Bubble Wall Velocities Jonathan Kozaczuk ACFI, UMass Amherst ACFI - PowerPoint PPT Presentation

Bubble Wall Velocities Jonathan Kozaczuk ACFI, UMass Amherst ACFI EWPT Workshop , 4/7/17

Why should we care about bubble wall dynamics? 2

Electroweak baryogenesis Γ B = 0 / Γ B 6 = 0 , ¯ , ¯ f f f f h φ i 6 = 0 / h φ i 6 = 0 / B / B B 6 = 0 Bubble wall catches up with diffusing current, freezing in B 3

Electroweak baryogenesis Requiring sufficient B-violation during diffusion requires relatively slow bubble walls Wall velocities conventionally required to be subsonic: v w <0.58 (subtle and there are exceptions; see e.g. No, 2011; Caprini + No, 2011; Katz+Riotto, 2016 ) Resulting asymmetry can strongly depend on v w , depending on the form of the primary CP-violating source Huber et al, 2001 7 60 H 1 + H 2 H 1 − H 2 6 50 5 η ↑ 40 4 30 3 20 2 10 1 0 0 –4 –3 –2 –1 –4 –3 –2 –1 4 ln 10 ( v w ) → ln 10 ( v w ) →

Gravitational waves Spectrum, and prospects for detection, depend on the wall velocity 3 ✓ 0 . 11 v 3 ◆ 2 ✓ κα ◆ 1 ◆ 2 ✓ 100 ✓ H ∗ ◆ h 2 Ω env ( f ) 1 . 67 × 10 − 5 = w S env ( f ) 1 + α g ∗ 0 . 42 + v 2 β w ↵ (0 . 73 + 0 . 083 p ↵ + ↵ ) ( − 1 v w ⇠ 1  v ' − 1 , w 6 . 9 ↵ (1 . 36 � 0 . 037 p ↵ + ↵ ) v 6 / 5 v w . 0 . 1 Caprini et al, 2015 5

Guiding questions How do we compute the wall velocity? What are the theoretical challenges involved in these calculations? 6

Master equation Scalar field EOM for one scalar D.O.F. at finite T: See e.g. Moore+Prokopec, 1996; X dm 2 d 3 k Z ⇤ φ + V 0 ( φ ) + Konstandin et al, 2014 (2 π ) 3 2 E f ( k, z ) = 0 d φ Distribution functions of all particles T=0 effective potential coupled to Higgs 7

Master equation Scalar field EOM for one scalar D.O.F. at finite T: See e.g. Moore+Prokopec, 1996; X dm 2 d 3 k Z ⇤ φ + V 0 ( φ ) + Konstandin et al, 2014 (2 π ) 3 2 E f ( k, z ) = 0 d φ Distribution functions of all particles T=0 effective potential coupled to Higgs Rewrite in terms of finite-T effective potential: X dm 2 d 3 k Z ⇤ φ + V 0 ( φ , T ) + (2 π ) 3 2 E δ f ( k, z ) = 0 d φ 8

Master equation Scalar field EOM for one scalar D.O.F. at finite T: See e.g. Moore+Prokopec, 1996; X dm 2 d 3 k Z ⇤ φ + V 0 ( φ ) + Konstandin et al, 2014 (2 π ) 3 2 E f ( k, z ) = 0 d φ Distribution functions of all particles T=0 effective potential coupled to Higgs Rewrite in terms of finite-T effective potential: X dm 2 d 3 k Z ⇤ φ + V 0 ( φ , T ) + (2 π ) 3 2 E δ f ( k, z ) = 0 d φ X dm 2 d 3 k Z  � Z ⇤ φ + V 0 ( φ , T ) + φ 0 ( z ) dz = ∆ p (2 π ) 3 2 E δ f ( k, z ) d φ 9

Master equation Scalar field EOM for one scalar D.O.F. at finite T: See e.g. Moore+Prokopec, 1996; X dm 2 d 3 k Z ⇤ φ + V 0 ( φ ) + Konstandin et al, 2014 (2 π ) 3 2 E f ( k, z ) = 0 d φ Distribution functions of all particles T=0 effective potential coupled to Higgs Rewrite in terms of finite-T effective potential: X dm 2 d 3 k Z ⇤ φ + V 0 ( φ , T ) + (2 π ) 3 2 E δ f ( k, z ) = 0 d φ X dm 2 d 3 k Z  � Z ⇤ φ + V 0 ( φ , T ) + φ 0 ( z ) dz = ∆ p (2 π ) 3 2 E δ f ( k, z ) d φ Z X dm 2 d 3 k Z Non-accelerating wall: (2 π ) 3 2 E δ f ( k, z ) φ 0 ( z ) dz ∆ V ( T ) = − d φ 10

Master equation Boils down to drawing a free body diagram Z X dm 2 d 3 k Z (2 π ) 3 2 E δ f ( k, z ) φ 0 ( z ) dz = − ∆ V ( T ) d φ h φ i = 0 h φ i 6 = 0 Friction ∆ V T =0 v w Two questions: enough friction to stop the wall from accelerating? ∃ If so, what is the terminal velocity and bubble profile satisfying the master equation above? 11

"Runaway Bubbles" Is there enough friction to stop the wall from accelerating once it’s moving ultra-relativistically? Bodeker + Moore, 2009 Z X dm 2 d 3 k Z (2 π ) 3 2 E f ( k, z ) φ 0 ( z ) dz ∆ V vac = − d φ 12

"Runaway Bubbles" Is there enough friction to stop the wall from accelerating once it’s moving ultra-relativistically? Bodeker + Moore, 2009 Z X dm 2 d 3 k Z (2 π ) 3 2 E f ( k, z ) φ 0 ( z ) dz ∆ V vac = − d φ For g >> 1: Equilibrium distributions ⇤ Z dm 2 Z X dm 2 d 3 k d 3 k Z Z X ⇥ m 2 i ( h 2 ) � m 2 (2 π ) 3 2 E f ( k, z ) φ 0 ( z ) dz ' i ( h 1 ) (2 π ) 3 2 E f ( k ) | h 1 d φ d φ High-T expansion (m/T<<1) X a i T 2 ⇥ m 2 i ( h 2 ) − m 2 ⇤ + O (1 / γ 2 ) i ( h 1 ) ≈ 13

"Runaway Bubbles" Is there enough friction to stop the wall from accelerating once it’s moving ultra-relativistically? Bodeker + Moore, 2009 Z X dm 2 d 3 k Z (2 π ) 3 2 E f ( k, z ) φ 0 ( z ) dz ∆ V vac = − d φ For g >> 1: Equilibrium distributions ⇤ Z dm 2 Z X dm 2 d 3 k d 3 k Z Z X ⇥ m 2 i ( h 2 ) � m 2 (2 π ) 3 2 E f ( k, z ) φ 0 ( z ) dz ' i ( h 1 ) (2 π ) 3 2 E f ( k ) | h 1 d φ d φ High-T expansion (m/T<<1) X a i T 2 ⇥ m 2 i ( h 2 ) − m 2 ⇤ + O (1 / γ 2 ) i ( h 1 ) ≈ Z X dm 2 d 3 k vacuum energy difference Z (2 π ) 3 2 E f ( k, z ) φ 0 ( z ) dz ∆ V vac < − overwhelms the friction d φ 14

"Runaway Bubbles" X Runaway a i T 2 [ m 2 i ( h 2 ) − m 2 ∆ V vac + i ( h 1 )] < 0 Bodeker + Moore, 2009 In the high-T approximation, the difference between vacua of the finite-T effective potential is given by X X a i T 2 ⇥ m 2 i ( h 2 ) − m 2 ⇤ ⇥ m 3 i ( h 2 ) − m 3 ⇤ ∆ V ( T ) ≈ ∆ V vac + i ( h 1 ) i ( h 1 ) b i T − Runaway condition can be interpreted in terms of finite-T effective potential with no cubic term: If, after dropping thermal cubic terms, it is energetically favorable to tunnel to the broken phase, the bubble can run away 15

"Runaway Bubbles" X Runaway a i T 2 [ m 2 i ( h 2 ) − m 2 ∆ V vac + i ( h 1 )] < 0 Bodeker + Moore, 2009 From JK et al, 2014 for the NMSSM If, after dropping thermal cubic terms, it is energetically favorable to tunnel to the broken phase, the bubble can run away 16

Some comments Important (and simple) criterion to check when doing pheno studies Theories with tree-level cubic terms (e.g. singlet models) are especially susceptible to runaways Recent progress and outstanding theoretical questions associated with friction in the ultra-relativistic limit: Bodeker+Moore, 2017 -NLO (e.g. 1 à 2) processes enhance the friction, tend to prevent runaway! Important for gravitational wave spectra -Effect is dominated by soft emission (in the wall frame). Full calculation is challenging -Sphalerons behind the wall?! 17

Beyond runaway What about non-relativistic walls? How do we get a number out? Some references: Moore + Prokopec, 1995 & 1996 John + Schmidt, 2000 Konstandin et al, 2014 JK, 2016 18

Beyond runaway Ultimate goal: find a wall velocity and bubble profile that solves the scalar field EOMs ∂ m 2 j ( φ i ) d 3 p i + ∂ V ( φ i ) Z X � (1 � v 2 w ) φ 00 + f j ( p, x ) = 0 (2 π ) 3 2 E j ∂φ i ∂φ i j (in the wall frame, neglecting sphericity, assuming stationary solution) Simplification: look for configurations satisfying constraint equations Z 2 3 5 · d ~ d 3 p Z � � 00 + r φ V ( � i ) + w ) ~ X 4 � (1 � v 2 r φ m 2 (Vanishing pressure) j ( � i ) f j ( p, x ) dx dx = 0 (2 ⇡ ) 3 2 E j j Z 2 3 5 · d 2 ~ d 3 p Z � (Vanishing pressure � 00 + r φ V ( � i ) + w ) ~ X 4 � (1 � v 2 r φ m 2 j ( � i ) f j ( p, x ) dx 2 dx = 0 gradient) (2 ⇡ ) 3 2 E j j 19

Wall velocities with singlets Illustrate in the real singlet extension of SM ∂ m 2 j ( φ i ) d 3 p i + ∂ V ( φ i ) Z X � (1 � v 2 w ) φ 00 + f j ( p, x ) = 0 (2 π ) 3 2 E j ∂φ i ∂φ i j First, write down Boltzmann equations for distributions ✓ ∂ d ◆ z ∂ ∂ dtf i ⌘ ∂ t + ˙ ∂ z + ˙ p z f i = � C [ f ] i ∂ p z d 3 kd 3 p 0 d 3 k 0 1 1 Z � 2 (2 π ) 4 δ ( p + k � p 0 � k 0 ) X � � � M ij ! mn ( p, k ; p 0 , k 0 ) C [ f ] i = (2 π ) 9 2 E k 2 E p 0 2 E k 0 2 N i 2 E p jmn ⇥ P ij ! mn [ f i ( p ) , f j ( k ) , f m ( p 0 ) , f n ( k 0 )] (4.3) the plasma E � 1 Utilize effective kinetic theory for excitations: , ange p & gT , L w e find for th Infrared modes with (e.g. for gauge bosons) dealt with ta p ⌧ T separately erly describ 20

Wall velocities with singlets Classify particles depending on strength of interaction with the condensate: Top quarks, SU(2) L gauge bosons, Higgs and singlet fields “feel” the passage of the wall the most; dominate the contribution in EOM v fluid All other particles treated as in local thermal equilibrium at common (space-time dependent) temperature and fluid velocity (determined from T − T + T n v + bulk properties of the PT: see Jose Miguel’s talk!) v w r/t Ansatz for relevant distributions: ⌘ � 1 δ j = � µ j � E ⇣ e ( E + δ a ) /T ± 1 , f a = T ( δ T j + δ T bg ) � p z ( δ v j + v bg ) is study, we will work to linear order in th all ( µ j /T , δ T j /T , δ T bg /T , δ v j , v bg ⌧ 1 ). 21 ong phase transitions, and we verify the va