Continuum limit of fishnet graphs and AdS sigma model

Continuum limit of fishnet graphs and AdS sigma model Benjamin Basso LPTENS 15th Workshop on Non-Perturbative QCD, IAP, Paris, June 2018 based on work done in collaboration with De-liang Zhong

Motivation Understand dynamics of planar graphs and its relation to sigma models [ t Hooft] Best possible starting point: N=4 SYM [Maldacena 97] String dual is believed to be known Theory is believed to be integrable (meaning we have methods for re-summing planar graphs) Use solution to gain knowledge about other models by deforming / twisting the theory Partial re-summation of planar graphs (Reduce complexity, but maintain as many important properties as possible: conformal symmetry, integrability, etc.)

Fishnet theory Baby version of N = 4 SYM A theory for matrix scalar fields with quartic coupling L fishnet = Ntr apple [Gurdogan,Kazakov 15] [Caetano,Gurdogan,Kazakov 16] @ µ 1 @ µ 1 + @ µ 2 @ µ 2 +(4 g) 2 1 2 1 2 It can be obtained by twisting N=4 SYM theory, so-called sending the deformation parameter to i-infinity while taking YM coupling to zero deformation, [Frolov 05] [Lunin,Maldacena 05] [Grabner,Gromov,Kazakov,Korchemsky 17] [Sieg,Wilhelm 16] 1. Gluons and gauginos decouple 2. Gauge group becomes a flavour group 3. Conformal symmetry is preserved for any coupling (at least in planar limit and for fine-tuned double-trace couplings) 4. Integrability is retained

Fishnet theory Baby version of N = 4 SYM A theory for matrix scalar fields with quartic coupling L fishnet = Ntr apple [Gurdogan,Kazakov 15] [Caetano,Gurdogan,Kazakov 16] @ µ 1 @ µ 1 + @ µ 2 @ µ 2 +(4 g) 2 1 2 1 2 All planar graphs locally look the same Integrability is not mysterious here and links directly to basic property of the coupling in d=4 Win: simplicity, many fewer graphs Lose: unitarity 4 bulk graph [Zamolodchikov 80] [Isaev 03] [Gromov,Kazakov,Korchemsky,Negro,Sizov 17] [Chicherin,Kazakov,Loebbert,Muller,Zhong 16]

Continuum limit & string? What about duality to string in AdS? Extremal twisting procedure forces the YM coupling to be small String in highly curved AdS? Question: what is the continuum limit of the fishnet graphs? Important observation concerning large order behaviour [Zamolodchikov 80] Zamolodchikov s thermodynamical scaling log Z L,T = L T log g 2 cr g cr = (3/4) p (5/4) =0.7... Determine a critical coupling where graphs become dense

Plan Study continuum limit using integrable methods borrowed from N=4 SYM Probe: scaling dimension of BMN vacuum operator O =tr L 1 Qualitative picture of as function of coupling when L!1 g 2 L 2 1/4 g 2 cr g 2 Main claim : continuum limit is given by the 2d AdS5 sigma model BMN operator maps to tachyon tr L 1 $ V e i t Relation to sigma model energy log g 2L = log g 2L cr + E 2d (,L)

Graphs versus integrability Computation of anomalous dimension BMN vacuum tr L 1 (not protected) = L + [Gurdogan,Kazakov 15] [Caetano,Gurdogan,Kazakov 16] Graphs: loop corrections come from the wheel diagrams Z = 1 + + +... wave-function renormalization 1 wheel 2 wheels g 2L g 4L Depends on cut off R log cut off Anomalous dimension controls the logarithmic dependence on cut off log Z R

Graphs versus integrability Computation of anomalous dimension BMN vacuum tr L 1 (not protected) = L + [Gurdogan,Kazakov 15] [Caetano,Gurdogan,Kazakov 16] Integrability: Free energy of a gas of magnons at temperature 1/L Partition function on R S L R Z L,R = X T >0 g 2LT L Each magnon carries a rapidity u, which is a momentum along euclidean time direction, and a discrete label a, which enumerates harmonics on 3-sphere log Z L,R = LR +... Weak coupling expansion: magnon = wheel u 1... u T Integrability: direct quantum mechanical interpretation of the graphs

Thermodynamical Bethe Ansatz Factorized scattering allows us to obtain free energy from TBA eqs = L 2 X Z du 2 Y a(u) a>1 X Z du 2 Y2 a(u) 2 X Z dudv (2 ) 2 Y a(u)k a,b (u, v)y b (v)+o(y 3 ) a>1 a,b>1 Boltzmann weight: Y a (u) =a 2 e Lh L a(u) 1 Mechanical energy: a (u) = log (u 2 + a 2 /4) Scattering kernel: K a,b (u, v) = @ i@u log S a,b(u, v) + matrix part Coupling constant is just a fugacity for the magnons (wheels) g 2 = e h with h the chemical potential

Thermodynamical limit Thermodynamical limit L!1 Interesting when chemical potential gets bigger than mass of lightest magnon that is for h>log (u = 0) = log 1/4 g>1/2 h A Fermi sea forms All states below the Fermi rapidity are filled Increasing coupling amounts to increasing B occupied states Comment: only the s-wave (lightest) magnons condense (higher Lorentz harmonics decouple) Fermi rapidity

Linear integral equation In the thermodynamical regime the TBA eqs linearize Integral equation for the rapidity distribution of energy levels (u) =C BC: (u = ±B) =0 (u)+ Z B Effective chemical potential: C = log g 2 B du 2 K(u v) (v) Kernel: K(u) =2 (1 + iu)+2 (1 iu)+ 2 1+u 2 Z B B du 2 k(u) (u) Scaling dimension: /L =1 Z B B du (u)

Critical regime Small B : dilute gas, density of magnons is small j = df/dh 0 " = f + hj 1 Critical regime relates to large magnon density B!1 " j log g 2 cr f 0 All energy levels are filled Equation is solved in Fourier space cr(u) =C cr One verifies: (u)+ Z 1 1 (i) Scaling dimension vanishes dv 2 K(u v) cr(v) ) /L =0 cr = log with (ii) Chemical potential (coupling) approaches predicted value p 2 cosh +1 p 2 cosh 1 = u/2 cr e ) C cr =0 ) g cr = (3/4)/ p (5/4)

Numerical check Integral equation can be solved numerically 1.0 DêL 0.8 0.6 0.4 0.2 0.55 0.60 0.65 0.70 0.75 g g thr =0.5 g cr =0.76276 thermodynamical window

Near-critical regime Particle-hole transformation: Fermi sea of magnons dual Fermi sea Equation for dual excitations: Dualize kernel by means of K = K 1 K = K K K... Act on both sides of the equation with 1 K

Dual equation: No chemical potential but extra BC: Dual equations Dual energy formula: log g 2 = log g 2 cr + 1) Dual kernel: K( ) = ( ) =E( )+ @ i@ log 2) Dual dispersion relation: Z 2 >B 2 d 0 2 K( 0 ) ( 0 ) Z 2 >B 2 i (1 + 2 ) ( 1 2 d 2 P 0 ( ) ( ) ( ) 2 log = /L = charge density E( ) = i 2 ) ( 3 4 + i (1 i 2 ) ( 1 2 + i 2 ) ( 3 4 cr ( ) = log i 2 ) 2 ) ( 1 4 i 2 ) ( 1 4 + i 2 ) applep 2 cosh +1 p 2 cosh 1 m 2 e P ( ) = ie( + i 2 )=i log applep 2 sinh + i p 2 sinh i m 2 e where m =4 p 2

Interpretation What is the dual system describing? 1) Kernel: K = i@ log S O(6) Excitations scatter as particles in 2d O(6) non-linear sigma model [Zamolodchikov&Zamolodchikov] 2) Dispersion relation: Energy sinh 2 ( 1 2 E)=sin2 ( 1 2 P ) 1.5 Gapless excitations (unlike O(6) model) Besides, E decreases when increases Support is non-compact and density is not normalizable (cannot count excitations) 1.0 0.5-3 -2-1 1 2 3 No mass gap + continuous spectrum Suggest: sigma model with non-compact target space Proposal: integrable lattice completion of sigma model AdS 5 Momentum

2d non-linear sigma model Sigma model with curved target space Weak coupling (large AdS radius) Beta function related to Ricci scalar 1 L = 2e 2 GAB @ a X A @ a X B e 2 1 For : AdS d+1 One loop running: Alternatively: Y 2 0 + Y 2? Y 2 d+1 = 1 µ @ @µ e2 (µ) = d 2 e4 (µ)+... 1 e 2 (µ) = d 2 log ( /µ) 1. Theory is weakly coupled in IR 2. There is no mass gap 3. There is no isolated vacuum

Perturbative tachyon Consider sigma model on cylinder of radius L Interested in 2d ground state energy : tachyon (best candidate for extremum of energy at given charge = global time energy) V e i t Classically, it corresponds to solution Y 0 ± iy d+1 = e ±ih Y? =0 Charge density /L = 1 e 2 (Y 0 Ẏ d+1 Y d+1 Ẏ 0 )= H/e 2 Energy density E/L = 1 2e 2 (Ẏ 0 Ẏ 0 + Ẏ d+1 Ẏ d+1 )= H 2 /(2e 2 ) Classical result : (c-o-m energy) E = e2 2 Same as in O(d+2) model if not for the sign of the coupling e 2 $ e 2 2L

Back to numerics Quadratic Casimir scaling near critical point fits numerics DêL 1.0 0.8 0.6 log g 2 /g 2 cr = e2 2 2L 2 0.4 0.2 0.55 0.60 0.65 0.70 0.75 g g thr =0.5 g cr =0.76276 thermodynamical window

Back to equation Compact case ( ) =m cosh + Z B B d 0 2 K O(6)( 0 ) ( 0 ) Eq. describes gas of massive excitations at finite particle density Systematic expansion is known in the large density regime B 1 [Volin 09] Non-compact case ( ) = m 2 e + Z ( 0 ) 2 >B 2 d 0 2 K O(6)( 0 ) ( 0 ) Eq. describes gas of massless excitations at finite charge density Despite being different, two problems are identical, to any order in if we flip the sign of Fermi rapidity B! B 1/B

All-order result Fermi parameter B 1/e 2 log (L/ ) 1 e 2 = coupling of sigma model at energy scale = /L Flipping the sign of the coupling e 2 $ e 2 Same as changing the sign of curvature non-compact L $ compact L Same as changing direction of RG flow Owing to this connection to the compact sigma model, the dual integral equation describes the tachyon of the AdS model to all orders in perturbation theory

TBA equations Original magnon TBA (massive + chemical potential) log Y 1 = L log g 2 L + K log (1 + Y 1 )+... coupling = input output : = L 1X a=1 (all blacks nodes contribute to energy) Z du log (1 + Y a) massive TBA Dual TBA (massless + no chemical potential) log Y 1 = LE K O(6) log (1 + 1/Y 1 )+... = input (tachyon rep) massless TBA (only 1 momentum carrier) coupling = output (sigma model energy) log g 2L /g 2L cr = Z d 2 P 0 ( ) log (1 + 1/Y 1 )

Central charge Analysis in CFT limit ( 1/L effect aka Casimir energy) Split into left and right movers = scale-invariant (kink) solution [Zamolodchikov 90s] [Klassen-Melzer 90s] Kink is characterized by its asymptotic values on far left and far right Standard analysis gives TBA central charge c = c 0 c 1 where c? = X i L( Y? i 1+Y? i with Rogers dilogarithm L(x) = 6 2 (Li 2(x)+ 1 2 ) log x log (1 x)) Stationary solutions to Y-system are known c 0 =7 c 1 =2 symmetric phase O(6) broken phase O(4) Central charge is as it should be c =5

CFT analysis Central charge agrees with perturbative analysis in the AdS sigma model = count of the number of Goldstone bosons Close to IR fixed point, i.e. large L, the 2d CFT gives information about the behaviour of the energy level Operator-state correspondence: energy maps to 2d anomalous dimension of vertex operator V e i t E 2d = c eff (L) 6L e 2 ( d) 2L + O(e 4 ) Effective central charge at distance L: c eff (L) =5+O(e 2 ) Running coupling at distance L: e 2 2 d log L

Conclusion Conjecture : 4d planar fishnet graphs define an integrable lattice regularization of the 2d AdS5 sigma model When all characteristic fishnet length scales are large, the sigma model description is weakly coupled This is applicable at large L and small quantum numbers

Outlook String worldsheet or not? We found a marginality condition of sort 0=Lµ + E 2d (L) with cosmological constant µ = log g 2 cr/g 2 Non-critical strings with a tunable intercept exist in flat space, at least classically Could it be that we are dealing with an AdS version of it? What seems to be clear is that the on-shell condition comes from the sum over the wheels X (g/g cr ) 2LT e TE 2d(L, ) = T >0 1 1 (g/g cr ) 2L e E 2d(L, ) with T acting as a discrete proper time (Schwinger parameter)

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