CDD ambiguity and irrelevant CDD ambiguity and irrelevant deformations of 2D QFTdeformations of 2D QFT

Tateo RobertoBased on:

M. Caselle, D. Fioravanti, F. Gliozzi, JHEP [arXiv:1305.1278] A. Cavaglià, S. Negro, I. Szécsényi, JHEP [arXiv:1608.05534]

Torino

IGST2017, Paris

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Other relevant referencesA.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];

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Main motivationsThe effective string theory for the quark-antiquark potential;

Emergence of singularities in RG/TBA flows with irrelevant perturbations;

Relation between irrelevant perturbations and S-matrix CDD (scalar phase factor) ambiguity;

AdS3/CFT

2 duality;

CFTUV

CFTIR

CFTIR

?

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Exact S-matrix and CDD ambiguity

Consider a relativistic integrable field theory with factorised scattering:

The simplest possibility, consistent with the crossing and unitarity relations is:

Castillejo-Dalitz-Dyson ambiguity:

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The sine-Gordon NLIE

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 complicatedcontours appear for excited states.

and

[A. Klumper, M. T. Batchelor and P. A. Pearce; C. Destri, H. DeVega]

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Zero-momentum case:

which allows to compute the exact form of the t-deformed energy level once its R-dependence is known at t = 0.

replacing

we get

with

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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

,

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we now have an implicit form of the solution of the inviscid Burgers equation with a source term:

again the undeformed energy E(R,0) plays the role of initial condition at t = 0.

source term

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Shock singularitiesTo 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:

for

,

.

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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 domain in a finite time, producing a shock in the physical solution.

Therefore:

and

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Typical t=0 finite-volume spectrum:

R

E(R)

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The energy levels display a pole at R = 0:

where, ceff

= 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.

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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) andt = 0.025 (solid line), for c

eff = −1

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An extra CDD factor couples left (-) with right (+) movers scattering, any NLIE or TBA equation leads to a pair of coupled algebraic equations:

ceff

= c – 24 Δ(primary), obtained by an energy-dependent shift:

which matches the form of the (D=26, cef=24) Nambu Goto spectrum, for generic

CFT, with t=1/(2, where is the string tension.

The CFT case

The total energy:

.

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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.

Best figure from:

A. Athenodoroua and M. TeperarXiv:1602.07634

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Identification of the perturbing operator

Start from the equation

and use the standard relations

then

with

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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:

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Therefore, up to total derivatives:

Putting all this information together:

and

then

and sending we get the desired factorisation result.

which generalises the near CFT perturbative result:

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Classical Action: one bosonic field

Starting from:

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Nambu Goto action in 3D target space:

and

with

in the static gauge:

N free massless boson →Nambu Goto in (N+2) taget space:

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The Lagrangian is very complicated, but we already know that its quantum spectrum is:

Single boson field with generic potential

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Generalisations

Many known scattering models differ only by CDD factors

CDD factors admit an expansion in terms of the spin of the local integrals of motion:

with

Thermally perturbed Ising model (c=1/2) Sinh-Gordon (c=1)

ADE minimal Scattering models Simply-Laced Affine Toda models

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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 )

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Conclusions● Adding CDD factors generates interesting, often UV incomplete, QFTs.

● 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 flux tube.