NLO BFKL and anomalous dimensions of light-ray operators
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- Ανάκλητος Δαγκλής
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1 NLO BFKL and anomalous dimensions of light-ray operators I. Balitsky JLAB & ODU (Non)Perturbative QFT 13 June 013 (Non)Perturbative QFT 13 June /
2 Outline Light-ray operators. Regge limit in the coordinate space. BFKL representation of 4-point correlation function in N = 4 SYM. DGLAP representation of 4-point correlation function. Anomalous dimensions from DGAP vs BFKL representations. (Non)Perturbative QFT 13 June 013 /
3 Problem Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) (Non)Perturbative QFT 13 June /
4 Problem Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) BFKL gives γ(nα s ) at the non-physical point n 1 γ n = [ A LO BFKL n NLO BFKL + ωbn +... ] ( α s N c ) n ω n 1 πω LO: Jaroszewicz (1980), NLO: Lipatov, Fadin, Camici, Ciafaloni (1998) (Non)Perturbative QFT 13 June /
5 Problem Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) BFKL gives γ(nα s ) at the non-physical point n 1 γ n = [ A LO BFKL n NLO BFKL + ωbn +... ] ( α s N c ) n ω n 1 πω LO: Jaroszewicz (1980), NLO: Lipatov, Fadin, Camici, Ciafaloni (1998) ω = n 1 0 corresponds to F i a ω 1 Fai - some non-local operator. (Non)Perturbative QFT 13 June /
6 Problem Gluon operators of leading twist O g n F i a n Fai Anomalous dimension (in gluodynamics) γ n = π α [ 1 sn c n(n 1) 1 (n + 1)(n + ) + ψ(n + 1) + γ E 11 ] + O(α 1 s ) BFKL gives γ(nα s ) at the non-physical point n 1 γ n = [ A LO BFKL n NLO BFKL + ωbn +... ] ( α s N c ) n ω n 1 πω LO: Jaroszewicz (1980), NLO: Lipatov, Fadin, Camici, Ciafaloni (1998) ω = n 1 0 corresponds to F i a ω 1 Fai - some non-local operator. Q: which one? A: gluon light-ray (LR) operator (Non)Perturbative QFT 13 June /
7 Light-ray operators and parton densities Gluon light-ray (LR) operator of twist F a i(x + + x )[x +, x + ] ab F b i (x + + x ) Forward matrix element - gluon parton density z µ z ν p Fµξ(z)[z, a 0] ab Fν bξ (0) p µ z =0 = (pz) dx B x B D g (x B, µ) cos(pz)x B Evolution equation (in gluodynamics) µ d dµ Fa i(x + + x )[x +, x + ] ab F b i (x + + x ) = x + x + dz + z + x + 0 dz + K(x +, x + ; z +, z + ; α s )F a i(z + + x )[z +, z + ] ab F b i (z + + x ) Forward LR operator F(L +, x ) = dx + F i(l a + + x + + x )[L + + x +, x + ] ab F (x bi + + x ) (Non)Perturbative QFT 13 June /
8 Light-ray operators and parton densities Expansion in ( forward ) local operators F(L +, x ) = n= Evolution equation for F(L +, x ) L n + (n )! Og n(x ), O g n dx + F i a n Fai (x +, x ) µ d dµ F(L +, x ) = γ n (α s ) = u 1 K gg - DGLAP kernel 0 0 du K gg (u, α s )F(uL +, x ) du u n K gg (u, α s ) µ d dµ Og n = γ n (α s )O g n u 1 K gg (u) = α ( [ sn c 1 ūu + π ūu] + 11 ) + 1 δ(ū) + higher orders in α s (Non)Perturbative QFT 13 June /
9 Light-ray operators Conformal LR operator (j = 1 + iν) F µ (L +, x ) = F µ j (x ) = 0 dν π (L +) 3 +iν F µ 1 dl + L 1 j + F µ (L +, x ) +iν(x ) Evolution equation for forward conformal light-ray operators µ d dµ F j(z ) = γ j (α s ) is an analytical continuation of γ n (α s ) 0 du K gg (u, α s )u j F j (z ) (Non)Perturbative QFT 13 June /
10 Conformal four-point amplitude in N = 4 SYM 4-point correlation function (CF) A(x, y, x, y ) = [(x y) (x y ) ] +γ k N c O(x)O(y)O(x )O(y ) O = φ a I φa I - Konishi operator (γ k - anomalous dimension) In a conformal theory the amplitude is a function of two conformal ratios A = F(R, R ) R = (x y) (x y ) (x x ) (y y ), R = (x y) (x y ) (x y ) (x y) At large N c A(x, y, x, y ) = A(g N c ) g N c = λ t Hooft coupling (Non)Perturbative QFT 13 June /
11 Regge limit in the coordinate space Regge limit: x + ρx +, x + ρx +, y ρ y, y ρ y ρ, ρ z_ x +, x _, y _ y + z_ z + x, x _ y _, y _ Full 4-dim conformal group: A = F(R, r) R = (x y) (x y ) (x x ) (y y ) ρ ρ x + x +y y (x x ) (y y ) r = [(x y) (x y ) (x y) (x y ) ] (x x ) (y y ) (x y) (x y ) [(x y ) x +y + x +y (x y) + x +y (x y) + x +y (x y ) ] (x x ) (y y ) x +x + y y (Non)Perturbative QFT 13 June /
12 4-dim conformal group versus SL(, C) Regge limit: x + ρx +, x + ρx +, y ρ y, y ρ y ρ, ρ z_ x +, x _, y _ y + z_ z + x, x _ y _, y _ Regge limit symmetry: -dim conformal group SL(, C) formed from P 1, P, M 1, D, K 1 and K which leave the plane (0, 0, z ) invariant. (Non)Perturbative QFT 13 June /
13 Pomeron in a conformal theory f + (ω) = eiπω 1 sin πω Ω(r, ν) = A(x, y; x, y ) s = i dν f + (ℵ(λ, ν))f(λ, ν)ω(r, ν)r ℵ(λ,ν)/ - signature factor sin νρ sinh ρ, cosh ρ = r - solution of the eqn ( H3 + ν + 1)Ω(r, ν) = 0 The dynamics is described by: ℵ(λ, ν) - pomeron intercept, and F(λ, ν) - pomeron residue. L. Cornalba (007) (Non)Perturbative QFT 13 June /
14 Pomeron in the conformal theory A(x, y; x, y ) s = i dν f + (ℵ(λ, ν))f(λ, ν)ω(r, ν)r ℵ(λ,ν)/ Pomeron intercept: ℵ(ν, λ) = λ [ λ χ(ν) + 4π 16π δ(ν)] + O(λ 3 ) χ(ν) = ψ(1) ψ( 1 + iν) ψ( 1 iν) = S 1( 1 + iν) + c.c. - BFKL intercept, δ(ν) = 3 ζ(3)+π ln + π 3 S 1( 1 +iν)+s 3( 1 +iν)+π S 1 ( 1 +iν) 4S,1( 1 +iν)+c.c. - NLO BFKL intercept S i (x) - harmonic sums Lipatov, Kotikov (000) Pomeron residue F(ν, λ): F 0 (ν) = π sinh πν 4ν cosh 3 πν F(ν, λ) = λ F 0 (ν) + λ 3 F 1 (ν) +... Cornalba, Costa, Penedones (007) F 1 (ν) = see below G. Chirilli and I.B. (009) (Non)Perturbative QFT 13 June /
15 NLO Amplitude in N =4 SYM theory: factorization in rapidity x y x y x y Y A Y B <Y< Y A Y B x y x y x y [(x y) (x y ) ] +γk T{Ô(x)Ô(y)Ô(x )Ô(y )} = d z 1 d z d z 1 d z IF a0 (x, y; z 1, z )[DD] a0,b0 (z 1, z ; z 1, z )IF b0 (x, y ; z 1, z ) a 0 = x+y+ (x y), b 0 = x y (x y ) impact factors do not scale with energy all energy dependence is contained in [DD] a0,b0 (a 0 b 0 = R) (Non)Perturbative QFT 13 June /
16 NLO Amplitude in N =4 SYM theory: factorization in rapidity x y x y x y Y A Y B <Y< Y A Y B x y x y x y [(x y) (x y ) ] +γk T{Ô(x)Ô(y)Ô(x )Ô(y )} = d z 1 d z d z 1 d z IF a0 (x, y; z 1, z )[DD] a0,b0 (z 1, z ; z 1, z )IF b0 (x, y ; z 1, z ) Dipole-dipole scattering χ(γ) C ψ(γ) ψ(1 γ) ( z ) 1 1 [DD] = dν dz +iν ( z ) 1 1 iν 0 z 10 z 0 z D( 1 + iν; λ)rω(ν)/ 10 z 0 D(γ; λ) = Γ( γ)γ(γ 1) Γ(1 + γ)γ( γ) { 1 λ [ χ(γ) 4π γ(1 γ) π 3 ] } + O(λ ) (Non)Perturbative QFT 13 June /
17 NLO Amplitude in N =4 SYM theory: factorization in rapidity x y x y x y Y A Y B <Y< Y A Y B x y x y x y [(x y) (x y ) ] +γk T{Ô(x)Ô(y)Ô(x )Ô(y )} = d z 1 d z d z 1 d z IF a0 (x, y; z 1, z )[DD] a0,b0 (z 1, z ; z 1, z )IF b0 (x, y ; z 1, z ) Result : (G.A. Chirilli and I.B., 010) F(ν) = N c 4π 4 α { s Nc 1 cosh 1 + α [ sn c π πν π π cosh πν ν ] } + O(αs ) (Non)Perturbative QFT 13 June /
18 Light-cone OPE in Cornalba s formula [x1x 34] +γk O(x 1 )O(x )O(x 3 )O(x 4 ) = i dν tanh πν F(ν)Ω(r, ν)r 1 ℵ(ν) f + (ℵ(ν)) ν ℵ(ν) - pomeron intercept, f + (ω) = (e iπω 1)/ sin πω - signature factor Ω(r, ν) = ν sin νρ r π sinh ρ, cosh ρ = In the double limit (Regge) + (x 1 0) R = x 13 x x 1 x 34 x 1 + x + x 3 x 4 x 1 x 34 r x 1 + (x 3 x 14 x 4 x 13 ) x 1+ x + x 3 x 4 x 1 x 34 Ω(r, ν) ν ( r 1 π +iν r 1 iν) i (Non)Perturbative QFT 13 June /
19 Light-cone limit of Cornalba s formula Cornalba s formula as x 1 0 [x 1x 34] +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 3, x )O(x 3, x 3 ) = iα s 8 π L + L 0 tanh πν ( x dudv dν 1 x 34 ūu vv ) 1 +iν ( L + L ūu vv ν cosh πν [x13 v + x 14 v] x 1 x 34 ) ℵ(ν)/f+ = iα s 8 π +i dv dξ f ( ℵ(ξ) ) ξ i + iν ℵ(ξ) ( vv)1 ξ+ cos πξ B ( ξ + ℵ(ξ) ) ( x ) 1 x ξ 34 (ξ 1 ) sin3 πξ [x13 v + x 14 v]+ℵ(ξ) [x13 v + x 14 v] ℵ(ξ) (L + L ) 1+ℵ(ξ) = BFKL representation of the amplitude in the (Regge) + (x1 0) limit (Non)Perturbative QFT 13 June /
20 Supermultiplet of twist- operators SU 4 singlet operators. (Korchemsky et al, 003) S 1n (z) = Og(z) n + n 1 n(n 1) 8 On λ (z) Oφ n 8 (z) S n (z) = Og(z) n 1 n(n + 1) 8 On λ (z) + Oφ n (z) S 3n (z) = Og(z) n n + (n + 1)(n + ) 4 On λ (z) Oφ n 8 (z) Oλ n (x ) = dx + i λ a n 1 σ λ a (x +, x ) Oφ n (z) = dx φa,i + n φ a,i (x +, x ) All operators have the same anomalous dimension γ S 1 n (α s ) γ n (α s ) = α s π N c[ψ(n 1) + C] + O(α s ), γ S n = γ S 1 n+1, γs 3 n = γ S 1 n+ (Non)Perturbative QFT 13 June /
21 Supermultiplet of LR operators Gluino and scalar LR operators Λ(L +, x ) = i dx + [ λ a (L + + x + + x )[x + + x +, x + ] ab σ λ b (x + + x ) + c.c] Φ(L +, x ) = dx + φ a,i (L + + x + + x )[x + + x +, x + ] ab φ b,i (x + + x ) Λ j (x ) = 0 SU 4 singlet LR operators. dl + L j+1 + Λ(L +, x ), Φ j (x ) = 0 dl + L j+1 + Φ(L +, x ) S 1j (x ) = F j (x ) + j 1 8 Λ j(j 1) j(x ) Φ j (x ) 8 S j (x ) = F j (x ) 1 8 Λ j(j + 1) j(x ) + Φ j (x ) S 3j (x ) = F j (x ) j + 4 Λ (j + 1)(j + ) j(x ) Φ j (x ) 8 All operators have the same anomalous dimension γ S 1 j (α s ) γ j (α s ) = α s π N c[ψ(j 1) + C] + O(αs ), γ S j = γ S 1 j+1, γs 3 j = γ S 1 j+ (Non)Perturbative QFT 13 June /
22 Three-point CF of LR operator and two locals Three-point correlation function of local operators (O = φ a I φa I - Konishi operator) (µ z 3) γ K S 1n (z 1 )O(z )O(z 3 ) = c n [1 + ( 1) n ] z 1 z 13 z 3 ( z1 z 1 z 13 z 13 x3 (µ x3 ) γ K dx 1+ S 1n (x 1+, x 1 )O(x, x )O(x 3, x 3 ) ) n ( µ z 3 z 1 z 13 ) 1 γ(n,αs) = πi c n[1 + ( 1) n ] Γ(1 + n + γ n ) ( x3 ) ( 1+ γn x 1 x3 4 Γ (1 + n + 1 γ + x ) 13 1 n γn n) x x 3 x x 3 CF of light-ray operator and two local operators x 3 (µ x 3 ) γ K S 1j (x 1 )O(x, x )O(x 3, x 3 ) = πi c j[1 + e iπj ] Γ(1 + j + γ j ) ( x3 ) γ 1+ j ( x x3 4 Γ (1 + j γ + x ) 13 1 j γj j) x x 3 x x 3 (Non)Perturbative QFT 13 June /
23 Three-point CF of LR operator and two locals Proof: compare conformal Ward identities for dx + S 1n (x +, x ) and S 1j (x ) i[d, dx + S 1n (x +, x )] = vs i[d, Φ(x +, x ) Φ(x +, x )] = i[d, Φ j (x )] = [ j (x x 0 ) i ( dx + x+ + n + + (x x 0 ) i x + [ 4 + x + + x + x + x i ] Φj (x ) x + x i + (x x 0 ) i ) S1n = (n + 1) dx + S 1n (x +, x ) x i ] Φ(x +, x ) Φ(x +, x ) Similarly i[k +, dx + Φ(x ) n Φ(x )] = 4 [ n + (x x 0 ) i x i ] dx + x + Φ(x ) n Φ(x ) vs i[k +, dx + ds + (s + ) j+1 Φ(x + + s x ) Φ(x + s + + x )] = 4 [ j + (x x 0 ) ] i dx + x + ds + (s + ) j+1 Φ(x + + s + + x )Φ(x + s + + x ) x i 0 (Non)Perturbative QFT 13 June /
24 Light-cone OPE CF of light-ray operator and two local operators in forward kinematics x3 (µ x3 ) γ K dx 3 S µ j (x 1 )O(L + x 3, x )O(x 3, x 3 ) = 0 du c(j, α s)[1 + e iπj ]L j [ (ūu)j ( ūuµ x ) 1 3 γ(j,αs) x 1 u + x13 ū] 1+j [x1 u + x13 ū] (Non)Perturbative QFT 13 June /
25 Light-cone OPE CF of light-ray operator and two local operators in forward kinematics x3 (µ x3 ) γ K dx 3 S µ j (x 1 )O(L + x 3, x )O(x 3, x 3 ) = 0 du c(j, α s)[1 + e iπj ]L j [ (ūu)j ( ūuµ x ) 1 3 γ(j,αs) x 1 u + x13 ū] 1+j [x1 u + x13 ū] Limit x 3 0 (µ ) γ K dx 3 S j + (x 1 )O(L + x 3, x )O(x 3, x 3 ) = Γ( 1 + j + γ ) c(j, α s )[1 + e iπj ]L j Γ( + j + γ) (x1 ) 1+j (µ x1 (µ )γ(j,αs) ) γ(j,αs) (Non)Perturbative QFT 13 June /
26 Light-cone OPE Light-ray operators in (+) and (-) directions: Φ + (L +, x ) = dx + φ a,i (L + + x + + x )[x + + x +, x + ] ab φ b,i (x + + x ) Φ + j (x ) = dl + L j+1 + Φ(L +, x ) 0 Φ (L, x ) = dx + φ a,i (L + x + x )[x + x, x ] ab + φ b,i (x + x ) Φ j (x ) = 0 dl L j+1 + Φ(L, x ) and similarly for Λ j s and G j s CF of two LRs : S + j= 3 +iν(x 1 )S j = 3 +iν (x 3 ) = δ(ν ν )a(j, α s ) (x 13 ) j+1 (x 13 µ ) γ(j,αs) (Non)Perturbative QFT 13 June /
27 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) (Non)Perturbative QFT 13 June /
28 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) DGLAP result (at x 1 0) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) (Non)Perturbative QFT 13 June /
29 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) DGLAP result (at x 1 0) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = +i dj C(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) 1 γ(j,αs) S µ j (x )O(L + x 4, x 3 )O(x 4, x 4 ) (Non)Perturbative QFT 13 June /
30 DGLAP represenation of 4-point CFs Light-cone OPE x1 (µ x1 ) γk dx 1+ O(L + + x +, x 1 )O(x +, x ) = +i dj c(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) γ(j,αs) S j + (x 1 ) DGLAP result (at x 1 0) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = = +i dj C(j, α s )[1 + e iπj ]L+(µ j x 1 i 1 ) 1 +i dj C(j, α s ) 1 i 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j γ(j,αs) S µ j (x )O(L + x 4, x 3 )O(x 4, x 4 ) x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) (Non)Perturbative QFT 13 June /
31 DGLAP vs BFKL for 4-point CFs DGLAP result (leading twist) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = +i dj C(j, α s ) 1 i 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) (Non)Perturbative QFT 13 June 013 /
32 DGLAP vs BFKL for 4-point CFs DGLAP result (leading twist) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = +i dj C(j, α s ) 1 i 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) BFKL result (ℵ(ξ, α s ) = αsnc π χ(ξ) pomeron intercept) [x1x 34] +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 3, x )O(x 3, x 3 ) = iα s 8 π +i dv dξ f ( ℵ(ξ) ) ξ i + iν ℵ(ξ) ( vv)1 ξ+ cos πξ B ( ξ + ℵ(ξ) ) ( x ) 1 x ξ 34 (ξ 1 ) sin3 πξ [x13 v + x 14 v]+ℵ(ξ) [x13 v + x 14 v] ℵ(ξ) (L + L ) 1+ℵ(ξ) (Non)Perturbative QFT 13 June 013 /
33 DGLAP vs BFKL for 4-point CFs DGLAP result (leading twist) µ 4 (µ 4 x 1x 34) +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 4, x 3 )O(x 4, x 4 ) = +i dj C(j, α s ) 1 i 0 dv [1 + eiπj ](L + L ) [ j ( x 1 v + x13 v] 1+j x 1 x 34 [x 13 v + x 14 v] ) 1 γ(j,αs) ( vv) j+ 1 γ(j,αs) BFKL result (ℵ(ξ, α s ) = αsnc π χ(ξ) pomeron intercept) [x1x 34] +γk dx + dx 3 O(L + + x +, x 1 )O(x +, x )O(L + x 3, x )O(x 3, x 3 ) = iα s 8 π +i dv dξ f ( ℵ(ξ) ) ξ i + iν ℵ(ξ) ( vv)1 ξ+ cos πξ B ( ξ + ℵ(ξ) ) ( x ) 1 x ξ 34 (ξ 1 ) sin3 πξ [x13 v + x 14 v]+ℵ(ξ) [x13 v + x 14 v] We compare these results around ξ j 1 0 ℵ(ξ) ℵ(ξ)ℵ (ξ) α sn c πξ +ζ(3)α s ξ 1 + ℵ(ξ, α s ) = j and γ(j, α s ) = ξ ℵ(ξ) +... γ j = α sn c π(j 1) ℵ(ξ) (L + L ) 1+ℵ(ξ) s N c α + [0 + ζ(3)(j 1)]( π(j 1) )) (Non)Perturbative QFT 13 June 013 /
34 Result s N c γ j = α sn c α + [0 + ζ(3)(j 1)]( π(j 1) π(j 1) )) is an anomalous dimension of light-ray operator F j F(x ) (Non)Perturbative QFT 13 June /
35 Conclusions and Outlook 1 Conclusions: NLO BFKL gives anomalous dimensions of light-ray operators F +i ω 1 F+ i as ω 0 (in all orders in α s ) (Non)Perturbative QFT 13 June 013 /
36 Conclusions and Outlook 1 Conclusions: NLO BFKL gives anomalous dimensions of light-ray operators F +i ω 1 F+ i as ω 0 (in all orders in α s ) Outlook In QCD 3-point CF of F ω1 1 F(x 1 ) F ω 1 F(x ) F ω3 1 F(x 3 ) as ω 1, ω, ω 3 0 (joint project with V. Kazakov and E. Sobko): (Non)Perturbative QFT 13 June 013 /
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![MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)](/thumbs/81/84712331.jpg "MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)")
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Α Ρ Ι Θ Μ Ο Σ : 6.913
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