Bootstrap for Large N Confining Gauge Theories Alexander Zhiboedov, - PowerPoint PPT Presentation

Bootstrap (Leading Order) For s,t large and positive a thermodynamic picture emerges lim log A ((1 + i ✏ ) s, (1 + i ✏ ) t ) s, t → ∞ s/t fixed s We are here − t distribution of excess zeros ρ poles zeros

Bootstrap (Leading Order) The amplitude takes the form Z d 2 z ρ ( t ; z, ¯ log A = z ) log( z − s )

Bootstrap (Leading Order) The amplitude takes the form Z d 2 z ρ ( t ; z, ¯ log A = z ) log( z − s ) In the large t limit we can write for the distribution z ) = j ( t ) ρ ( t ; z, ¯ t 2 ρ ( z/t, ¯ z/t ) dimensional analysis at large s,t

Bootstrap (Leading Order) The amplitude takes the form Z d 2 z ρ ( t ; z, ¯ log A = z ) log( z − s ) In the large t limit we can write for the distribution z ) = j ( t ) ρ ( t ; z, ¯ t 2 ρ ( z/t, ¯ z/t ) dimensional analysis at large s,t z ) log(1 − s Z d 2 z ρ ( z, ¯ log A ( s, t ) = j ( t ) tz ) ⇒ Z d 2 z ρ ( z, ¯ ρ ( z, ¯ z ) ≥ 0 z ) = 1 unitarity normalization

Bootstrap (Leading Order)

Unitarity

Unitarity The distribution of the zeros comes from a sum of Legendre polynomials with positive coefficients j ( t ) X C 2 n P n (1 + 2 β ) n =0

Unitarity The distribution of the zeros comes from a sum of Legendre polynomials with positive coefficients j ( t ) X C 2 n P n (1 + 2 β ) n =0 together with the Regge limit (3pt couplings cannot be too small) implies finite support of the excess zeros β = s 0.015 t � � � β + 1 � � � ≤ 1 0 -1 � � 2 -0.015

Bootstrap (Leading Order) z ) log(1 − s Z d 2 z ρ ( z, ¯ log A ( s, t ) = j ( t ) tz ) ⇒ Z d 2 z ρ ( z, ¯ ρ ( z, ¯ z ) ≥ 0 z ) = 1 unitarity normalization j ( t ) = t k Assume

Bootstrap (Leading Order) 2d “electric potential” for a positive V ( β ) = log A ( s, t ) Z z ) log(1 − β d 2 z ρ ( z, ¯ = z ) = j ( t ) ρ ( z, ¯ z ) distribution of charge s/t d 2 z ρ ( z, ¯ z ) Z 2d “electric field” E ( β ) = t ∂ s V ( s/t ) = β − z =

Bootstrap (Leading Order) 2d “electric potential” for a positive V ( β ) = log A ( s, t ) Z z ) log(1 − β d 2 z ρ ( z, ¯ = z ) = j ( t ) ρ ( z, ¯ z ) distribution of charge s/t d 2 z ρ ( z, ¯ z ) Z 2d “electric field” E ( β ) = t ∂ s V ( s/t ) = β − z = E ( β ) analytic where ρ = 0 k 2 E ( β ) = 1 β 2 F 1 ( k, k, k + 1; − 1 / β ) = 1 1 + ⇒ β 2 + . . . β − crossing k + 1 M 1

Bootstrap (Leading Order) 2d “electric potential” for a positive V ( β ) = log A ( s, t ) Z z ) log(1 − β d 2 z ρ ( z, ¯ = z ) = j ( t ) ρ ( z, ¯ z ) distribution of charge s/t d 2 z ρ ( z, ¯ z ) Z 2d “electric field” E ( β ) = t ∂ s V ( s/t ) = β − z = E ( β ) analytic where ρ = 0 k 2 E ( β ) = 1 β 2 F 1 ( k, k, k + 1; − 1 / β ) = 1 1 + ⇒ β 2 + . . . β − crossing k + 1 M 1 k ≤ 1 β →∞ E ( β ) = 1 β → 0 E ( β ) = − k β k − 1 log β + . . . lim β lim β + . . . k ≤ 1 ⇒ ⇒ crossing

Bootstrap (Leading Order) M 1 ≥ 1 X ∂ 2 C 2 θ log n P n (cos θ ) ≥ 0 k = 1 ⇒ ⇒ k ≥ 1 ⇒ 2 n mathematical identity

Bootstrap (Leading Order) M 1 ≥ 1 X ∂ 2 C 2 θ log n P n (cos θ ) ≥ 0 k = 1 ⇒ ⇒ k ≥ 1 ⇒ 2 n mathematical identity for ρ ( x, x ) = 1 − 1 < x < 0 ⇒ s The unique solution is 1 Z 1 + s ⇣ ⌘ − t log A ( s, t ) = α 0 t dx ρ ( x ) log tx 0 = α 0 [( s + t ) log( s + t ) − s log s − t log t ] = classical string theory

Part II Universal Correction to the Veneziano Amplitude

Result Part II The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form ∼ E 2 lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ s/t fixed

Result Part II The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form ∼ E 2 lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ s/t fixed ✓ s t ◆ 1 − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t

Result Part II The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form ∼ E 2 lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ s/t fixed 1 ∼ E 2 ✓ s t ◆ 1 − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t

Result Part II The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form ∼ E 2 lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ s/t fixed 1 ∼ E 2 ✓ s t ◆ 1 − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t elliptic integral of the first kind EllipticK[x]

Result Part II The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form ∼ E 2 lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ s/t fixed 1 ∼ E 2 ✓ s t ◆ 1 − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t elliptic integral of the first kind correction due to the slowdown of the string EllipticK[x] (massive endpoints)/spectrum non-degeneracy

Result Part II The high energy limit of WIHS amplitudes at imaginary scattering angles takes the form ∼ E 2 lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ s/t fixed corrections are O(1) 1 ∼ E 2 ✓ s t ◆ 1 − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t elliptic integral of the first kind correction due to the slowdown of the string EllipticK[x] (massive endpoints)/spectrum non-degeneracy

Result ✓ s t ◆ 1 δ log A ( s, t ) = − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t

Result ✓ s t ◆ 1 δ log A ( s, t ) = − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t • worldsheet: slowdown of the string endpoints t − 8 √ π ✓ ◆ m 3 / 2 t 1 / 4 + ... j ( t ) = α 0 m m 3 [Chodos, Thorn, 74’] [Baker, Steinke] [Wilczek] [Sonnenschein et al.]

Result ✓ s t ◆ 1 δ log A ( s, t ) = − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t • worldsheet: slowdown of the string endpoints t − 8 √ π ✓ ◆ m 3 / 2 t 1 / 4 + ... j ( t ) = α 0 m m 3 [Chodos, Thorn, 74’] [Baker, Steinke] [Wilczek] [Sonnenschein et al.] • bootstrap: removal of the spectrum degeneracy j sub − leading ( t ) 6 = j leading ( t ) + integer

• Scattering of Strings With Massive Endpoints • Universality (Holography & EFT of Long Strings) • Bootstrap

Worldsheet Computation (review) A ( s, t ) = e − S E ( s,t ) lim [Gross, Mende] [Gross, Mañes] | s | , | t | → ∞ [Alday, Maldacena] s/t fixed

Worldsheet Computation (review) A ( s, t ) = e − S E ( s,t ) lim [Gross, Mende] [Gross, Mañes] | s | , | t | → ∞ [Alday, Maldacena] s/t fixed • real scattering angles (amplitude is small) S E � 1

Worldsheet Computation (review) A ( s, t ) = e − S E ( s,t ) lim [Gross, Mende] [Gross, Mañes] | s | , | t | → ∞ [Alday, Maldacena] s/t fixed • real scattering angles (amplitude is small) S E � 1 • imaginary scattering angles (amplitude is large) � S E � 1

Worldsheet Computation (review)

Worldsheet Computation (review) 1 Z Flat space d 2 z ∂ x · ¯ X S E = k j · x ( σ j ) ∂ x − i 2 πα 0 j

Worldsheet Computation (review) 1 Z Flat space d 2 z ∂ x · ¯ X S E = k j · x ( σ j ) ∂ x − i 2 πα 0 j x µ X k µ i log | z − σ i | 2 • general solution 0 = i i

Worldsheet Computation (review) 1 Z Flat space d 2 z ∂ x · ¯ X S E = k j · x ( σ j ) ∂ x − i 2 πα 0 j x µ X k µ i log | z − σ i | 2 • general solution 0 = i i k i · k j X • Virasoro (scattering equations) = 0 σ i − σ j j

Worldsheet Computation (review) 1 Z Flat space d 2 z ∂ x · ¯ X S E = k j · x ( σ j ) ∂ x − i 2 πα 0 j x µ X k µ i log | z − σ i | 2 • general solution 0 = i i k i · k j X • Virasoro (scattering equations) = 0 σ i − σ j j log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log s − t log t ] ⇒ s, t > 0

Adding The Mass

Adding The Mass 1 Z Z d 2 z ∂ x · ¯ X p S E = ∂ x + m | ∂ σ x | 2 − i k j · x ( σ j ) d σ 2 πα 0 j [Chodos, Thorn]

Adding The Mass 1 Z Z d 2 z ∂ x · ¯ X p S E = ∂ x + m | ∂ σ x | 2 − i k j · x ( σ j ) d σ 2 πα 0 j [Chodos, Thorn] Modified boundary condition: 1 ∂ σ x X 2 πα 0 ∂ τ x + m ∂ σ √ ∂ σ x · ∂ σ x = i k j δ ( σ − σ j ) j

Adding The Mass 1 Z Z d 2 z ∂ x · ¯ X p S E = ∂ x + m | ∂ σ x | 2 − i k j · x ( σ j ) d σ 2 πα 0 j [Chodos, Thorn] Modified boundary condition: 1 ∂ σ x X 2 πα 0 ∂ τ x + m ∂ σ √ ∂ σ x · ∂ σ x = i k j δ ( σ − σ j ) j ∂ σ x 0 · ∂ σ x 0 = 0 is zero for a free string!

Adding The Mass 1 Z Z d 2 z ∂ x · ¯ X p S E = ∂ x + m | ∂ σ x | 2 − i k j · x ( σ j ) d σ 2 πα 0 j [Chodos, Thorn] Modified boundary condition: 1 ∂ σ x X 2 πα 0 ∂ τ x + m ∂ σ √ ∂ σ x · ∂ σ x = i k j δ ( σ − σ j ) j ∂ σ x 0 · ∂ σ x 0 = 0 is zero for a free string! √ m The expansion reorganizes itself in terms of : x µ = x µ S = S 0 + √ mS 1 + mS 2 + m 3 / 2 S 3 0 + √ m x µ 1 + ...

Adding The Mass 1 Z Z d 2 z ∂ x · ¯ X p S E = ∂ x + m | ∂ σ x | 2 − i k j · x ( σ j ) d σ 2 πα 0 j [Chodos, Thorn] Modified boundary condition: 1 ∂ σ x X 2 πα 0 ∂ τ x + m ∂ σ √ ∂ σ x · ∂ σ x = i k j δ ( σ − σ j ) j ∂ σ x 0 · ∂ σ x 0 = 0 is zero for a free string! √ m The expansion reorganizes itself in terms of : x µ = x µ S = S 0 + √ mS 1 + mS 2 + m 3 / 2 S 3 0 + √ m x µ 1 + ...

Adding The Mass The on-shell action evaluates to S E = S GM + 2 3 mL b + ... Z 2 πα 0 m √ σ x 0 ) 1 / 4 d σ ( ∂ 2 σ x 0 · ∂ 2 L b =

Adding The Mass The on-shell action evaluates to S E = S GM + 2 Gross-Mende solution 3 mL b + ... Z 2 πα 0 m √ σ x 0 ) 1 / 4 d σ ( ∂ 2 σ x 0 · ∂ 2 L b =

Adding The Mass The on-shell action evaluates to S E = S GM + 2 Gross-Mende solution 3 mL b + ... Z 2 πα 0 m √ σ x 0 ) 1 / 4 d σ ( ∂ 2 σ x 0 · ∂ 2 L b = reparameterization invariant

Adding The Mass The on-shell action evaluates to S E = S GM + 2 Gross-Mende solution 3 mL b + ... Z 2 πα 0 m √ σ x 0 ) 1 / 4 d σ ( ∂ 2 σ x 0 · ∂ 2 L b = reparameterization invariant For four external particles ✓ s t ◆ 1 δ log A ( s, t ) = − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 + O ( m 5 / 2 ) K + K 3 s + t s + t s + t

Adding The Mass The on-shell action evaluates to S E = S GM + 2 Gross-Mende solution 3 mL b + ... Z 2 πα 0 m √ σ x 0 ) 1 / 4 d σ ( ∂ 2 σ x 0 · ∂ 2 L b = reparameterization invariant For four external particles ✓ s t ◆ 1 δ log A ( s, t ) = − 16 √ π 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 + O ( m 5 / 2 ) K + K 3 s + t s + t s + t non-universal O ( t − 1 / 4 )

Emergent s-u Crossing Symmetry lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ ✓ s t ◆ 1 − 16 √ π s/t fixed 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t

Emergent s-u Crossing Symmetry lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ ✓ s t ◆ 1 − 16 √ π s/t fixed 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t log A ( s, t ) = log A ( t, s ) The s-t crossing is manifest:

Emergent s-u Crossing Symmetry lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ ✓ s t ◆ 1 − 16 √ π s/t fixed 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t log A ( s, t ) = log A ( t, s ) The s-t crossing is manifest: What about the s-u crossing?

Emergent s-u Crossing Symmetry lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ ✓ s t ◆ 1 − 16 √ π s/t fixed 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t log A ( s, t ) = log A ( t, s ) The s-t crossing is manifest: What about the s-u crossing? log A ( s, t ) = Re[log A ( u, t )] u = − s − t

Emergent s-u Crossing Symmetry lim log A ( s, t ) = α 0 [( s + t ) log( s + t ) − s log( s ) − t log( t )] s, t → ∞ ✓ s t ◆ 1 − 16 √ π s/t fixed 4  ✓ ◆ ✓ ◆� s t α 0 m 3 / 2 K + K + . . . 3 s + t s + t s + t log A ( s, t ) = log A ( t, s ) The s-t crossing is manifest: What about the s-u crossing? log A ( s, t ) = Re[log A ( u, t )] u = − s − t 2 3 ??? 4 4 3 2 1 1

Emergent s-u Crossing Symmetry [Komatsu] 2 3 ??? 4 4 3 2 1 1

Emergent s-u Crossing Symmetry [Komatsu] 2 3 ??? 4 4 3 2 1 1

Asymptotic s-u Crossing Equivalently, the asymptotic s-u crossing is: dDisc s log A ( s, t ) ≡ log A ( − s − t + i ✏ , t ) + log A ( − s − t − i ✏ , t ) − 2 log A ( s, t ) = 0 Double discontinuity is zero!

Why is the correction universal?

Why is the correction universal? Why is the massive ends model physical?