1604.03377 Renormalizable SU(5) Unification C.M and P. Fileviez - PowerPoint PPT Presentation

1604.03377 Renormalizable SU(5) Unification C.M and P. Fileviez Perez IFIC, Universitat de Valencia-CSIC BSM Journal Club

Motivation Theorem for model builders: B eauty - S implicity - P redictability • Aesthetics: why three gauge groups? • Simplicity: why three different strengths? why so many representations? are quarks and leptons that different? • Predictability: why so many inputs? why arbitrary charges? what about the Yukawas? and the Weinberg angle?

Motivation Theorem for model builders: B eauty - S implicity - P redictability • Aesthetics: why three gauge groups? • Simplicity: why three different strengths? why so many representations? are quarks and leptons that different? • Predictability: why so many inputs? why arbitrary charges? what about the Yukawas? and the Weinberg angle? SU(5) is the only rank 4 candidate able to embed SM!

The simplest GUT: SU(5)

Simplest GUT: SU(5) • Matter content: ¯ 5 ∼ ( 1 , ¯ ⊕ (¯ 10 ∼ (¯ 2 , − 1 / 2 ) 3 , 1 , 1 / 3 ) 3 , 1 , − 2 / 3 ) ⊕ ( 3 , 2 , 1 / 6 ) ⊕ ( 1 , 1 , 1 ) � �� � � �� � � �� � � �� � � �� � ( d c ) L ( u c ) L q L ( e c ) L l L     d c u c − u c 0 u 1 d 1 1 3 2 d c − u c u c 0 u 2 d 2     2 3 1     ¯ d c u c − u c 5 = 10 = 0    u 3 d 3  3 2 1     e + e − u 1 − u 2 − u 3 0     − e + − ν − d 1 − d 2 − d 3 0

Simplest GUT: SU(5) • Matter content: ¯ 5 ∼ ( 1 , ¯ ⊕ (¯ 10 ∼ (¯ 2 , − 1 / 2 ) 3 , 1 , 1 / 3 ) 3 , 1 , − 2 / 3 ) ⊕ ( 3 , 2 , 1 / 6 ) ⊕ ( 1 , 1 , 1 ) � �� � � �� � � �� � � �� � � �� � ( d c ) L ( u c ) L q L ( e c ) L l L     d c u c − u c 0 u 1 d 1 1 3 2 d c − u c u c 0 u 2 d 2     2 3 1     ¯ d c u c − u c 5 = 10 = 0    u 3 d 3  3 2 1     e + e − u 1 − u 2 − u 3 0     − e + − ν − d 1 − d 2 − d 3 0 ⇒ The 15 SM Weyl d.o.f. perfectly fit in the two simplest SU(5) representations!

Simplest GUT: SU(5) � � { T R a , T R b } , T T • What about anomalies? A ( R ) d abc = Tr c } A ( anti-fund ) + A ( anti-symmetric ) = − 1 2 + N − 4 For SU(N), 2

Simplest GUT: SU(5) � � { T R a , T R b } , T T • What about anomalies? A ( R ) d abc = Tr c } A ( anti-fund ) + A ( anti-symmetric ) = − 1 2 + N − 4 For SU(N), 2 A (¯ for N=5, 5 ) + A ( 10 ) = 0

Simplest GUT: SU(5) � � { T R a , T R b } , T T • What about anomalies? A ( R ) d abc = Tr c } A ( anti-fund ) + A ( anti-symmetric ) = − 1 2 + N − 4 For SU(N), 2 A (¯ for N=5, 5 ) + A ( 10 ) = 0 • Hypercharge generator:   − 1 0 0 0 0 0 − 1 0 0 0   1   T 24 = √ 0 0 − 1 0 0 ∝ Y     15 3 0 0 0 0   2 3 0 0 0 0 2 Fixing the normalisation: � 5 Y = 3 T 24

Simplest GUT: SU(5) � � { T R a , T R b } , T T • What about anomalies? A ( R ) d abc = Tr c } A ( anti-fund ) + A ( anti-symmetric ) = − 1 2 + N − 4 For SU(N), 2 A (¯ for N=5, 5 ) + A ( 10 ) = 0 • Hypercharge generator:   − 1 0 0 0 0 0 − 1 0 0 0   1   T 24 = √ 0 0 − 1 0 0 ∝ Y     15 3 0 0 0 0   2 3 0 0 0 0 2 Fixing the normalisation: All hypercharges predicted! � 5 Y = 3 T 24

Simplest GUT: SU(5) • And the gauge bosons? ⊕ (¯ V 24 ∼ ( 8 , 1 , 0 ) ⊕ ( 1 , 3 , 0 ) ⊕ ( 3 , 2 , − 5 / 6 ) 3 , 2 , 5 / 6 ) ⊕ ( 1 , 1 , 0 ) � �� � � �� � � �� � � �� � � �� � G µ W µ X µ Y µ γ µ

Simplest GUT: SU(5) • And the gauge bosons? ⊕ (¯ V 24 ∼ ( 8 , 1 , 0 ) ⊕ ( 1 , 3 , 0 ) ⊕ ( 3 , 2 , − 5 / 6 ) 3 , 2 , 5 / 6 ) ⊕ ( 1 , 1 , 0 ) � �� � � �� � � �� � � �� � � �� � G µ W µ X µ Y µ γ µ τ a 24 � Va V µ = = µ 2 a = 1 G 1 µ + 2 B µ  G 1 G 1 XC 1 YC 1  √ µ µ 2 µ 3 µ 30  2 µ + 2 B µ   G 2 G 2 G 2 XC 2 YC 2  √   1 µ 3 µ µ µ  30    3 µ + 2 B µ G 3 G 3 G 3 XC 3 YC 3 1   √  µ µ  = 1 µ 2 µ √ 30     2 W 3 � 3   µ X 1 X 2 X 3 W +   √ 2 − 10 B µ  µ µ µ µ     W 3  � 3  Y 1 Y 2 Y 3 W − µ  − √ 2 − 10 B µ µ µ µ µ

SU(5): scalar sector 24 H • Breaking I: SU ( 5 ) → SU ( 3 ) ⊗ SU ( 2 ) ⊗ U ( 1 ) Y � Σ 8 � Σ ( 3 , 2 ) 24 H = + Σ S λ 24 , Σ (¯ Σ 3 3 , 2 ) � 24 H � = v 24 √  15diag ( 2 , 2 , 2 , − 3 , − 3 ) : breaks to SM       � 24 H � = v 24 diag ( 1 , 1 , 1 , 1 , − 4 ) : breaks to SU ( 4 ) ⊗ U ( 1 )       � 24 H � = diag ( 0 , 0 , 0 , 0 , 0 ): no breaking

SU(5): scalar sector 24 H • Breaking I: SU ( 5 ) → SU ( 3 ) ⊗ SU ( 2 ) ⊗ U ( 1 ) Y � Σ 8 � Σ ( 3 , 2 ) 24 H = + Σ S λ 24 , Σ (¯ Σ 3 3 , 2 ) � 24 H � = v 24 √  15diag ( 2 , 2 , 2 , − 3 , − 3 ) : breaks to SM       � 24 H � = v 24 diag ( 1 , 1 , 1 , 1 , − 4 ) : breaks to SU ( 4 ) ⊗ U ( 1 )       � 24 H � = diag ( 0 , 0 , 0 , 0 , 0 ): no breaking 5 H • Breaking II: SU ( 3 ) ⊗ SU ( 2 ) ⊗ U ( 1 ) Y → SU ( 3 ) ⊗ U ( 1 ) Q     T 1 0 T 2 0         SSB T 3 5 H = → � 5 H � = 0         H + 0     √ 1 H 0 v 5 / 2 1

But it is too predictive... Predictability may be its most beautiful feature but it is also its Achilles’ heel

Fermion masses • Yukawa Lagrangian ( i , j , k are SU(3) indices and α, β are SU(2) indices) Y 1 ¯ 5 10 5 ∗ L Y ⊃ H + Y 3 10 10 5 H ǫ 5

Fermion masses • Yukawa Lagrangian ( i , j , k are SU(3) indices and α, β are SU(2) indices) Y 1 ¯ 5 10 5 ∗ L Y ⊃ H + Y 3 10 10 5 H ǫ 5 5 i 10 i α + ¯ H α + Y 3 ( 10 ij 10 k α + 10 i α 10 jk ) 5 β Y 1 (¯ 5 β 10 βα ) 5 ∗ ⊃ H ǫ ijk αβ √ • After SSB: � 5 H � = v 5 / 2, v ∗ 3 ) v i d i + ee C ) + 4 ( Y 3 + Y T ( d C u C i u i . L Y ⊃ Y 1 √ √ 2 2 • Fermion masses: v ∗ M d = M T 5 e = Y 1 √ 2 3 ) v 5 M u = 4 ( Y 3 + Y T √ 2

Fermion masses • Yukawa Lagrangian ( i , j , k are SU(3) indices and α, β are SU(2) indices) Y 1 ¯ 5 10 5 ∗ L Y ⊃ H + Y 3 10 10 5 H ǫ 5

Fermion masses • Yukawa Lagrangian ( i , j , k are SU(3) indices and α, β are SU(2) indices) Y 1 ¯ 5 10 5 ∗ L Y ⊃ H + Y 3 10 10 5 H ǫ 5 5 i 10 i α + ¯ H α + Y 3 ( 10 ij 10 k α + 10 i α 10 jk ) 5 β Y 1 (¯ 5 β 10 βα ) 5 ∗ ⊃ H ǫ ijk αβ √ • After SSB: � 5 H � = v 5 / 2, v ∗ 3 ) v i d i + ee C ) + 4 ( Y 3 + Y T ( d C u C i u i . L Y ⊃ Y 1 √ √ 2 2 • Fermion masses: v ∗ M d = M T 5 e = Y 1 √ 2 3 ) v 5 M u = 4 ( Y 3 + Y T √ 2

Unification constraints • RGEs: �  B i = b SM + b I i r I � M GUT  GUT + B i i α − 1 ( M Z ) = α − 1 r I = Log ( M GUT / M I ) 2 π Log i Log ( M GUT / M Z ) , M Z < M I < M GUT M Z  • In Yang-Mills theories:    1 for R scalar ,   1 � �  , S ( R ) b i = S ( R ) T i ( R ) dim j ( R ) 2 for R chiral fermion , 3  − 11 for R gauge boson . R j � = i • Table of B ij ≡ B i − B j contributions to the running: ¯ 5 10 V 24 5 H 24 H ( dc ) L ( uc ) L ( ec ) L bi / Bij lL qL G µ W µ H 1 T Σ 8 Σ 3 3 2 8 1 6 1 1 b 1 0 0 15 rT 0 0 5 5 5 5 5 10 − 22 1 1 b 2 1 0 0 3 0 0 0 0 3 r Σ 3 3 6 1 1 b 3 0 1 1 2 0 − 11 0 0 6 rT 2 r Σ 8 0 − 4 2 8 − 44 − 2 22 − 1 1 − 1 B 12 0 15 rT 0 3 r Σ 3 5 15 15 15 5 3 15 − 22 1 − 1 − 1 1 B 23 1 -1 -1 1 0 11 6 rT 2 r Σ 8 3 r Σ 3 3 6

Unification constraints • Introducing B ij = B i − B j , sin 2 θ W ( M Z ) − α ( M Z ) /α 3 ( M Z ) B 23 5 = 3 / 8 − sin 2 θ W ( M Z ) 8 B 12 3 / 8 − sin 2 θ W ( M Z ) � M GUT � 16 π = Log 5 α ( M Z ) M Z B 12

Unification constraints • Introducing B ij = B i − B j , sin 2 θ W ( M Z ) − α ( M Z ) /α 3 ( M Z ) B 23 5 = 3 / 8 − sin 2 θ W ( M Z ) 8 B 12 3 / 8 − sin 2 θ W ( M Z ) � M GUT � 16 π = Log 5 α ( M Z ) M Z B 12 • Input: sin 2 θ W ( M Z ) = 0.231, α − 1 ( M Z ) = 127.94, α s ( M Z ) = 0.1185

Unification constraints • Introducing B ij = B i − B j , sin 2 θ W ( M Z ) − α ( M Z ) /α 3 ( M Z ) B 23 5 = 3 / 8 − sin 2 θ W ( M Z ) 8 B 12 3 / 8 − sin 2 θ W ( M Z ) � M GUT � 16 π = Log 5 α ( M Z ) M Z B 12 • Input: sin 2 θ W ( M Z ) = 0.231, α − 1 ( M Z ) = 127.94, α s ( M Z ) = 0.1185 • To achieve unification: B 23 = 0.718 B 12 � M GUT � 184.87 = Log M Z B 12

Unification constraints • SM contribution (assuming only the spitting of 5 H ): B SM 23 = 0.53 B SM 12

Unification constraints • SM contribution (assuming only the spitting of 5 H ): B SM 23 = 0.53 B SM 12 • SU(5) contribution (assuming splitting of 24 H too): B SU ( 5 ) B SU ( 5 ) = B SM 23 + 1 3 r Σ 3 − 1 6 r T − 1 2 r Σ 8 most optimistic case 23 23 → � 0.6 B SU ( 5 ) B SM 12 − 1 3 r Σ 3 + 1 B SU ( 5 ) 15 r T 12 12