Bounds on Amplitudes and EFTs Brando Bellazzini IPhT - CEA/Saclay - PowerPoint PPT Presentation

Bounds on Amplitudes and EFTs Brando Bellazzini IPhT - CEA/Saclay based on 1605.06111 and work in progress Eltville, Burg Crass, ‘EFTs for collider physics, Flavor phenomena and EWSB’, Sept 15th 2016

Hierarchy of scales Λ UV a blessing… g ρ f π Λ IR

Hierarchy of scales Λ UV a blessing… g ρ f π - small parameter 
 E/ Λ UV Λ IR - emerging patterns 
 O ( x ) L IR = L ∆ ≤ 4 + X - suppress dangerous operators 
 Λ ∆ − 4 UV O

Hierarchy of scales Λ UV a blessing… or a curse g ρ f π - small parameter 
 E/ Λ UV Λ IR - emerging patterns 
 O ( x ) L IR = L ∆ ≤ 4 + X - suppress dangerous operators 
 Λ ∆ − 4 UV O

Hierarchy of scales Λ UV a blessing… or a curse g ρ f π - small parameter 
 E/ Λ UV Λ IR - emerging patterns 
 O ( x ) L IR = L ∆ ≤ 4 + X - suppress dangerous operators 
 Λ ∆ − 4 UV O large couplings from a strong sector may help L = g 2 ( ∂ H 2 ) 2 e.g. in CHM: ∗ m 2 ∗

The EFT paradigm UV O i ( x ) X L IR = IR c i Λ ∆ − 4 UV i EFT encodes UV-info via c i Finite set of C’s is needed at any order in E/ Λ UV Power counting = understanding = symmetries

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Example ψ i ∂ψ − m ∗ ¯ ¯ ψψ + . . . (1)

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Example ψ i ∂ψ − m ∗ ¯ ¯ i @ − ✏ · m ∗ ¯ ¯ ψψ + . . . + . . . (1) χ –sym

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Example ψ i ∂ψ − m ∗ ¯ ¯ i @ − ✏ · m ∗ ¯ ¯ ψψ + . . . + . . . (1) χ –sym 1-to-1 amplitude dominated by a less-relevant operator M (1 → 1) ∼ 1 ✏ · m ∗ < E < m ∗ E

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Example ψ i ∂ψ − gA µ ¯ ¯ (2) ψγ µ ψ + . . .

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Example � µ + g 2 � µ ) 2 + . . . i @ − ✏ · g ∗ A µ ¯ ¯ ( ¯ ψ i ∂ψ − gA µ ¯ ¯ (2) ψγ µ ψ + . . . ∗ m 2 g ⌧ 1 ∗

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Example � µ + g 2 � µ ) 2 + . . . i @ − ✏ · g ∗ A µ ¯ ¯ ( ¯ ψ i ∂ψ − gA µ ¯ ¯ (2) ψγ µ ψ + . . . ∗ m 2 g ⌧ 1 ∗ dominated by dim-6 at intermediate energy ✏ · m ∗ < E < m ∗ g SM g SM g ∗ g ∗ M (2 → 2) = g 2 E 2 ✓ ◆ 1 + 1 SM E 2 ✏ 2 m 2 ∗ 1 /m 2 1 /E 2 ∗ Amplitude runs fast within the validity of EFT

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Examples ∗ π 2 + g 2 ∗ π 4 + . . . ( ∂π ) 2 − m 2 (3)

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Examples ∗ ⇡ 4 ) + g 2 ∗ π 2 + g 2 ∗ π 4 + . . . ∗ ⇡ 2 + ✏ 2 g 2 ( @⇡ ) 4 + . . . ( ∂π ) 2 − m 2 (3) ( @⇡ ) 2 − ✏ 2 ( m 2 ∗ m 4 π → π + c ∗

Symmetry <—> Softness Higher dim-operators may dominate the amplitude within EFT just suppress relevant, marginal and less-irrelevant operators by symmetries Examples ∗ ⇡ 4 ) + g 2 ∗ π 2 + g 2 ∗ π 4 + . . . ∗ ⇡ 2 + ✏ 2 g 2 ( @⇡ ) 4 + . . . ( ∂π ) 2 − m 2 (3) ( @⇡ ) 2 − ✏ 2 ( m 2 ∗ m 4 π → π + c ∗ E 4 ✓ ◆ M (2 → 2) = g 2 ∗ ✏ 4 1 + Amplitude runs fast within the validity of EFT: m 4 ∗ ✏ 4 ✏ · m ∗ < E < m ∗

Running Coupling g ∗ = M 4 E 2 E 4 g SM E m ∗

Running Coupling LHC g ∗ = M 4 E 2 LEP E 4 g SM E m ∗

Running Coupling LHC g ∗ = M 4 E 2 LEP E 4 g SM E m ∗ V T E 4 g 2 W 4 ∼ g 2 ∗ ‘remedios’: strongly int. transv. vectors m 4 µ ν ∗ m 4 1603.03064 Liu, Pomarol, Rattazzi, Riva ∗ ∗ V T

Running Coupling LHC g ∗ = M 4 E 2 Freeze-out E 4 g SM E m ∗ V T E 4 g 2 W 4 ∼ g 2 ∗ ‘remedios’: strongly int. transv. vectors m 4 µ ν ∗ m 4 1603.03064 Liu, Pomarol, Rattazzi, Riva ∗ ∗ V T

Running Coupling LHC g ∗ = M 4 E 2 Freeze-out E 4 g SM E m ∗ V T E 4 g 2 W 4 ∼ g 2 ∗ ‘remedios’: strongly int. transv. vectors m 4 µ ν ∗ m 4 1603.03064 Liu, Pomarol, Rattazzi, Riva ∗ ∗ V T DM as light pseudo-goldstino 1607.02474 Brugisser, Riva, Urbano *

Running Coupling FCC g ∗ = M 4 E 2 LHC E 4 g SM E m ∗ V T E 4 g 2 W 4 ∼ g 2 ∗ ‘remedios’: strongly int. transv. vectors m 4 µ ν ∗ m 4 1603.03064 Liu, Pomarol, Rattazzi, Riva ∗ ∗ V T DM as light pseudo-goldstino 1607.02474 Brugisser, Riva, Urbano *

How fast? ( ∂π ) 2 π 2 Goldstones ∼ E 2 ( ¯ ψγ µ ψ ) 2 4-Fermions … can amplitudes be softer than E 4 ? (within an EFT) ( ∂σ ) 4 dilaton ψ 2 ⇤ ψ 2 ¯ Goldstino ∼ E 4 F 4 remedios µ ν …

UV-IR connection (well known) 
 e.g. hep-th/0602178 For spin-0 particles the answer is: No! Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi

UV-IR connection (well known) 
 e.g. hep-th/0602178 For spin-0 particles the answer is: No! Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 1 3 Analyticity, Crossing, and Unitarity M 4 2 4

UV-IR connection (well known) 
 e.g. hep-th/0602178 For spin-0 particles the answer is: No! Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 1 3 Analyticity, Crossing, and Unitarity M 4 2 4 Im s ˜ C C ● μ 2 Re s ● ● u IR =( m 1 - m 2 ) 2 s IR =( m 1 + m 2 ) 2 s - plane

UV-IR connection (well known) 
 e.g. hep-th/0602178 For spin-0 particles the answer is: No! Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 1 3 Analyticity, Crossing, and Unitarity M 4 2 4 small circle=big circle Im s ˜ C C ● μ 2 Re s ● ● u IR =( m 1 - m 2 ) 2 s IR =( m 1 + m 2 ) 2 s - plane

UV-IR connection (well known) 
 e.g. hep-th/0602178 For spin-0 particles the answer is: No! Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 1 3 Analyticity, Crossing, and Unitarity M 4 2 4 s ↔ u small circle=big circle Disc s = Disc u Im s ˜ C C ● μ 2 Re s ● ● u IR =( m 1 - m 2 ) 2 s IR =( m 1 + m 2 ) 2 s - plane

UV-IR connection (well known) 
 e.g. hep-th/0602178 For spin-0 particles the answer is: No! Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 1 3 Analyticity, Crossing, and Unitarity M 4 2 4 s ↔ u Disc ∼ s σ T ot ( s ) > 0 small circle=big circle Disc s = Disc u t=0 Im s ˜ C C ● μ 2 Re s ● ● u IR =( m 1 - m 2 ) 2 s IR =( m 1 + m 2 ) 2 s - plane

UV-IR connection (well known) 
 e.g. hep-th/0602178 For spin-0 particles the answer is: No! Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi 1 3 Analyticity, Crossing, and Unitarity M 4 2 4 s ↔ u Disc ∼ s σ T ot ( s ) > 0 small circle=big circle Disc s = Disc u t=0 Im s ˜ C C ● Z 1 μ 2 ds � M 00 (2 → 2) IR = s 3 σ 12 ! anything ( s ) > 0 � Re s 0 ● ● u IR =( m 1 - m 2 ) 2 s IR =( m 1 + m 2 ) 2 IR-side UV-side E 4 -terms are strictly positive s - plane

Example L = 1 2( ∂ µ π ) 2 + c Λ 4 ( ∂ µ π ) 4 + . . . π → π + const

Example L = 1 2( ∂ µ π ) 2 + c Λ 4 ( ∂ µ π ) 4 + . . . π → π + const Z 1 ππ ! ππ ( s = 0) = 4 ds dispersion relation: M 00 s 3 σ ππ ! anything ( s ) > 0 π ππ → ππ 0 IR-side UV-side

Example L = 1 2( ∂ µ π ) 2 + c Λ 4 ( ∂ µ π ) 4 + . . . π → π + const Z 1 ππ ! ππ ( s = 0) = 4 ds dispersion relation: M 00 s 3 σ ππ ! anything ( s ) > 0 π ππ → ππ 0 IR-side UV-side calculable within the EFT c > 0

Example L = 1 2( ∂ µ π ) 2 + c Λ 4 ( ∂ µ π ) 4 + . . . π → π + const Z 1 ππ ! ππ ( s = 0) = 4 ds dispersion relation: M 00 s 3 σ ππ ! anything ( s ) > 0 π ππ → ππ 0 IR-side UV-side calculable within the EFT c > 0 This interacting theory can’t be softer than E 4

Spinning Particles?

Spinning Particles? (1) Amplitudes not Lorentz inv. but Little-group covariant

Spinning Particles? (1) Amplitudes not Lorentz inv. but Little-group covariant 1 s, t, u α v σ u σ + polarizations α ε σ µ … (not amplitudes squared)

Spinning Particles? (1) Amplitudes not Lorentz inv. but Little-group covariant σ 1 σ 3 α 1 1 1 3 s, t, u α v σ u σ u σ ) − 1 ( x = M 4 + polarizations α α αα 1 ε σ 2 4 µ … σ 4 σ 2 h Φ α 1 ( p 1 ) Φ β 1 ( p 2 ) . . . i (not amplitudes squared)