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  1. ❚❤❡ t✇❡♥t✐❡t❤ ✇♦r❦s❤♦♣ ❲❤❛t ❈♦♠❡s ❇❡②♦♥❞ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧s❄ ❏✉❧② ✾✕✶✼✱ ✷✵✶✼✱ ❇❧❡❞✱ ❙❧♦✈❡♥✐❛ ❑✳❱✳❙t❡♣❛♥②❛♥t③ ▼♦s❝♦✇ ❙t❛t❡ ❯♥✐✈❡rs✐t②✱ P❤②s✐❝❛❧ ❋❛❝✉❧t②✱ ❉❡♣❛rt♠❡♥t ♦❢ ❚❤❡♦r❡t✐❝❛❧ P❤②s✐❝s ❙tr✉❝t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ N = 1 s✉♣❡rs②♠♠❡tr✐❝ ❣❛✉❣❡ t❤❡♦r✐❡s ✶ ❑✳❱✳❙t❡♣❛♥②❛♥t③ ❙tr✉❝t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ N = 1

  2. ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❚❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ✭❙▼✮ ✐s ❛ ❣❛✉❣❡ t❤❡♦r② ✇✐t❤ t❤❡ ❣r♦✉♣ SU (3) × SU (2) × U (1) ❛♥❞ ❝❤✐r❛❧ ❢❡r♠✐♦♥s✱ ✇❤✐❝❤ ✈❡r② s✉❝❝❡ss❢✉❧❧② ❞❡s❝r✐❜❡s t❤❡ str♦♥❣ ❛♥❞ ❡❧❡❝tr♦✇❡❛❦ ✐♥t❡r❛❝t✐♦♥s✳ ❚❤❡ ❣❛✉❣❡ s②♠♠❡tr② ✐s s♣♦♥t❛♥❡♦✉s❧② ❜r♦❦❡♥ ❜② t❤❡ ❤❡❧♣ ♦❢ t❤❡ ❍✐❣❣s ♠❡❝❤❛♥✐s♠✱ SU (3) × SU (2) × U (1) → SU (3) × U (1) em , ♣r♦❞✉❝✐♥❣ t❤❡ ♠❛ss✐✈❡ Z ✲ ❛♥❞ W ± ✲❜♦s♦♥s t♦❣❡t❤❡r ✇✐t❤ t❤❡ ❍✐❣❣s ❜♦s♦♥✳ ❚❤❡✐r ❡①♣❡r✐♠❡♥t❛❧ ❞✐s❝♦✈❡r✐❡s ❤❛✈❡ ❜❡❡♥ ❡①❝❡❧❧❡♥t ❝♦♥✜r♠❛t✐♦♥s ♦❢ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧✳ ❋r♦♠ t❤❡ t❤❡♦r❡t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ ✐t ✐s ❛ r❡♥♦r♠❛❧✐③❛❜❧❡ ♠♦❞❡❧✱ ✐♥ ✇❤✐❝❤ ❛♥♦♠❛❧✐❡s ✐♥tr♦❞✉❝❡❞ ❜② ❝❤✐r❛❧ ❢❡r♠✐♦♥s ❝❛♥❝❡❧✳ ❍♦✇❡✈❡r✱ ✐t ✐s ✇✐❞❡❧② ❜❡❧✐❡✈❡❞ t❤❛t t❤❡ ❙▼ ✐s ♥♦t ❛♥ ✉❧t✐♠❛t❡ t❤❡♦r②✳ ◗✉❛♥t✉♠ ♥✉♠❜❡rs ♦❢ q✉❛r❦s ❛♥❞ ❧❡♣t♦♥s ✐♠♣❧② t❤❛t t❤❡ ❣❛✉❣❡ ❣r♦✉♣ SU (3) × SU (2) × U (1) ❝❛♥ ❜❡ ❛ r❡♠♥❛♥t ♦❢ ❛ ✇✐❞❡r ❣❛✉❣❡ ❣r♦✉♣✱ ❡✳❣✳✱ SU (3) × SU (2) × U (1) ⊂ SU (5) . ❚❤❡ SO (10) ❣r♦✉♣ ❛❧❧♦✇s t♦ ♣❧❛❝❡ ❛❧❧ ❢❡r♠✐♦♥s ♦❢ ♦♥❡ ❣❡♥❡r❛t✐♦♥ ✭✐♥❝❧✉❞✐♥❣ t❤❡ r✐❣❤t ♥❡✉tr✐♥♦s✮ t♦ t❤❡ ✐rr❡❞✉❝✐❜❧❡ r❡♣r❡s❡♥t❛t✐♦♥ 16 ✳ ✷ ❑✳❱✳❙t❡♣❛♥②❛♥t③ ❙tr✉❝t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ N = 1

  3. ■♥❝♦♠♣❧❡t❡♥❡ss ♦❢ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❍♦✇❡✈❡r✱ t❤❡ ❡✈♦❧✉t✐♦♥ ♦❢ r✉♥♥✐♥❣ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥ts ✐♥ ❙▼ ✐s ♥♦t ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ t❤❡ ♣r❡❞✐❝t✐♦♥ ♦❢ ●r❛♥❞ ❯♥✐✜❝❛t✐♦♥ ❚❤❡♦r✐❡s✳ ▼♦r❡♦✈❡r✱ t❤❡ ✉♥✐✜❝❛t✐♦♥ ♠❛ss M X ∼ 10 15 GeV ❧❡❛❞s t♦ ✉♥❛❝❝❡♣t❛❜❧② r❛♣✐❞ ♣r♦t♦♥ ❞❡❝❛②✳ ❋r♦♠ t❤❡ t❤❡♦r❡t✐❝❛❧ ♣♦✐♥t ♦❢ ✈✐❡✇✱ q✉❛❞r❛t✐❝❛❧❧② ❞✐✈❡r❣❡♥t q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ♠❛ss ♦❢ t❤❡ ❍✐❣❣s ❜♦s♦♥ ♣r♦❞✉❝❡ t❤❡ ♣r♦❜❧❡♠ ♦❢ ❍✐❣❣s ♠❛ss ✜♥❡ t✉♥✐♥❣✳ α − 1 i 60 50 40 30 20 10 ln µ/ 1 GeV 0 2 4 6 8 10 12 14 16 18 ✸ ❑✳❱✳❙t❡♣❛♥②❛♥t③ ❙tr✉❝t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ N = 1

  4. ❙✉♣❡rs②♠♠❡tr② ❛♥❞ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❆ ✈❡r② ♣r♦♠✐s✐♥❣ ✇❛② t♦ s♦❧✈❡ t❤❡ ❛❜♦✈❡ ♣r♦❜❧❡♠s ✐s t♦ ❝♦♥s✐❞❡r N = 1 s✉♣❡rs②♠♠❡tr✐❝ ❡①t❡♥s✐♦♥s ♦❢ ❙▼✳ ■♥ t❤❡ s✉♣❡rs②♠♠❡tr✐❝ ✈❡rs✐♦♥ ♦❢ ❙▼ r✉♥♥✐♥❣ ♦❢ t❤❡ ❣❛✉❣❡ ❝♦✉♣❧✐♥❣ ❝♦♥st❛♥ts ❛❣r❡❡ ✇✐t❤ t❤❡ ♣r❡❞✐❝t✐♦♥s ♦❢ ●r❛♥❞ ❯♥✐✜❡❞ ❚❤❡♦r✐❡s ✭●❯❚✮✳ ■♥❝r❡❛s✐♥❣ ♦❢ t❤❡ ✉♥✐✜❝❛t✐♦♥ ♠❛ss ❡ss❡♥t✐❛❧❧② ✐♥❝r❡❛s❡s t❤❡ ♣r♦t♦♥ ❧✐❢❡ t✐♠❡✱ τ ∼ M 4 X ✳ ❚❤❡r❡ ❛r❡ ♥♦ q✉❛❞r❛t✐❝❛❧❧② ❞✐✈❡r❣❡♥t q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s t♦ t❤❡ ❍✐❣❣s ♠❛ss✳ α − 1 i 60 50 40 30 20 10 ln µ/ 1 GeV 0 2 4 6 8 10 12 14 16 18 ✹ ❑✳❱✳❙t❡♣❛♥②❛♥t③ ❙tr✉❝t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ N = 1

  5. ❙✉♣❡rs②♠♠❡tr② ❛♥❞ t❤❡ ❙t❛♥❞❛r❞ ▼♦❞❡❧ ❙✉♣❡rs②♠♠❡tr✐❝ ♠♦❞❡❧s ♣r❡❞✐❝t ❛ ❧♦t ♦❢ ♥❡✇ ♣❛rt✐❝❧❡s✳ ❚❤❡r❡ ❛r❡ s✉♣❡r♣❛rt♥❡rs ♦❢ q✉❛r❦s✱ ❧❡♣t♦♥s✱ ❛♥❞ ❣❛✉❣❡ ❜♦s♦♥s✳ ❙✉♣❡rs②♠♠❡tr② ❛❧s♦ r❡q✉✐r❡s t✇♦ ❍✐❣❣s ❞♦✉❜❧❡ts✱ ✇❤✐❝❤ ♣r♦❞✉❝❡s 2 × 2 × 2 − 3 = 5 ❍✐❣❣s ❜♦s♦♥s✳ ❚♦ ♠❛❦❡ ♠❛ss❡s ♦❢ s✉♣❡r♣❛rt♥❡rs s✉✣❝✐❡♥t❧② ❧❛r❣❡✱ ✐t ✐s ♥❡❝❡ss❛r② t♦ ❜r❡❛❦ s✉♣❡rs②♠♠❡tr②✳ ❆❧t❤♦✉❣❤ ✐t ✐s ❤✐❣❤❧② ❞❡s✐r❛❜❧❡ t♦ ❜r❡❛❦ s✉♣❡rs②♠♠❡tr② s♣♦♥t❛♥❡♦✉s❧②✱ t❤❡ s✐♠♣❧❡st ♠♦❞❡❧s ✭❧✐❦❡ ▼❙❙▼✮ ✐♥❝❧✉❞❡ s♦❢t t❡r♠s✱ ✇❤✐❝❤ ❡①♣❧✐❝✐t❧② ❜r❡❛❦ s✉♣❡rs②♠♠❡tr②✱ ❜✉t ❞♦ ♥♦t ♣r♦❞✉❝❡ q✉❛❞r❛t✐❝ ❞✐✈❡r❣❡♥❝❡s✳ ❚❤❡r❡ ❛r❡ ✹ t②♣❡s ♦❢ t❤❡ s♦❢t t❡r♠s✱ ✐♥❝❧✉❞✐♥❣✱ ❡✳❣✳✱ ❣❛✉❣✐♥♦ ♠❛ss❡s✳ ■♥ ●❯❚ t❤❡♦r✐❡s r✉♥♥✐♥❣ ❣❛✉❣✐♥♦ ♠❛ss❡s ❝❛♥ ❜❡ ✉♥✐✜❡❞✳ ■♥✈❡st✐❣❛t✐♦♥ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ ❙❯❙❨ t❤❡♦r✐❡s ❛♥❞ t❤❡♦r✐❡s ✇✐t❤ s♦❢t❧② ❜r♦❦❡♥ ❙❯❙❨ ❛♥❞ ❝♦♠♣❛r✐♥❣ t❤❡♠ ✇✐t❤ ❡①♣❡r✐♠❡♥t❛❧ ❞❛t❛ ❝❛♥ ❣✐✈❡ ✐♥❢♦r♠❛t✐♦♥ ❛❜♦✉t ♣❤②s✐❝s ❜❡②♦♥❞ ❙▼✳ ✺ ❑✳❱✳❙t❡♣❛♥②❛♥t③ ❙tr✉❝t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ N = 1

  6. N = 1 s✉♣❡rs②♠♠❡tr✐❝ ❣❛✉❣❡ t❤❡♦r✐❡s ■t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ❞❡s❝r✐❜❡ N = 1 ❙❯❙❨ t❤❡♦r✐❡s ✉s✐♥❣ N = 1 s✉♣❡r✜❡❧❞s✱ ❜❡❝❛✉s❡ ✐♥ t❤✐s ❝❛s❡ s✉♣❡rs②♠♠❡tr② ✐s ♠❛♥✐❢❡st✳ ■♥ t❤✐s ❧❛♥❣✉❛❣❡✱ t❤❡ N = 1 ❙❨▼ t❤❡♦r② ✐s ❣✐✈❡♥ ❜② t❤❡ ❛❝t✐♦♥ � � 1 d 4 x d 2 θ W a W a + 1 d 4 x d 4 θ φ ∗ i ( e 2 V ) i j φ j S = ❘❡ tr 2 e 2 4 0 � � � 1 � � 0 φ i φ j + 1 4 m ij 6 λ ijk d 4 x d 2 θ + 0 φ i φ j φ k + ❝✳❝✳ , ✇❤❡r❡ θ ❞❡♥♦t❡s ❛♥ ❛✉①✐❧✐❛r② ●r❛ss♠❛♥♥✐❛♥ ❝♦♦r❞✐♥❛t❡s✳ V ( x, θ, ¯ θ ) ✐s t❤❡ ❣❛✉❣❡ s✉♣❡r✜❡❧❞ ❛♥❞ t❤❡ s✉♣❡rs②♠♠❡tr✐❝ ❣❛✉❣❡ ✜❡❧❞ str❡♥❣t❤ ✐s ❞❡✜♥❡❞ ❛s D 2 � e − 2 V D a e 2 V � W a = 1 ¯ . 8 a φ i = 0 ✱ ✇❤❡r❡ y µ ≡ x µ + φ i ( y µ , θ ) ❛r❡ ❝❤✐r❛❧ ♠❛tt❡r s✉♣❡r✜❡❧❞s✱ ¯ D ˙ i ¯ ab θ b ✐s t❤❡ ❝❤✐r❛❧ ❝♦♦r❞✐♥❛t❡ ❛♥❞ D a ❛♥❞ ¯ θ ˙ a ( γ µ ) ˙ D ˙ a ❛r❡ t❤❡ r✐❣❤t ❛♥❞ ❧❡❢t s✉♣❡rs②♠♠❡tr✐❝ ❝♦✈❛r✐❛♥t ❞❡r✐✈❛t✐✈❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ✻ ❑✳❱✳❙t❡♣❛♥②❛♥t③ ❙tr✉❝t✉r❡ ♦❢ q✉❛♥t✉♠ ❝♦rr❡❝t✐♦♥s ✐♥ N = 1

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