CDD ambiguity and irrelevant CDD ambiguity and irrelevant deformations of 2D QFT deformations of 2D QFT IGST2017, Paris Tateo Roberto Based on: M. Caselle, D. Fioravanti, F. Gliozzi, JHEP [arXiv:1305.1278] A. Cavaglià, S. Negro, I. Szécsényi, JHEP [arXiv:1608.05534] Torino
Other relevant references A.B. Zamolodchikov, Expectation value of composite field TT in two- dimensional quantum field theory , [hep-th/0401146]; S. Dubovsky, R. Flauger and V. Gorbenko, Solving the simplest theory of quantum gravity , JHEP 2012 (2012); S. Dubovsky, V. Gorbenko and M. Mirbabayi, Natural tuning: towards a proof of concept , [hep-th/1305.6939]; F. A. Smirnov and A. B. Zamolodchikov, On space of integrable quantum field theories, Nucl.Phys. B915 (2017), [hep-th/1608.05499]; L. McGough, M. Mezei, H. Verlinde, Moving the CFT into the bulk with TT , [hep-th/1611.03470]; A. Giveon, N. Itzhaki, D. Kutasov, TT and LST, [hep- th/1701.05576]; 2
Main motivations The effective string theory for the quark-antiquark potential; Emergence of singularities in RG/TBA flows with irrelevant perturbations; ? CFT UV CFT IR CFT IR Relation between irrelevant perturbations and S-matrix CDD (scalar phase factor) ambiguity; 3 AdS 3 /CFT 2 duality;
Exact S-matrix and CDD ambiguity Consider a relativistic integrable field theory with factorised scattering: C astillejo- D alitz- D yson ambiguity: The simplest possibility, consistent with the crossing and unitarity relations is: 4
The sine-Gordon NLIE [A. Klủmper, M. T. Batchelor and P. A. Pearce; C. Destri, H. DeVega] The finite-size properties of the sine-Gordon model are encoded in the single counting function f(θ), solution to the following nonlinear integral equation: For the ground state and , but more more complicated contours appear for excited states. and 5
replacing we get with Zero-momentum case: which allows to compute the exact form of the t-deformed energy level once its 6 R-dependence is known at t = 0.
The latter equation is the implicit form of a solution of a well-known hydrodynamic equation, the invicid Burgers equation: , the deformation parameter t plays the role of “time” variable, and the undeformed energy level serves as initial condition at t = 0. For the general case: with Then and 7
we now have an implicit form of the solution of the inviscid Burgers equation with a source term: source term again the undeformed energy E(R,0) plays the role of initial condition at t = 0. 8
Shock singularities To see the emergence of the wave-breaking phenomenon in the inviscid Burgers equation, let’s consider the P = 0 case. For a generic initial condition, E(R,0): with , the map has in general a number of square-root branch points in the complex R-plane. To find their location, consider the inverse map: . Then, a singularity is characterised by the condition Indeed, around a solution of this equation, we have: 9 for
Therefore: and In typical hydrodynamic applications, the initial profile is smooth on the real-R axis, and for short times all branch points lie in the complex plane. The time evolution however in general brings one of the singularities on the real 10 domain in a finite time, producing a shock in the physical solution.
Typical t=0 finite-volume spectrum: E(R) R 11
The energy levels display a pole at R = 0: where, c eff = c – 24Δ is the “effective central charge” of the UV CFT state. This behaviour implies that, for small times t > 0, the equation has two solutions very close to and correspondingly the solution is singular at In other words, as soon as t > 0, the pole at R = 0 resolves into a pair of branch points. 12
Real part of E(R, t) for t = 0 (dashed line) and t = 0.025 (solid line), for c eff = 1 Real part of E(R, t) for t = 0 (dashed line) and t = 0.025 (solid line), for c eff = −1 13
The CFT case An extra CDD factor couples left (-) with right (+) movers scattering, any NLIE or TBA equation leads to a pair of coupled algebraic equations: c eff = c – 24 Δ(primary), obtained by an energy-dependent shift: . The total energy: which matches the form of the (D=26, c ef =24) Nambu Goto spectrum, for generic 14 CFT, with t=1/(2 σ) , where σ is the string tension.
Best figure from: A. Athenodoroua and M. Teper arXiv:1602.07634 SU(2) at β = 16. Solid curves are continuum NG; dashed curves are NG with a ‘lattice’ dispersion relation. Thick vertical line is deconfining transition; thin vertical line is NG tachyonic transition. 15
Identification of the perturbing operator Start from the equation and use the standard relations then with 16
Use Zamolodchikov's definition for the composite operator: which fulfils the following factorization property: Proof: Consider that together with the conservation laws: keeping z and z’ separated: 17
then and sending we get the desired factorisation result. Putting all this information together: and Therefore, up to total derivatives: which generalises the near CFT perturbative result: 18
Classical Action: one bosonic field Starting from: 19
with Nambu Goto action in 3D target space: in the static gauge: and N free massless boson →Nambu Goto in (N+2) taget space: 20
Single boson field with generic potential The Lagrangian is very complicated, but we already know that its quantum spectrum is: 21
Generalisations Many known scattering models differ only by CDD factors Thermally perturbed Ising model (c=1/2) Sinh-Gordon (c=1) ADE minimal Scattering models Simply-Laced Affine Toda models with CDD factors admit an expansion in terms of the spin of the local integrals of motion: 22
F. Smirnov, A. Zamolodchikov: adding CDD factors leads to well-identified effective field theories. The corresponding perturbing fields are the composite operators: Generalised Burgers equation (i.e. a Generalized Gibbs ensemble ) 23
Conclusions ● Adding CDD factors generates interesting, often UV incomplete, QFT s. ● While the study of generic irrelevant perturbations in QFT remains very problematic, the perturbation appears to be surprisingly easy to treat also in non-integrable models. ● The research on this topic may clarify important aspects concerning the appearance of singularities in efective QFT (Landau pole and tachyon singularity). ● It may be useful in the efective string framework since contributions from higher-dimension integrable composite operators can be added (Virasoro, W-algebra ...). ● It may help to understand the origin of massive modes propagating on the fmux tube. 24
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