Lüscher corrections for non-diagonal form factors in integrable QFTs

Transcription

1 Lüscher corrections for non-diagonal form factors in integrable QFTs Supervisor: Zoltán Bajnok Márton Lájer ELFT Seminar, 11 April 018

2 Outline Introduction to integrability Form factors Finite volume physics Nondiagonal 1-particle form factor Conclusions

3 Recent Work [arxiv: ] Z. Bajnok, J. Balog, M. Lájer, C. Wu, Field theoretical derivation of Lüscher's formula and calculation of nite volume form factors, (018)

4 Outline Introduction to integrability Form factors Finite volume physics Nondiagonal 1-particle form factor Conclusions

5 Integrable QFT Dening property: number of (higher spin) coserved charges Examples: Free theories L = 1 ( µ φ ) m φ L = i ψγ µ µ ψ + m ψψ some 1+1D theories with nontrivial scattering: L = 1 ( µ φ ) m (coshbφ 1) b sinh-gordon L = 1 ( µ φ ) m (coshβφ 1) β sine-gordon L = 1 ( µ φ ) ( m e gφ + e gφ 6g Bullough-Dodd ) q i (e βα i φ 1 L = 1 ( µ φ ) ( µ φ) µ 8π 16λ r+1 i=1 Toda hierarchy

6 Conserved charges (cl. sinh-gordon) Lightcone coordinates: x ± = 1 (t ± x) sinh-gordon EOM (m = 1,φ bφ) Conserved currents: Conserved charges: µ J (s)µ = 0 + φ = sinhφ + J (s) + J (s) + Q (s) = dxj (s) 0

7 Conserved charges (cl. sinh-gordon) Bäcklund transformation: Let φ be a solution of the EOM. Then ˆφ also a solution if ) [ 1 ( ) (φ ] ˆφ = ε sinh φ + ˆφ ) + (φ + ˆφ = [ 1 ( ) ] ε sinh ˆφ φ Expand ˆφ (x +,x,ε) in ε and solve order by order ˆφ ˆφ (x +,x,ε) = n=0 ˆφ n (x +,x )ε n ˆφ 0 = 0 ˆφ 1 = + φ ˆφ 4 = 4 +φ ( + φ) +φ ˆφ = +φ ˆφ 3 = 3 +φ 1 3 ( +φ) 3

8 Conserved charges (cl. sinh-gordon) An useful identity (for φ solving the EOM): ( 1 [ +φ] ) + + (coshφ 1) = 0 Substitute series expansion for ˆφ: conserved current for each power of ε J (1) + T = 1 ( +φ) ( J (3) + T 4 = +φ) + + φ +φ 3 ( ) J (5) + T ( 6 = +φ φ + + φ 5 +φ ) 4 +φ 6 At quantum level: [ Q (s),q (s ) ] = 0 ( +φ) ( + φ) ( + φ) 3 3 +φ

9 S-matrix generalities S-matrix denition: S = f,out i,in = f,in S i,in Unitarity: ψ = c n n, n c n = 1 out = S 1 in 1 = m S ψ = c S n n 1 c n n 1 S m n 1 n = S S = 1 c n1 Separation of interaction part: S = δ + i (π) d δ d (p f p i )T Implication of unitarity (optical theorem): T T if = i (π)d n H δ d (p n p i )T fn T in

10 S-matrix elements scattering matrix elements: i = p 1,p, f = p 3,p 4 1 p 3,p 4 (T T ) p 1,p = (π) d δ d (p n p i ) i n H p 3,p 4 T n n T p1,p d = : separate a δ from the S-matrix element p 3,p 4 S p 1,p (π) δ (p 1 + p p 3 p 4 )S (s,t,u) s = (p 1 + p ), t = (p 1 p 3 ), u = (p 1 p 4 ) Purely elastic scattering: p 1 = p 4, S (s,t,u) S (s,t) Mandelstam variable identity: s + t + u = 4 i=1 m i, so S (s,t) S (s) Crossing symmetry: ) S (s + iε) = S (t + iε) = S (m 1 + s iε m Time reversal: S (s + iε) = S (s iε) 1

11 elastic S-matrix analytic structure Implication of optical theorem for S (s): ReS (s) = dk n δ (p 1 + p p n )Γ n 1 Γ34 n + dk n1 dk n δ (p 1 + p p n1 p n )S ({p 1,p,p n1,p n })S ({p n1,p n,p 3,p 4 })+ dk n1 dk n dk n3... s Bound states 4m 1 S i ε (m + m ) 1 4m 1 m m iε + S

12 Rapidity, states Ecient paramtetrization of particle momenta: p (0) = m i i coshθ i p (1) = m i i sinhθ i Action of Lorentz transform: θ θ + α n-particle state: A a1 (θ 1 )A a (θ )...A an (θ n ) in state ordering: θ 1 > θ > > θ n out state ordering: θ 1 < θ < < θ n customary normalization: Ai (θ 1 ) A j (θ ) = πδ ij δ (θ 1 θ )

13 Eect of higher spin charges Conserved charges: Q 1 = P = P (0) + P (1) Q 1 = P = P (0) P (1) All charges commute Common eigensystem: [Q s,q s ] = 0 Q s A a (θ) = ω (a) s A a (θ) Lorentz transform: Q s transforms as s copies of P, Q s as s copies of P χ (a) s ω (a) s is an attribute of particle type a (θ) = χ (a) s e sθ

14 Eect of higher spin charges Coleman-Mandula theorem (3+1D): higher spin conserved charges = S = 1 1+1D: severe constraints N a of particles with mass m a conserved a set of nal momenta coincide with the set of initial momenta (purely elastic scattering) n-particle scattering completely factorises into 1 n (n 1) two-particle scatterings Reason for elasticity: Q s A a1 (θ 1 )...A an (θ n ) = ( n i=1 ) χ (a i ) s e sθ i A a1 (θ 1 )...A an (θ n ) χ a i i in dq s dt = 0 s esθ i = j out χ a j s e sθ j Only solution: sets of rapidities equal (permutations among particles with same mass and χs a possible)

15 Eect of higher spin charges (Heuristical) reason of factorisation: use (χ (a) s = 1 for simplicity) Take a localised wavepacket e icq s A a (p) = e icps A a (p) ψ (x) = dpe a(p p 0) e ip(x x 0) inspect the eect of symmetry transform generated by Q s on ψ (x): e icq s ψ (x) = dpe a(p p 0) e ip(x x 0) e icp s Saddle-point approximation = particle localised at x = x 0 scp s 1 momentum-dependent transformation for s

16 Eect of higher spin charges t = = Yang-Baxter equations S ({p 1,p })S ({p,p 3 })S (p 1,p 3 ) = S ({p,p 3 })S ({p 3,p 1 })S ({p 1,p }) This is necessary and sucient for the complete factorization of n-particle scattering S-matrix theory drastically simplied -particle S-matrices can be found exactly

17 S-matrix properties d S-matrix elements: Ai (θ 1 )A j (θ ) = S kl ij (θ) A k (θ )A l (θ 1 ), θ 1 > θ θ θ 1 θ Dierent normalization of states = dierent normalization of S-matrix elements (earlier) S kl ij (s) = 4m i m j sinhθs kl ij (θ) Mandelstam variables (θ ij θ i θ j ): s ( θ ij ) = ( pi + p j ) = m i + m j + m i m j coshθ ij t ( θ ij ) = ( pi p j ) = s ( iπ θij ) S kl ij (θ): r types of particles = r 4 functions

18 S-matrix properties Discrete symmetries Yang-Baxter equations: P : C : T : S kl ij S kl ij S kl ij = S lk ji = S k l ī j = S ji lk S ab ij (θ 1 )S cd bk (θ 13)Sac nm n m l (θ 3 ) = S ab n jk (θ 3)S nc ia (θ 13)S ml cb (θ 1) m l c a b i k j 1 3 = c b i a j k 1 3

19 S-matrix properties Analytic properties: only 1 pair of branch cuts: s ( ), ( ) m i + m j s mi m j (no particle production) [ ] S is a real analytic function S kl ij (s ) = (s) S kl ij S kl ij (s) S kl ij (θ): ( ) s θ ij = m i + m j + m i m j coshθ ij θ S crossing 1 unitarity = unitarity crossing 1

20 S-matrix properties Unitarity (s + s + i0, s > ( m i + m j ) S kl ij ( s + )[ S mn kl ( s + )] = δ m i δ n j S kl ij (θ)s mn kl ( θ) = δ m i δ n j m n m n θ k l i θ j i j Crossing S kl ij ( s + ) ( = S k j i l m i m j s+) S kl ij (θ) = S k j (iπ θ) i l

21 Bound states S-matrix poles S kl R (n) ij (θ) i θ iu n ij ( ) s u n ij mn = m i + m j + m i m j cosu n ij Particles appear as bound states of each other u n ij + uj in + ui jn = π S-matrix bootstrap (diagonal scattering for simplicity): ( ) S il (θ) = S ij (θ + iū k jl )S ik θ iū j, ū c lk ab = π uc ab k k l = θ l j i j i

22 Exact S-matrices Building block: f x (θ) = sinhθ + i sinπx sinhθ i sinπx Sinh-Gordon model (r = 1, no bound state) L = 1 ( µ φ ) m (coshbφ 1) b S (θ) = f B (θ), B (b) = b 8π + b Bullough-Dodd model (r=1, self bound state) L = 1 ( µ φ ) ( ) m e gφ + e gφ 6g S (θ) = f 3 (θ)f G 3 (θ)f G 3 (θ), G (g) = g 4π + g

23 Exact S-matrices sine-gordon model: complicated spectrum (kink K, antikink K, breathers B n ) L = 1 ( µ φ ) m (cosβφ 1) β ξ = β 8π β Kink/antikink scattering nondiagonal S SG (θ) = S S T S R S R S T S, S (θ) = exp i t dt 0 t t(ξ θ) sinh ξ sinh t πt cosh sinθt πθ sinh ξ S T (θ) = sinh π(iπ θ) S (θ) ξ Number of breathers: N = exactly S R (θ) = i sin π ξ sinh π(iπ θ) ξ S (θ) π ξ, KB n, K Bn, B n B m S-matrices also known

24 Outline Introduction to integrability Form factors Finite volume physics Nondiagonal 1-particle form factor Conclusions

25 Motivation How to calculate correlation functions (of local, scalar operators)? In D a good strategy is to insert complete systems (of in/out states) 1 = O (x)o (0) = n=0 n=0 dθ1...dθ n n!(π) n θ 1,...,θ n θ 1,...,θ n dθ 1...dθ n n!(π) n 0 O (x) θ 1...θ n in θ 1...θ n O (0) 0 This series has very good convergence properties. A form factor is a matrix element between an in and an out state θ m+1...θ n O (x) θ 1...θ m = 0 O (x) θ 1...θ m,θ m+1 iπ,...θ n iπ + disc. The elementary form factor F O n is dened as F O n (θ 1...θ n ) := 0 O (0) θ 1...θ n

26 Form factor axioms 1. Maximal analycity: F O n is a meromorphic function, and its poles have physical origin. permutation property F n (θ 1,...,θ i,θ i+1,...,θ n ) = S (θ i θ i+1 )F n (θ 1,...,θ i+1,θ i,...,θ n ) = 1 i i+1 n 1 i i+1 n 3. periodicity property F n (θ 1 + πi,θ...θ n ) = F n (θ,...,θ n,θ 1 ) = n i= S ( θ i θ j ) Fn (θ 1,...,θ n ) = 1 i i+1 n 1 i i+1 n

27 Form factor axioms 4. kinematical singularity ( i lim θ θ ) ( ( F θ n+ θ ) + iπ,θ,θ 1,...,θ n = 1 θ n i=1 S (θ θ i ) ) F n (θ 1...θ n ) θ θ i Res = + θ θ 1 n θ 1 n θ 1 n 5. dynamical singularity: if the S-matrix has a simple pole ( ) ( ), θ iu k ij S ij (θ) = Γ k ij then i lim θ iu k ij ( ) i lim F n+1 θ + iū j ε 0 ik ε,θ iūi jk + ε,θ 1...θ n 1 = Γ k ij F n (θ,θ 1,...θ n 1 ) i Res = Γ n 1 n 1 1 1

28 Form factor bootstrap Axioms 1-3 satised by taking F O n in the following form F O n = K O ( ) n (θ 1,...θ n ) F min θij i<j where F min (θ) is the minimal -particle FF, with the following properties: analytic in 0 Imθ < π mildest possible behavior as θ Solution to axioms -3 (Watson equations) with neither zeroes or poles in 0 < Imθ < π The above properties uniquely determine F min up to a normalization factor. Parametrizing the S-matrix as dt S (θ) = exp tθ f (t)sinh t iπ 0 dt F min (θ) = N exp t f t ˆθ, ˆθ = iπ θ (t)sin iπ 0

29 Form factor bootstrap K O n factors: solve Watson equations with S (θ) = 1 completely symmetric in each variable+period πi = functions of coshθ ij contain all physical poles: K O n (θ 1...θ n ) = QO n (θ 1...θ n ) D n (θ 1...θ n ) D n only depends on the theory, not the operator. All information about O is contained in Q O n

30 sinh-gordon form factors Calculation of the minimal -particle form factor sinhθ i sinπb S (θ) = sinhθ + i sinπb ( dx sinh = exp 8 x dx F min (θ,b) = N exp 8 x choosing the normalization as dx N = exp 4 x xb 4 ( sinh ) sinh xb 4 ( sinh xb 4 ( x ) sinh ) sinh ( ( 1 ))sinh B x sinh sinhx ( x sinh x ( x ( )) 1 B sinh x sinh x ( )) 1 B sinh x xθ iπ ) ( ) x ˆθ sin π results in lim F min (θ,b) = 1 θ Since there are only kinematical poles (no bound states), the form factor Ansatz can be written ( ) F min θij F n (θ 1...θ n ) = H n Q n (x 1...x n ), x x i<j i + i e θ i x j

31 sinh-gordon form factors Since there are only kinematical poles (no bound states), the form factor Ansatz can be written ( ) F min θij F n (θ 1...θ n ) = H n Q n (x 1...x n ), x x i<j i + i e θ i x j This has poles exactly at θ i = θ j + iπ Kinematical singularity condition is translated into a recursion on Q n ( 1) n Q n+ ( x,x,x 1...x n ) = xc n (x,x 1,...,x n )Q n (x 1,...,x n ) ( n C n = i ) n ( ) 4sin πb (x + ωx i ) (x ) ω 1 x i (x ωx i ) x + ω 1 x i i=1 i=1 ( ) iπb ω = exp Normalization H n xed (up to H 1 and H ) by ( H n+1 = H 1 µ n, H n = H µ n, µ = 4sin πb F min (iπ,b) ) 1

32 sinh-gordon form factors This recursion can be solved! A remarkable class of solutions is the elementary solution (k Z) Q n (k) = detm (k) [ ] M ij (k) = σ (n) sin (i j + k) B i j sin B where σ (n) is the completely symmetric polynomial in n variables of total k degree k (but linear in each variable). Form factors of φ, T µν and e kφ can be obtained from this solution, corresponding to dierent choices of H 1, H and k.

33 Outline Introduction to integrability Form factors Finite volume physics Nondiagonal 1-particle form factor Conclusions

34 Finite size eects Finite volume: space is compactied via ϕ (x + L) = ±ϕ (x) Lüscher: nite volume spectrum of H innite volume physical quantities Leading order: Bethe-Yang corrections Polynomial in L 1 due to momentum quantization (BC) Lüscher corrections exponentially small due to vacuum polarization (virtual particles)

35 Finite volume spectrum 5 Luscher Bethe Yang ml 4 3 1

36 Polynomial corrections to spectrum Momentum quantization (Bethe-Yang): Dimensionless volume large compared to particle number Particles move freely most of the time: e ip i L Particle wave functions pick up phase upon interacting e ip i L S ij (θ i θ j ) = ±1, j:j i ( ) m i Lsinhθ i + δ ij θi θ j = πni, j i {n i } determine a state of energy i = 1...N n i Z E(L) = N n=1 m n coshθ n (L) ± depends on periodic/antiperiodic nature of wave function

37 Lüscher corrections to particle mass virtual processes (Lüscher) Leading order for vacuum: lightest (massm) particle-antiparticle pair appears, travels around the world and annihilate dθ coshθ E 0 (L) = m coshθ e ml π Leading order for 1-particle state a: F-term: virtual pair bb from vacuum scattering on a annihilate: F m a = m b dθ π coshθ (S ba(iπ/ + θ) 1)e m b L coshθ µ-term: residue of F-term at pole bound state c: µ m a = θ(ma m b c )µ c m ab ( i)ress ab(θ)e µ c L ab b,c

38 TBA for ground state Start from Euclidean eld theory on the torus, periodic BC, L equivalent ways of quantization: Z (R,L) = Tre LH R (L channel) = Tre RH L (R channel) H R = 1 π H L = 1 π T yy dx T xx dy (x : coordinate along R axis, y: coordinate along L axis) L R L R L channel R channel

39 TBA for ground state L : Tr e LH R e LE 0(R) Tr e RH L LRf (R) e R-channel limit: TD limit of an 1D quantum system with T = R 1 TD limit obtained from the Asymptotic Bethe Ansatz (=Bethe-Yang equations) in the limit L, N, N/L nite. from this, f (R) = 1 R n a=1 dθ π m a coshθ log [1 ] ± e ε a(θ) where ε a (θ) is the pseudo-energy, which satises ε a (θ) = m a R coshθ n b=1 dθ π ϕ ( ab θ θ ) ) log(1 ± e ε (θ b ) d ϕ ab (θ) = i dθ log S ij (θ)

40 Determining the exact spectrum methods: attempt to analytically continue the ground state TBA nd an integrable lattice regularization and investigate its continuum limit For sinh-gordon model, the exact nite-volume spectrum is known (in the form of integral equations) (proof: Teschner) For an N-particle state {n 1,n,...,n N }, the pseudo-energy is given by ε (θ) = mr coshθ + N j=1 Quantization condition: And the energy is ε log S ( θ j + iπ E N (R) = m i ( θ θ j iπ ) ) = i ( n j + 1 ) π, j = 1...N coshθ i m dθ π ϕ ( ab θ θ ) ) log(1 + e ε(θ ) dθ π coshθ log ( 1 + e ε(θ))

41 Finite volume form factors Polynomial corrections (Pozsgay-Takács): take Euclidean two-point function OO and use that O (τ,0)o (0,0) O (τ,0)o (0,0) L O (e µl ) insert innite volume resp. nite volume complete system; dierent integration measures enforce corrections to nite volume matrix elements. The result is ( θ 1,..., θ n ) F O 0 O (0,0) {I 1,...,I n } i1,...,i n,l = n i 1...i n ρi1...i n( θ 1,..., θ n) ) + O (e µl where and ρ i1...i n (θ 1,...,θ n ) = detj (n) (θ 1,...,θ n ) i1...i n J (n) = Q k (θ 1...θ n ), k,l = 1...n kl θ l Q k (θ 1...θ n ) = m ik Lsinhθ k + δ ik i l (θ k θ l ) l k

42 Outline Introduction to integrability Form factors Finite volume physics Nondiagonal 1-particle form factor Conclusions

43 The idea The idea is to calculate the Euclidean torus two-point function in the limit when one of the radii is sent to innity: [Dφ]O (x,t)o (0,0)e S[φ] O (x,t)o L = [Dφ]e S[φ] We then use the nite volume nite temperature duality O (x,t)o L = Θ(t) 0 O (x,t)o 0 L + Θ( t) 0 OO (x,t) 0 L Tr [O ] (0,t)e H x Oe H (L x) = Θ(x) Tr [ ] e + H L Tr [Oe ] H x O (0,t)e H (L+x) Θ( x) Tr [ ] e H L and extract the relevant information from the poles of the analytically continued momentum-space two-point function

44 The idea Γ(ω,q) = 1 L L/ L/ dx dt e i(ωt+qx) O (x,t)o L { δq+pn (L) = 0 O θ 1,...,θ N L E N N (L) iω + δ } q P N (L) E N (L) + iω = π ν O µ e E ν L δ ( P µ P ν + ω ){ 1 ZL µ,ν E µ E ν iq + 1 E µ E ν + iq with Z = Tr [ e H L ] } O(x,t) t O(x,t) O(0,0) x O(0,0) x t

45 The idea Observe that: Finite volume energies can be obtained from positions of poles residues are related to nite volume form factors Expansion in ν corresponds to expansion in Lüscher orders Exact determination of the two-point function is hopeless, but any systematic expansion leads to a systematic expansion (e.g. in Lüscher orders or coupling) of both the energy levels and form factors.

46 Realization for 0 O q L We will focus on the one-particle nite volume pole Γ(ω,q) = F (q) E (q) + iω +...; F (q) = 0 O q L where E (q) is the exact nite volume energy with momentum q m sinhθ 1 We can expand Γ around the large volume Bethe-Yang pole at ω = ie (q) im coshθ 1. At leading Lüscher order, we obtain Γ(ω,q) = πf 1 (q) LE (q) Leading exponential corrections read { E (q) = E (q) 1 + L πf 1 F (q) = i ω ie (q) + L 0 (q) (ω ie (q)) + L 1 (q) ω ie (q) + regular { πf1 LE 1 + (q) } L 0 (q) +... ile (q) 4πF 1 } L 1 (q) +...

47 Realization for 0 O q L Expansion of the partition function: Z = ν ν ν e E ν L = 1 + δ (0) due ml cosh u +... Leading (0 th order) term in the Lüscher expansion of Γ(ω,q): ν = 0, µ one-particle state 1 st order term: ν one-particle state; µ = 0 or µ = β 1,β. The potentially singular part can be written π L sing (ω,q) = due ml cosh u m L [ F ] (δ (0) + δ (u ψ)) 1 + j (u,ψ,q) + (q q) coshψ (coshψ i ˆq) j (u,ψ,q) = β 1 dβ 1 where q = mˆq, ω = m sinhψ dβ u O β 1,β δ (sinhβ 1 + sinhβ sinhu sinhψ) coshβ 1 + coshβ coshu i ˆq

48 Realization for 0 O q L Relation of the matrix element u O β 1,β to the form factor (from LSZ reduction): u O β 1,β = δ (u β 1 )F 1 +S (β 1 β )δ (u β )F 1 +F 3 (u + iπ iε,β 1,β ) The integral of its square is divergent and needs to be regularized. Regularization: δ (x) = i ( 1 π x + iε 1 ) x iε (Shown to be equivalent to nite-volume regularization of the form factor)

49 Regularization and analytic continuation We integrate out the δ function in j (u,ψ,q) then shift integration contour. Poles of the regularized matrix elements are then taken into account by the residue theorem. After this, the ε 0 limit is performed. We then analytically continue j (u,ψ,q) towards ω = ie (q).this is done in two steps: First we extend it to a small region where ω is just above the real axis. During this, a double pole will cross the integration contour. After that, the remaining integral is regular; all singular terms are explicit and analytic in ω.

50 Result This method gives back the rst Lüscher correction to the one-particle energy (in agreement with TBA). For the Lüscher correction of the form factor, we obtain F (q) = π ρ (1) 1 F 1 + dθ F reg 3 ( θ + iπ,θ,θ (0) F reg (θ,β 3 1,β ) = F 3 (θ,β 1,β ) if 1 [1 S (β 1 β )] θ β 1 iπ ) ( ρ 1 = i (1) θ ε (1) θ (1) + iπ 1 1 iπ ) e ml coshθ if 1 S (β 1 β ) Freg t x

51 Outline Introduction to integrability Form factors Finite volume physics Nondiagonal 1-particle form factor Conclusions

52 Conclusions we initiated a programme to calculate systematically both the nite volume energy levels and the nite volume form factors performed two dierent expansions of this nite volume two-point function: nite volume L coupling constant (second order nite-volume Hamiltonian and Lagrangian PT in sinh-gordon theory) These expansions were done explicitely for a moving one-particle state Energy expansion was also compared to excited state TBA result Future: Lüscher corrections for many-particle states

53 Conclusions Thank you for your attention