Search for Sphalerons at LHC and IceCube Kazuki Sakurai IPPP, - PowerPoint PPT Presentation

Search for Sphalerons at LHC and IceCube Kazuki Sakurai IPPP, Durham In collaboration with: John Ellis and Michael Spannowsky 3/8/2016 @ IPMU

Plan • Introduction • Review of Sphalerons • Sphalerons at the LHC • Sphalerons at IceCube • Summary 2

How well do we know about EW theory? • Remarkable agreement between experimental results and perturbative calculations. • How about non-perturbative part of EW theory? σ [pb] 10 11 ATLAS Preliminary 80 µ b − 1 Theory √ s = 7, 8, 13 TeV Run 1,2 LHC pp √ s = 7 TeV 10 6 Data 4.5 − 4.9 fb − 1 LHC pp √ s = 8 TeV 10 5 Data 20.3 fb − 1 35 pb − 1 LHC pp √ s = 13 TeV 35 pb − 1 10 4 Data 85 pb − 1 10 3 10 2 13.0 fb − 1 total 10 1 2.0 fb − 1 VBF 1 VH t ¯ tH 10 − 1 pp t¯ t t − chan t s − chan t¯ t¯ W Z t WW H Wt WZ ZZ tW tZ 3

Vacua of EW theory 1 � action: d 4 xF µ ν F µ ν F µ ν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] S EW = 2 g 2 gauge trans.: A µ → U † A µ U + U † ∂ µ U U = a + i ( b · σ ) U ∈ SU (2) a 2 + b 2 = 1 S EW → S EW a vacuum: A = U † ∂ µ U A µ = 0 ↔ At a given t, U ( x ) is a function that maps from x ∈ R 3 to U ∈ SU (2) . The space of vacua is equivalent to the space of these maps. 4

Vacua of EW theory 1 � action: d 4 xF µ ν F µ ν F µ ν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] S EW = 2 g 2 gauge trans.: A µ → U † A µ U + U † ∂ µ U U = a + i ( b · σ ) U ∈ SU (2) a 2 + b 2 = 1 S EW → S EW a vacuum: A = U † ∂ µ U A µ = 0 ↔ At a given t, U ( x ) is a function that maps from x ∈ R 3 to U ∈ SU (2) . The space of vacua is equivalent to the space of these maps. R 3 = S 3 + ( · ) , SU (2) = S 3 SU (2) π 3 ( S 3 ) = Z The map has distinctive sectors classified by the winding number! 5

perturbative E · · · n A n,µ ( x ) = U n ( x ) † ∂ µ U n ( x ) x · σ � � U n ( x ) = exp in π � x 2 − ρ 2 6

perturbative sphaleron E · · · n instanton A n,µ ( x ) = U n ( x ) † ∂ µ U n ( x ) x · σ � � U n ( x ) = exp in π � x 2 − ρ 2 7

finite energy condition: Instantons F µ ν ( x ) → 0 for x → ∞ Define a current K as 8 π 2 � µ νρσ tr A ν ( ∂ ρ A σ + 2 1 K µ = 3 A ρ A σ ) Then it follows 1 � F µ ν = ∂ µ K µ F µ ν = � µ νρσ F ρσ ˜ 16 π 2 F µ ν ˜ K 0 ( A n ( x )) d 3 x = n , , , therefore 1 � � h Z i t = ∞ 16 π 2 F µ ν ˜ F µ ν d 4 x = ∂ µ K µ d 3 xdt K 0 ( t, x ) d 3 x = t = −∞ = n ( t = ∞ ) − n ( t = −∞ ) = ∆ n There exist evolutions of field configuration that change the winding number. • What do such processes look like? • How large is the event rate? 8

The triangle anomaly gives 1 ψ ( i ) L γ µ ψ ( i ) 16 π 2 F µ ν ˜ = ¯ ∂ µ J ( i ) F µ ν J ( i ) with = µ µ L ψ ( i ) L , ˆ L = { ˆ u α L , ˆ c α t α L , ℓ e , ℓ µ , ℓ τ } � u L � � ν e � u L = ˆ , ℓ e = , · · · d L e L 9

The triangle anomaly gives 1 ψ ( i ) L γ µ ψ ( i ) 16 π 2 F µ ν ˜ = ¯ ∂ µ J ( i ) F µ ν J ( i ) with = µ µ L ψ ( i ) We have L , ˆ L = { ˆ u α L , ˆ c α t α L , ℓ e , ℓ µ , ℓ τ } � u L � � ν e � 1 � 16 π 2 F µ ν ˜ F µ ν d 4 x u L = ˆ , ℓ e = , · · · ∆ n = d L e L � t = ∞ � � J ( i ) 0 d 3 x = ∆ N ( i ) = F t = −∞ We find 12 related equalities ∆ n = ∆ N ˆ u L = ∆ N ˆ u L = ∆ N ˆ u L = ∆ N ˆ c L = · · · = ∆ N ˆ t L = · · · = ∆ N ℓ e = ∆ N ℓ µ = ∆ N ℓ τ 10

The triangle anomaly gives 1 ψ ( i ) L γ µ ψ ( i ) 16 π 2 F µ ν ˜ = ¯ ∂ µ J ( i ) F µ ν J ( i ) with = µ µ L ψ ( i ) We have L , ˆ L = { ˆ u α L , ˆ c α t α L , ℓ e , ℓ µ , ℓ τ } � u L � � ν e � 1 � 16 π 2 F µ ν ˜ F µ ν d 4 x u L = ˆ , ℓ e = , · · · ∆ n = d L e L � t = ∞ � � J ( i ) 0 d 3 x = ∆ N ( i ) The event looks like = F t = −∞ a fire ball! ˆ t L We find 12 related equalities ˆ c L ℓ e ˆ c L ˆ t L ∆ n = ∆ N ˆ u L = ∆ N ˆ u L = ∆ N ˆ u L = ∆ N ˆ c L = · · · ∆ n = 1 u L ˆ u L ˆ = ∆ N ˆ t L = · · · c L ˆ ℓ µ = ∆ N ℓ e = ∆ N ℓ µ = ∆ N ℓ τ ˆ ℓ τ t L u L ˆ 11

The tunnelling rate can be estimated using the WKB approximation as ⟨ n | n + ∆ n ⟩ ∼ e − ˆ S E S E is the Euclidean action at the stationary point, which is given by 1 � ˆ FFd 4 x S E = 2 g 2 12

The tunnelling rate can be estimated using the WKB approximation as ⟨ n | n + ∆ n ⟩ ∼ e − ˆ S E S E is the Euclidean action at the stationary point, which is given by 1 � Note that: ˆ FFd 4 x S E = 2 g 2 � ( F ± ˜ F ) 2 d 4 x ≥ 0 F ˜ FFd 4 x ≥ � Fd 4 x � � � � = ⇒ � 13

The tunnelling rate can be estimated using the WKB approximation as ⟨ n | n + ∆ n ⟩ ∼ e − ˆ S E S E is the Euclidean action at the stationary point, which is given by 1 � Note that: ˆ FFd 4 x S E = 2 g 2 � ( F ± ˜ F ) 2 d 4 x ≥ 0 � � � 1 � � F � Fd 4 x = � � F ˜ FFd 4 x ≥ � Fd 4 x � � � � = 2 g 2 ⇒ � = 8 π 2 � � � ∆ n � E g 2 n n + 1 The tunnelling rate is e − 4 π α W ∼ 10 − 170 unobservably small 14

The barrier hight was calculated by sphaleron F.R.Klinkhamer and N.S.Manton (1984) E E Sph � m H E Sph = 2 m W � B m W α W (for m H = 125GeV) ≃ 9 TeV n n + 1 • At high temperature, the sphaleron rate may be unsuppressed. − E Sph ( T ) � � Γ ∝ exp T It plays an important role in baryo(lepto)genesis. 15

• At high energy, the tunnelling exponent was calculated by a semi- classical approach (perturbation in the instanton background). c 4 π � � σ ( ∆ n = ± 1) ∝ exp S ( E ) ( c ≃ 2) α W � E � E � 4 � 2 S ( E ) = − 1 + 9 3 − 9 √ E 0 = 6 π m W / α W + · · · + · · · 8 E 0 16 E 0 ≃ 18 TeV instanton S ( E ) 0.5 E [TeV] 0.0 10 20 30 40 - 0.5 - 1.0 instanton 16

• At high energy, the tunnelling exponent was calculated by a semi- classical approach (perturbation in the instanton background). c 4 π � � σ ( ∆ n = ± 1) ∝ exp S ( E ) ( c ≃ 2) α W � E � E � 4 � 2 S ( E ) = − 1 + 9 3 − 9 √ E 0 = 6 π m W / α W + · · · + · · · 8 E 0 16 E 0 ≃ 18 TeV S.Khlebnikov, V.Rubakov, P.Tinyakov 1991, M.Porrati 1990, V.Zakharov 1992, … S ( E ) 0.5 E [TeV] 0.0 10 20 30 40 - 0.5 - 1.0 instanton 17

• At high energy, the tunnelling exponent was calculated by a semi- classical approach (perturbation in the instanton background). c 4 π � � σ ( ∆ n = ± 1) ∝ exp S ( E ) ( c ≃ 2) α W � E � E � 4 � 2 S ( E ) = − 1 + 9 3 − 9 √ E 0 = 6 π m W / α W + · · · 8 E 0 16 E 0 ≃ 18 TeV S.Khlebnikov, V.Rubakov, P.Tinyakov 1991, P.Arnold, M.Mattis 1991, A.Mueller 1991, M.Porrati 1990, V.Zakharov 1992, … D.Diakonov, V.Petrov 1991, … S ( E ) 0.5 E [TeV] 0.0 10 20 30 40 - 0.5 - 1.0 instanton 18

Recently Tye and Wong (TW) have pointed out that the periodic nature of the EW potential is important and this effect was not taken into account in the previous calculations. They evaluated the sphaleron rate by constructing a 1D quantum mechanical system [1505.03690]. ∂ 2 − 1 � � ∂ Q 2 + V ( Q ) Ψ ( Q ) = E Ψ ( Q ) 2 m where Q is related to the winding number n as Q = µ/m W n π = µ − sin(2 µ ) / 2 , By following a sphaleron trajectory in the original YM Lagrangian, they found: 1 . 31 sin 2 ( m W Q ) + 0 . 60 sin 4 ( m W Q ) � � � ! ! V ( Q ) ' 4 . 75 TeV 0 0 ˜ Φ = v (1 � h ( r )) U + h ( r ) , cos µ v ✓ ◆ , m = 17 . 1 TeV , A i = i g (1 � f ( r )) U ∂ i U † , ! � sin µ sin θ e i ϕ cos µ + i sin µ cos θ U = sin µ sin θ e � i ϕ cos µ � i sin µ cos θ f ( r ) lim = h (0) = 0 , f ( 1 ) = h ( 1 ) = 1 , r r ! 0 19

The wave functions in periodic potentials are given by Bloch waves. Ψ ( Q ) = e ikQ u k ( Q ) , u k ( Q ) = u k ( Q + m W ) π | Ψ ( Q ) | 2 = | Ψ ( Q + m W ) | 2 π ⇒ The spectrum exhibits a band structure. k 2 E E 2.5 2 m 2.0 1.5 1.0 0.5 k - 4 - 2 0 2 4 0 0 20

The wave functions in periodic potentials are given by Bloch waves. Ψ ( Q ) = e ikQ u k ( Q ) , u k ( Q ) = u k ( Q + m W ) π | Ψ ( Q ) | 2 = | Ψ ( Q + m W ) | 2 π ⇒ The spectrum exhibits a band structure. k 2 E 2.5 2 m 2.0 bands 1.5 1.0 gaps 0.5 k - 4 - 2 0 2 4 3 π − 3 π − 2 π − π 2 π 4 π − 4 π π 0 ( a = π /m W ) a a a a a a a a 21