❡ ♣✐♦♥s ❛♥❞ t❤❡ ♦t❤❡r ❤❛❞r♦♥s✳ ■t ✐s ✇❡❧❧ ❦♥♦✇ t❤❛t ✇❤❡♥ ❛ ❝♦♥t✐♥✉♦✉s s②♠♠❡tr② ♦❢ t❤❡ ▲❛❣r❛♥❣✐❛♥ ✐s ❜r♦❦❡♥✱ ❛♣♣❡❛r ✐♥ t❤❡ t❤❡♦r② s♦♠❡ ♠❛ss❧❡ss ♣❛rt✐❝❧❡s ❝❛❧❧❡❞ t❤❡ ●♦❧❞st♦♥❡

❈❍❆P❚❊❘ ✸✳ ❈❍■❘❆▲ ❙❨▼▼❊❚❘❨ ✶✽

❜♦s♦♥s ❬✸✼❪✳ ■♥ t❤❡ ♣✐❝t✉r❡ ♦❢ ❉❈❙❇ t❤❡ ♣✐♦♥s ❛r❡ t❤❡ ●♦❧❞st♦♥❡ ❜♦s♦♥s ♦❢ t❤❡ t❤❡♦r② ❛♥❞ t❤❡② ❛r❡ ♠❛ss✐✈❡ ❜❡❝❛✉s❡ t❤❡ ❝❤✐r❛❧ s②♠♠❡tr② ✐s ♥♦t ❛♥ ❡①❛❝t s②♠♠❡tr②✱ ❛♥❞ t❤✐s ❢❛❝t ✐s ✐❧❧✉str❛t❡❞ ❜② t❤❡ ✇❡❧❧ ❦♥♦✇♥ ●❡❧❧✲▼❛♥♥✲❖❛❦❡s✲❘❡♥♥❡r ❡q✉❛t✐♦♥ ❬✸✽❪ t❤❛t s❤♦✇ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ♣✐♦♥s ♠❛ss ❛♥❞ t❤❡ ❜❛r❡ q✉❛r❦ ♠❛ss✿

mq

¯

ψψ

=f_π2m2_π

❲❤❡r❡¯

ψψ

✐s t❤❡ q✉❛r❦ ❝♦♥❞❡♥s❛t❡✱ mπ t❤❡ ♣✐♦♥ ♠❛ss✱mq t❤❡ q✉❛r❦ ❜❛r❡ ♠❛ss ❛♥❞ fπ ✐s

t❤❡ ♣✐♦♥ ❞❡❝❛② ❝♦♥st❛♥t✳

✸✳✷ ◆❛♠❜✉✲❏♦♥❛✲▲❛s✐♥✐♦ ▼♦❞❡❧

❚❤❡ ◆❛♠❜✉✲❏♦♥❛✲▲❛s✐♥✐♦ ✐s ❛ str♦♥❣ ✐♥t❡r❛❝t✐♦♥ ♠♦❞❡❧ ✇✐t❤ ❛ ❢♦✉r ❢❡r♠✐♦♥ ✐♥t❡r❛❝t✐♦♥✳ ❚❤❡ ✐♠♣♦rt❛♥❝❡ ♦❢ t❤✐s ♠♦❞❡❧ ✐s t❤❛t ✐t ❝♦♥t❛✐♥s t❤❡ ♠❛✐♥ s②♠♠❡tr✐❡s r❡❧❛t❡❞ t♦ t❤❡ str♦♥❣ ✐♥t❡r❛❝t✐♦♥ ✐✳❡✳ ■s♦s♣✐♥ ✭✐♥ SU(2)✮ ❛♥❞ ❝❤✐r❛❧ s②♠♠❡tr②✳ ❚❤❡ ♠❛✐♥ s✉❝❝❡ss ♦❢ t❤❡ ♠♦❞❡❧ ✐s t❤❡ ♣♦ss✐❜✐❧✐t② ♦❢

❞❡s❝r✐❜❡ t❤❡ ❞②♥❛♠✐❝❛❧ ❝❤✐r❛❧ s②♠♠❡tr② ❜r❡❛❦✐♥❣ ❛♥❞ t♦ ❜❡ ❛❜❧❡ t♦ r❡♣r♦❞✉❝❡ t❤❡ ●♦❧❞❜❡r❣❡r✲ ❚r❡✐♠❛♥ ❛♥❞ t❤❡ ●❡❧❧✲▼❛♥♥✲❖❛❦❡s✲❘❡♥♥❡r r❡❧❛t✐♦♥s ✭❛s ♣♦✐♥t❡❞ ♦✉t ✐♥ ❬✸✾❪✮✳

❚❤❡ ◆❏▲ ❛❝t✐♦♥ ✐s✿

SN J L[ ¯ψψ] =

ˆ dd_x

¯

ψ(x) (γ.∂+m)ψ(x) +1 2G0

_¯

ψ(x)ψ(x)2

❲✐t❤ t❤✐s✱ ✇❡ ✉s❡ t❤❡ ❡q✉❛t✐♦♥ ✭✷✳✾✮✱ ❛♥❞✿

δSN J L

δψ¯(x) = (γ.∂+m)ψ(x) +G0

¯

ψ(x)ψ(x)

ψ(x)

❈❍❆P❚❊❘ ✸✳ ❈❍■❘❆▲ ❙❨▼▼❊❚❘❨ ✶✾

(γ.∂+m)S(x, y) + δ

4_W

δη(x)δη¯(x)δη(y)δη¯(x) =δ(x−y)

❚❤✐s ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ✇r✐tt❡♥ ❛s ✭✐♥ ♠♦♠❡♥t✉♠ s♣❛❝❡✮✿

S−1_{(p) =S}−1

0 (p) + ΣN J L(0)

❲❤❡r❡✿

ΣN J L(0) =G0

ˆ _dd_q

(2π)dT r[S(q)]

❯s✐♥❣ ❡q✉❛t✐♦♥ ✭✷✳✶✽✮ ❛♥❞ t❛❦✐♥❣ t❤❡ tr❛❝❡✱ ✇❡ ♦❜t❛✐♥ ❡q✉❛t✐♦♥s s✐♠✐❧❛r t♦ ✭✷✳✶✾✮ ❛♥❞ ✭✷✳✷✵✮✿

M(p2₎

Z(p2₎ =m+G0Nf

ˆ _dd_q

(2π)d

M(q2₎

q2_+M2_(q2₎Z(q

2_{) ✭✸✳✶✮}

1

Z(p2₎ = 1−G0Nf

ˆ _dd_q

(2π)d

Z(q2_{) (p} q)

p2_[q2_+M2_(q2_)] ✭✸✳✷✮

■♥ ❢♦✉r ❞✐♠❡♥s✐♦♥s ✇❡ ❤❛✈❡ t❤❡ ✐♥t❡❣r❛❧ ´π 0 dθsin

2_{θcosθ= 0 ✐♥ t❤❡ Z(p}2_{)t❡r♠✱ s♦ t❤❛t✿}

Z(p2) = 1

M(p2) =m+G0Nf

ˆ _d4_q

(2π)4

M(q2₎

q2_+M2_(q2₎

❆♥❞ t❤✐s ✐s t❤❡ ❣❛♣ ❡q✉❛t✐♦♥ ❢♦r t❤❡ ◆❏▲ ♠♦❞❡❧✳

❚❤✐s ❣❛♣ ❡q✉❛t✐♦♥ ♣r❡s❡♥ts ❛♥ ✐♠♣♦rt❛♥t r❡s✉❧t✱ ✐♥ t❤❡ ❝❤✐r❛❧ ❧✐♠✐t ✐✳❡✳ ✇❤❡♥ m = 0 t❤❡

❞②♥❛♠✐❝❛❧q✉❛r❦ ♠❛ss ❤❛s ❛ ♥♦♥③❡r♦ ✈❛❧✉❡ ❢♦r ❛♥② p✳ ❚❤✐s ✐s ❛♥ ❡①♣❧✐❝✐t ❢♦r♠ ♦❢ t❤❡ ❞②♥❛♠✐❝❛❧

❈❍❆P❚❊❘ ✸✳ ❈❍■❘❆▲ ❙❨▼▼❊❚❘❨ ✷✵

✸✳✸ ❉②♥❛♠✐❝❛❧ ●❧✉♦♥ ▼❛ss ❬✹✵❪

■♥ t❤❡ ❡❛r❧② ❡✐❣❤t✐❡s✱ ✇♦r❦✐♥❣ ✐♥ t❤❡ ▲❛♥❞❛✉ ❣❛✉❣❡ ❛♥❞ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡✱ ❈♦r♥✇❛❧❧ ♦❜t❛✐♥❡❞ ❛ ❣❛✉❣❡ ✐♥✈❛r✐❛♥t s♦❧✉t✐♦♥ ❢♦r t❤❡ ❣❧✉♦♥ ♣r♦♣❛❣❛t♦r t❤❛t ❜❡❤❛✈❡❞ ❛s 1/(k2_+m2_(k2_{)) ❬✹✶❪✳ ■♥ t❤✐s}

❝❛s❡✱ ❛s k2_{→ 0✱ t❤❡ ❢✉♥❝t✐♦♥ m}2_(k2_{)✇❛s ✐♥t❡r♣r❡t❡❞ ❛s ❛ ❞②♥❛♠✐❝❛❧ ❣❧✉♦♥ ♠❛ss ✇✐t❤ t❤❡ ❧✐♠✐t}

m2_(k2 _{→0) =m}2

g✳ ❚❤✐s s♦❧✉t✐♦♥✱ ✇❤✐❝❤ ❜❡❝❛♠❡ ❦♥♦✇♥ ❛s ❛ ✧❝♦♥✜♥❡❞ s♦❧✉t✐♦♥✧✱ r❡♣r♦❞✉❝❡s t❤❡

❡①♣❡❝t❡❞ ♣❡rt✉r❜❛t✐✈❡ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❣❧✉♦♥ ♣r♦♣❛❣❛t♦r ❛t ❧❛r❣❡ k2 _{❜❡❝❛✉s❡ m}2_(k2 _{→ ∞) = 0✳}

❋✐❣✉r❡ ✸✳✶✿ ❚❤❡ ♠❛ss✐✈❡ ❣❧✉♦♥ ♣r♦♣❛❣❛t♦r ✇✐t❤ nf = 3❛♥❞ ΛQCD = 300▼❡❱ ❬✷❪✳

❆ ❣r❡❛t st❡♣ ❛❤❡❛❞ ✐♥ t❤✐s ♣❤❡♥♦♠❡♥♦♥ ❤❛s ❛❧s♦ ❜❡❡♥ ♣r♦✈✐❞❡❞ ❜② t❤❡ ◗❈❉ ❧❛tt✐❝❡ s✐♠✉❧❛t✐♦♥s ✐♥ ▲❛♥❞❛✉ ❣❛✉❣❡✱ ✇❤✐❝❤ str♦♥❣❧② s✉♣♣♦rt t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛♥ ✐♥❢r❛r❡❞ ✜♥✐t❡ ❣❧✉♦♥ ♣r♦♣❛❣❛t♦r ❬✾✱ ✶✵✱ ✶✶❪✭❋✐❣✉r❡ ✸✳✶✮✳ ❚❤✐s ✐s ✐♥t❡r❡st✐♥❣ ❡♥♦✉❣❤✱ ❜❡❝❛✉s❡ ✐♥❞✐❝❛t❡s t❤❡ ❛♣♣❡❛r❛♥❝❡ ♦❢ ❛ ❞②♥❛♠✐❝❛❧ ♠❛ss s❝❛❧❡ ❢♦r t❤❡ ❣❧✉♦♥✱ ✇❤✐❝❤ ✐♠♣❧② ✐♥ t❤❡ ❡①✐st❡♥❝❡ ♦❢ ❛ ♥♦♥✲tr✐✈✐❛❧ ◗❈❉ ✐♥❢r❛r❡❞ ✜①❡❞ ♣♦✐♥t✱ ✐✳❡✳ t❤❡ ❢r❡❡③✐♥❣ ♦❢ t❤❡ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t ❛t t❤❡ ♦r✐❣✐♥ ♦❢ ♠♦♠❡♥t❛ ❬✹✷❪✳

❈❍❆P❚❊❘ ✸✳ ❈❍■❘❆▲ ❙❨▼▼❊❚❘❨ ✷✶

❈❤❛♣t❡r ✹

❉②♥❛♠✐❝❛❧ ◗✉❛r❦ ▼❛ss

✹✳✶ ❖♥❡✲❉r❡ss❡❞✲●❧✉♦♥ ❊①❝❤❛♥❣❡

❍❡r❡ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❣❛♣ ❡q✉❛t✐♦♥ ✭✷✳✷✺✮ ❢♦r ◗❈❉ ✐♥ t❤❡ ♠❛ss✐✈❡ ♦♥❡✲❞r❡ss❡❞✲❣❧✉♦♥ ❡①❝❤❛♥❣❡ ❝❛s❡ ✐♥❝❧✉❞✐♥❣ ❛ ❞②♥❛♠✐❝❛❧ ❣❧✉♦♥ ♠❛ss✿

M1g(p2) =C2

ˆ _d4_k

(2π)4

¯

g2_[(p−k)2_]

(p₋k)2_+m2

g

3M(k2₎

[k2_+M2_(k2_)] ✭✹✳✶✮

❲❤❡r❡ t❤❡ ❡✛❡❝t✐✈❡ ❝❤❛r❣❡ ✐s ❣✐✈❡♥ ❜②❬✹✶❪✿

¯

g2(k2) = 1

blnk2+4m2g

Λ2

QCD

✭✹✳✷✮

❲❤❡r❡✱ C2 ✐s t❤❡ q✉❛r❦ ❈❛s✐♠✐r ❡✐❣❡♥✈❛❧✉❡ (C2 = 4/3 ❢♦r q✉❛r❦s ✐♥ t❤❡ ❢✉♥❞❛♠❡♥t❛❧

r❡♣r❡s❡♥t❛t✐♦♥✮✱ b = 11N−2nf

48π2 ✐s t❤❡ ♦♥❡✲❧♦♦♣ ❝♦❡✣❝✐❡♥t ✐♥ t❤❡ ❜❡t❛✲❢✉♥❝t✐♦♥ ❢♦r t❤❡ SU(N) ❣r♦✉♣ ✇✐t❤ nf ✢❛✈♦rs✱ mg ✐s t❤❡ ❞②♥❛♠✐❝❛❧ ❣❧✉♦♥ ♠❛ss ✭❤❡r❡ ✇❡ ♥❡❣❧❡❝t t❤❡ r✉♥♥✐♥❣ ♦❢ t❤❡

❣❧✉♦♥ ♠❛ss✮ ❛♥❞ Λ2

QCD ✐s t❤❡ ◗❈❉ ♠❛ss s❝❛❧❡✳

❚♦ s♦❧✈❡ ✭✹✳✶✮ ✇❡ ✉s❡ t❤❡ ▲❆❑ ❛♣♣r♦①✐♠❛t✐♦♥ ✭✷✳✷✸✮ ❛♥❞ t❤❡ ❛♥❣❧❡ ❛♣♣r♦①✐♠❛t✐♦♥ ❬✹✹❪✿

ˆ dΩ4

1 (p₋k)2_+m2

g

≈2π2

_θ(p2_−k2₎

p2_+m2

g

+ θ(k

2_−p2₎

k2_+m2

g

❈❍❆P❚❊❘ ✹✳ ❉❨◆❆▼■❈❆▲ ◗❯❆❘❑ ▼❆❙❙ ✷✸

❙♦ ✇❡ ❤❛✈❡✿

M1g(p2) =

3C2

16π2 ˆ

dk2

_θ(p2_−k2₎

p2_+m2

g

¯

g2(p2) +θ(k

2_−p2₎

k2_+m2

g

¯

g2(k2)

_k2

[k2_+M2_(k2_)]M1g(k

2_{) ✭✹✳✸✮}

0.5 1.0 1.5 2.0

g

0.02 0.04 0.06 0.08

M

❋✐❣✉r❡ ✹✳✶✿ ❇✐❢✉r❝❛t✐♦♥ ♦❢ t❤❡ r✉♥♥✐♥❣ q✉❛r❦ ♠❛ss✳

❚❤✐s ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥ ❛❞♠✐t tr✐✈✐❛❧ s♦❧✉t✐♦♥s M(p2_{) = 0 ❛♥❞✭❜② ❛ss✉♠♣t✐♦♥✮ ❛ ♥♦♥tr✐✈✐❛❧}

♦♥❡ ✭❋✐❣✉r❡ ✹✳✶✮ ❛♥❞✇❤❡♥ ✇❡ ❞❡❛❧ ✇✐t❤ t❤❡ ❤✐❣❤ ♠♦♠❡♥t✉♠ r❡❣✐♦♥ ✐♥ t❤❡ ♥❡✐❣❤❜♦r❤♦♦❞♦❢ t❤❡ ❝r✐t✐❝❛❧ ♣♦✐♥t g0✱ ✐t ✐s s✉✣❝✐❡♥t t♦ ❝♦♥s✐❞❡r t❤❡ ❧✐♥❡❛r✐③❡❞ ✈❡rs✐♦♥ ♦❢ ✭✹✳✸✮ ❬✹✺✱ ✹✻❪ t♦ ✜♥❞ t❤❡

❝r✐t✐❝❛❧ ❜❡❤❛✈✐♦r✿

M1g(x) =λ

ˆ Λ2

0

dy

θ(x₋y)¯g2_(x)

x+m2

g

+

θ(y₋x)¯g2_(y)

y+m2

g

M1g(y) ✭✹✳✹✮

❲❤❡r❡ ✇❡ ❤❛✈❡ ✐♥tr♦❞✉❝❡❞ ❛♥ ❯❱ ❝✉t♦✛ Λ ❛♥❞✇❡ ✉s❡❞t❤❡ s✉❜st✐t✉t✐♦♥s✿

x=p2_{; y =k}2_{; λ=} 3C2

❈❍❆P❚❊❘ ✹✳ ❉❨◆❆▼■❈❆▲ ◗❯❆❘❑ ▼❆❙❙ ✷✹

✹✳✶✳✶ ❈♦♥st❛♥t ❈♦✉♣❧✐♥❣

❲❤❡♥ ❝♦♥s✐❞❡r✐♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t ✇❡ ✇r✐t❡ g¯2_{(x) ≈ g¯}2_{(0) ≡ g}

0 ❛♥❞ ✇❡ ❞❡✜♥❡✿ λ0 =

3g0C2/16π2✳ ❲✐t❤ t❤✐s ❛♣♣r♦①✐♠❛t✐♦♥✱ ❡q✉❛t✐♦♥ ✭✹✳✹✮ st❛♥❞✿

M1g(x) =λ0 ˆ Λ2

0

dy

θ(x₋y)

x+m2

g

+

θ(y₋x)

y+m2

g

M1g(y)

❚❤✐s ❡q✉❛t✐♦♥ ❝❛♥ ❜❡ ❡❛s✐❧② tr❛♥s❢♦r♠❡❞ ✐♥t♦ t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥✿

x+m2_g2

M′′

1g(x) + 2

x+m2_g

M′

1g(x) +λ0M1g(x) = 0

❲❤✐❝❤ ❛❞♠✐t t✇♦ ❛s②♠♣t♦t✐❝ s♦❧✉t✐♦♥s✿

M±

1g(x) =

x+m2

g

α±

❲❤❡r❡✿ α±=−(1/2)±

(1/4)₋λ0✳

■❢✇❡ ❤❛✈❡ ❛ tr✐✈✐❛❧ ❝♦✉♣❧✐♥❣✱ λ0 = 0 ❛♥❞ ✇❡ ❞♦ ♥♦t ❤❛✈❡ ❈❙❇✳ ❚❤❡ ❝❧❛ss✐❝❛❧ ❛♥❛❧②s✐s st❛t❡

t❤❛t ❢♦r s♦♠❡ ✈❛❧✉❡ ♦❢λ0 ✇❡ st❛rt ❤❛✈✐♥❣ ❈❙❇ ❛♥❞ t❤✐s ✈❛❧✉❡ ✐s r❡❧❛t❡❞ t♦ t❤❡ ✈❛❧✉❡ ✐♥ ✇❤✐❝❤ t❤❡

sq✉❛r❡ t✉r♥s t♦ ❛♥ ✐♠❛❣✐♥❛r② ♥✉♠❜❡r ❬✶✱ ✹✼❪✳ ❚❤✐s ❣✐✈❡ ✉s t❤❡ ❝♦♥❞✐t✐♦♥ ❢♦r ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥

λ0 ≥1/4✿

αs(0) =

¯

g(0) 4π ≥

π

3C2 ≥

0.8

❈❍❆P❚❊❘ ✹✳ ❉❨◆❆▼■❈❆▲ ◗❯❆❘❑ ▼❆❙❙ ✷✺

106

0.001 1 1000 106

q2MeV2 0.1

0.2 0.3 0.4 0.5 0.6

Α_q2

❋✐❣✉r❡ ✹✳✷✿ ❘✉♥♥✐♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t ❢♦r ◗❈❉ ✇✐t❤ ❞②♥❛♠✐❝❛❧ ❣❧✉♦♥ ♠❛ss ✭❡q✉❛t✐♦♥ ✭✹✳✷✮ ✇✐t❤

mg = 2ΛQCD ≈600M eV)✳

✹✳✶✳✷ ❘✉♥♥✐♥❣ ❈♦✉♣❧✐♥❣ ❈♦♥st❛♥t

❲❤❡♥ ✇❡ ❝♦♥s✐❞❡r t❤❡ r✉♥♥✐♥❣ ♦❢ t❤❡ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t ✇❡ ♦❜t❛✐♥ t❤❡ ✐♥t❡❣r❛❧ ❡q✉❛t✐♦♥✿

M1g(x) =a0 ˆ Λ2

0

dyK1(x, y)M1g(y) ✭✹✳✻✮

❲✐t❤ t❤❡ ❑❡r♥❡❧✿

K1(x, y) =

⎡

⎣

θ(x₋y)

x+m2

g

lnx+4m2g

Λ2

QCD

+

θ(y₋x)

y+m2

g

lny+4m2g

Λ2

QCD

⎤

⎦

❲❤❡r❡ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ ❡q✉❛t✐♦♥ ✭✹✳✷✮ ❛♥❞ a0 = 3C2/16bπ2 ✐s t❤❡ ▲❛♥❡ ❝♦♥st❛♥t ❬✺✵❪✳

❊q✉❛t✐♦♥ ✭✹✳✻✮ ✐s ❛ ❋r❡❞❤♦❧♠ ❡q✉❛t✐♦♥ ♦❢ t❤❡ ❢♦r♠✿ ϕ(x) = λ´b

a dyK(x, y)ϕ(y)✳ ❚❤❡

❝♦♥❞✐t✐♦♥ ❢♦r ♦❜t❛✐♥ ♥♦♥✲tr✐✈✐❛❧ s♦❧✉t✐♦♥s ✐s ❣✐✈❡♥ ❜② ❬✺✶❪✿

|λ_{| ≥} 1

´b

a

´b

aK2(x, y)dxdy

❈❍❆P❚❊❘ ✹✳ ❉❨◆❆▼■❈❆▲ ◗❯❆❘❑ ▼❆❙❙ ✷✻

❲❤✐❝❤ ✐s tr❛♥s❧❛t❡❞ ✐♥ ♦✉r ❝❛s❡ t♦✿

3C2

16bπ2 ≥

1 ! " " " " # ´b a ´b a ⎡ ⎢ ⎣

θ(x−y)

(x+m2

g)ln

x+4m2g

Λ2

QCD

+

θ(y−x)

(y+m2

g)ln

y+4m2g

Λ2 QCD ⎤ ⎥ ⎦ 2 dxdy

❋r♦♠ t❤✐s ❡q✉❛t✐♦♥ ✇❡ ♦❜t❛✐♥ ❛ ❝♦♥❞✐t✐♦♥ ❢♦r ❉❈❙❇✿

f(mg) =

3C2

16bπ2

! " " " # ˆ b a ˆ b a ⎡ ⎣

θ(x₋y)

x+m2

g

lnx+4m2g

Λ2

QCD

+

θ(y₋x)

y+m2

g

lny+4m2g

Λ2 QCD ⎤ ⎦ 2

dxdy_≥1

❲❤❡♥ t❤✐s ❝♦♥❞✐t✐♦♥ ✐s s❛t✐s✜❡❞✱ ✇❡ ♦❜t❛✐♥ ♥♦♥tr✐✈✐❛❧ s♦❧✉t✐♦♥ ❢♦r t❤❡ ❛s②♠♣t♦t✐❝ ❜❡❤❛✈✐♦r ♦❢ t❤❡ ❣❛♣❡q✉❛t✐♦♥ ❢♦r t❤❡ ♦♥❡✲❞r❡ss❡❞✲❣❧✉♦♥ ❡①❝❤❛♥❣❡✱ ❛♥❞ t❤✐s ♦❝❝✉r ❢♦r ❛ ✈❛❧✉❡ ♦❢ mg ♦❢ ❛❜♦✉t

150▼❡❱ ✭❋✐❣✉r❡ ✹✳✸✮✱ ✈❛❧✉❡ ✇❤✐❝❤ ✐s q✉✐t❡ ❢❛r ❢r♦♠ t❤❡ ❦♥♦✇♥ ✈❛❧✉❡ ♦❢ mg ≈600M eV ❬✺✷❪✳

0.0 0.2 0.4 0.6 0.8 1.0

m_g 0.2 0.4 0.6 0.8 1.0

fm_g

❈❍❆P❚❊❘ ✹✳ ❉❨◆❆▼■❈❆▲ ◗❯❆❘❑ ▼❆❙❙ ✷✼

✹✳✷ ❖♥❡✲❉r❡ss❡❞✲●❧✉♦♥ ❊①❝❤❛♥❣❡ ✰ ❈♦♥✜♥❡♠❡♥t

❙✐♥❝❡ t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❞②♥❛♠✐❝❛❧❧② ♠❛ss✐✈❡ ❣❧✉♦♥s ✐♥t♦ t❤❡ ●❊ ❞♦❡s ♥♦t ❧❡❛❞ t♦ t❤❡ ❡①♣❡❝t❡❞ ❉❈❙❇ ❢♦r q✉❛r❦s✱ ✐t ✐s ❝❧❡❛r t❤❛t s♦♠❡t❤✐♥❣ ✐s ♠✐ss✐♥❣ ✐♥ t❤❡ ❡q✉❛t✐♦♥ ✭✹✳✶✮✳ ■♥ ❛ r❡❝❡♥t ♣❛♣❡r ❬✶❪✱ ❈♦r♥✇❛❧❧ ❤❛s ♣r♦♣♦s❡❞ t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❛ ✧❈♦♥✜♥✐♥❣ ❊✛❡❝t✐✈❡ Pr♦♣❛❣❛t♦r✧ ♦❢ t❤❡ ❢♦r♠

Dµν_{ef f ≡}δµν_D

ef f(k)✱ ✇❤❡r❡✿

Def f(k) =

8πKF

(k2_+m2₎2 ✭✹✳✽✮

❋r♦♠ ✇❤✐❝❤ ✇❡ ❤❛✈❡ t❤❡ ●❊✿

Mc(p) =

ˆ _d4_k

(2π)4Def f(p−k)

4M(k2₎

[k2_+M2_(k2_)]

=

ˆ _d4_k

(2π)4

8πKF

(k2_+m2₎2

4M(k2₎

[k2_+M2_(k2_)]

✭✹✳✾✮

❲❤❡r❡✱ KF ✐s t❤❡ str✐♥❣ t❡♥s✐♦♥ ❛♥❞ m ✐s ❛ ♣❛r❛♠❡t❡r r❡❧❛t❡❞ t♦ t❤❡ ❞②♥❛♠✐❝❛❧ q✉❛r❦ ♠❛ss

✇❤✐❝❤ ♠✉st ❜❡ ♣r❡s❡♥t ❞✉❡ t♦ ❡♥tr♦♣✐❝ r❡❛s♦♥s ❬✶❪✳

❖♥❡ ♦❢ t❤❡ ♠❛✐♥ r❡❛s♦♥s t♦ ❥✉st✐❢② t❤❡ ✐♥❝❧✉s✐♦♥ ♦❢ ❛ t❡r♠ ❧✐❦❡ ✭✹✳✽✮ ✐s t❤❡ ❢❛❝t t❤❛t ✐t ❝❛♥ r❡♣r♦❞✉❝❡ t❤❡ ❧✐♥❡❛r t❡r♠ ♣r❡s❡♥t ✐♥ t❤❡ ♣❤❡♥♦♠❡♥♦❧♦❣✐❝❛❧ ♣♦t❡♥t✐❛❧ ❢♦r q✉❛r❦♦♥✐✉♠ ✭❡q✉❛t✐♦♥ ✭✷✳✷✮✮ ❇❡❝❛✉s❡ t❤❡ ♣♦t❡♥t✐❛❧ ❜❡t✇❡❡♥ st❛t✐❝ q✉❛r❦ ❝❤❛r❣❡s ✐s r❡❧❛t❡❞ t♦ t❤❡ ❋♦✉r✐❡r tr❛♥s❢♦r♠ ♦❢ t❤❡ t✐♠❡✲t✐♠❡ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡ ❢✉❧❧ ❣❧✉♦♥ ♣r♦♣❛❣❛t♦r✱ ❜② t❤❡ ❡q✉❛t✐♦♥✿

VF(r) =−

2C2

π ˆ

d3_qα

s(q2)∆00(q)expiq·r

❲❤❡r❡ αs(q2) ✐s t❤❡ r✉♥♥✐♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥t ❛♥❞ ∆00(q) t❤❡ ③❡r♦✲③❡r♦ ❝♦♠♣♦♥❡♥t ♦❢ t❤❡

❣❧✉♦♥ ♣r♦♣❛❣❛t♦r ✐♥ t❤❡ ♠♦♠❡♥t✉♠ ❝♦♥✜❣✉r❛t✐♦♥✳