We derive the GKP-Witten relation in terms of correlation functions by symmetry without referring to a Lagrangian or the large <italic>N</italic> expansion. By constructing bulk operators from boundary operators in conformal field theory (CFT) by the conformal smearing, we first determine bulk-boundary 2-pt functions for an arbitrary spin using both conformal and bulk symmetries, then evaluate their small <italic>z</italic> behaviors, where <italic>z</italic> is the <math altimg='si1.svg'><mo stretchy='false'>(</mo><mi>d</mi><mo linebreak='badbreak' linebreakstyle='after'>+</mo><mn>1</mn><mo stretchy='false'>)</mo></math>-th coordinate in the bulk. Next, we explicitly determine small <italic>z</italic> behaviors of bulk-boundary-boundary 3-pt functions also by the symmetries, while small <italic>z</italic> behaviors of correlation functions among one bulk and <italic>n</italic> boundary operators with <math altimg='si2.svg'><mi>n</mi><mo>&#8805;</mo><mn>3</mn></math> are fixed by the operator product expansion (OPE). Combining all results, we construct the GKP-Witten relation in terms of these correlation functions at all orders in an external source <italic>J</italic>. We compare our non-Lagrangian approach with the standard approach employing the bulk action. Our results indicate that the GKP-Witten relation holds not only for holographic CFTs but also for generic CFTs as long as certain conditions are satisfied.

Japan

Sinya Aoki

Hungary

Janos Balog

Kengo Shimada

Derivation of the GKP-Witten relation by symmetry without Lagrangian

We provide an improved definition of new conserved quantities derived from the energy–momentum tensor in curved spacetime by introducing an additional scalar function. We find that the conserved current and the associated conserved charge become geometric under a certain initial condition of the scalar function, and show that such a conserved geometric current generally exists in curved spacetime. Furthermore, we demonstrate that the geometric conserved current agrees with the entropy current in an effective theory of a perfect fluid, thus the conserved charge is the total entropy of the system. While the geometric charge can be regarded as the entropy for a nondissipative fluid, its physical meaning should be investigated for more general cases.

Yoshimasa Hidaka

South Korea

Kiyoharu Kawana

Geometric conservation in curved spacetime and entropy

We investigate a structure of a 4-dimensional bulk space constructed from the O(N) invariant critical φ 4 model in 3-dimension using the conformal smearing. We calculate a bulk metric corresponding to the information metric and the bulk-to-boundary propagator for a composite scalar field φ 2 in the large N expansion. We show that the bulk metric describes an asymptotic AdS space at both UV (near boundary) and IR (deep in the bulk) limits, which correspond to the asymptotic free UV fixed point and the Wilson-Fisher IR fixed point of the 3-dimensional φ 4 model, respectively. The bulk-to-boundary scalar propagator, on the other hand, encodes   <math> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </math>  $$ {\Delta }_{\varphi^2} $$  (the conformal dimension of φ 2) into its z (a coordinate in the extra direction of the AdS space) dependence. Namely it correctly reproduces not only   <math> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </math>  $$ {\Delta }_{\varphi^2} $$  = 1 at UV fixed point but also   <math> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </math>  $$ {\Delta }_{\varphi^2} $$  = 2 at the IR fixed point for the boundary theory. Moreover, we confirm consistency with the GKP-Witten relation in the interacting theory that the coefficient of the   <math> <msup> <mi>z</mi> <msub> <mtext>∆</mtext> <msup> <mi>φ</mi> <mn>2</mn> </msup> </msub> </msup> </math>  $$ {z}^{\Delta _{\varphi^2}} $$  term in z → 0 limit agrees exactly with the two-point function of φ 2 including an effect of the φ 4 interaction.

AdS/CFT correspondence for the O(N) invariant critical φ 4 model in 3-dimensions by the conformal smearing

We construct a bulk spacetime from a boundary conformal field theory, <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> free scalar model, at finite temperature using a smearing technique, called a conformal flow. The bulk metric is constructed from the thermal two-point function of the smeared boundary elementary field in such a way that it can be interpreted as an information metric associated with the boundary thermofield double state. Near the boundary (UV region), an asymptotically anti–de Sitter (AdS) spacetime is obtained with a leading order perturbation of scalar mode. Based on the falloff behavior of the perturbations and the <math><mi>O</mi><mo>(</mo><mi>N</mi><mo>)</mo></math> symmetry in the conformal field theory, we argue that the corresponding bulk theory is a modified gravity with scalar mode such as <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></math> gravity rather than Einstein’s general relativity coupled minimally to matter fields. Moving to the Einstein frame, we show that the metric is asymptotically the same as the AdS black brane solution. On the other hand, deep in the bulk (IR region), the spacetime turns out to be conformally equivalent to the near horizon limit of AdS extremal black brane, though it is no longer a solution of <math><mi>f</mi><mo>(</mo><mi>R</mi><mo>)</mo></math> gravity, and hence more general classes of modified gravity need to be considered.

János Balog

Bulk modified gravity from a thermal CFT by the conformal flow

The doubly charmed tetraquark <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math> recently discovered by the LHCb Collaboration is studied on the basis of (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor lattice QCD simulations of the <math><msup><mi>D</mi><mo>*</mo></msup><mi>D</mi></math> system with nearly physical pion mass <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>146</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. The interaction of <math><msup><mi>D</mi><mo>*</mo></msup><mi>D</mi></math> in the isoscalar and <math><mi>S</mi></math>-wave channel, derived from the hadronic spacetime correlation by the HAL QCD method, is attractive for all distances and leads to a near-threshold virtual state with a pole position <math><mrow><msub><mrow><mi>E</mi></mrow><mrow><mi>pole</mi></mrow></msub><mo>=</mo><mo>−</mo><mn>59</mn><msubsup><mrow><mo>(</mo></mrow><mrow><mo>−</mo><mn>99</mn></mrow><mrow><mo>+</mo><mn>53</mn></mrow></msubsup><mo>)</mo><msubsup><mrow><mo>(</mo></mrow><mrow><mo>−</mo><mn>67</mn></mrow><mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>keV</mi></mrow></math> and a large scattering length <math><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>0.05</mn><mo>(</mo><mn>5</mn><mo>)</mo><msubsup><mrow><mo>(</mo></mrow><mrow><mo>−</mo><mn>2</mn></mrow><mrow><mo>+</mo><mn>2</mn></mrow></msubsup><mo>)</mo><mtext> </mtext><mtext> </mtext><msup><mrow><mi>fm</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math>. The virtual state is shown to evolve into a loosely bound state as <math><msub><mi>m</mi><mi>π</mi></msub></math> decreases to its physical value by using a potential modified to <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>=</mo><mn>135</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> based on the pion-exchange interaction. Such a potential is found to give a semiquantitative description of the LHCb data on the <math><msup><mi>D</mi><mn>0</mn></msup><msup><mi>D</mi><mn>0</mn></msup><msup><mi>π</mi><mo>+</mo></msup></math> mass spectrum. Future study is necessary to perform physical-point simulations with the isospin-breaking and open three-body-channel effects taken into account.

China

Yan Lyu

Takumi Doi

Tetsuo Hatsuda

Yoichi Ikeda

Jie Meng

Doubly Charmed Tetraquark <math><msubsup><mi>T</mi><mrow><mi>c</mi><mi>c</mi></mrow><mo>+</mo></msubsup></math> from Lattice QCD near Physical Point

We study a doubly bottomed tetraquark state <math><mo>(</mo><mi>b</mi><mi>b</mi><mover><mi>u</mi><mo>¯</mo></mover><mover><mi>d</mi><mo>¯</mo></mover><mo>)</mo></math> with quantum number <math><mrow><mi>I</mi><mo>(</mo><msup><mrow><mi>J</mi></mrow><mrow><mi>P</mi></mrow></msup><mo>)</mo><mo>=</mo><mn>0</mn><mo>(</mo><msup><mrow><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msup><mo>)</mo></mrow></math>, denoted by <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math>, in lattice QCD with the nonrelativistic QCD (NRQCD) quark action for <math><mi>b</mi></math> quarks. Employing (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor gauge configurations at <math><mi>a</mi><mo>≈</mo><mn>0.09</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></math> on <math><msup><mn>32</mn><mn>3</mn></msup><mo>×</mo><mn>64</mn></math> lattices, we have extracted the coupled-channel potential between <math><mover><mi>B</mi><mo>¯</mo></mover><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> and <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> in the HAL QCD method, which predicts an existence of a bound <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> below the <math><mover><mi>B</mi><mo>¯</mo></mover><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> threshold. By extrapolating results at <math><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>410</mn></math>, 570, 700 MeV to the physical pion mass <math><msub><mi>m</mi><mi>π</mi></msub><mo>≈</mo><mn>140</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math>, we obtain a binding energy with its statistical error as <math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mtext>binding</mtext></mrow><mrow><mo>(</mo><mi>single</mi><mo>)</mo></mrow></msubsup><mo>=</mo><mn>155</mn><mo>(</mo><mn>17</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math> and <math><mrow><msubsup><mrow><mi>E</mi></mrow><mrow><mtext>binding</mtext></mrow><mrow><mo>(</mo><mi>coupled</mi><mo>)</mo></mrow></msubsup><mo>=</mo><mn>83</mn><mo>(</mo><mn>10</mn><mo>)</mo><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>, where “coupled” means that effects due to virtual <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> states are included through the coupled channel potential, while only a potential for a single <math><mover><mi>B</mi><mo>¯</mo></mover><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> channel is used in the analysis for “single.” A comparison shows that the effect from virtual <math><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup><msup><mover><mi>B</mi><mo>¯</mo></mover><mo>*</mo></msup></math> states is quite sizable to the binding energy of <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math>. We estimate systematic errors to be <math><mo>±</mo><mn>20</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></math> at most, which are mainly caused by the NRQCD approximation for <math><mi>b</mi></math> quarks.

Takafumi Aoki

Takashi Inoue

Lattice study on a tetraquark state <math><msub><mi>T</mi><mrow><mi>b</mi><mi>b</mi></mrow></msub></math> in the HAL QCD method

In this study, we extend the HAL QCD method to a case where a total momentum of a two-particle system is nonzero and apply it to the <math><mrow><mi>I</mi><mo>=</mo><mn>2</mn></mrow></math> S-wave <math><mi>π</mi><mi>π</mi></math> scattering in order to confirm its validity. We derive a fundamental relation of an energy-independent nonlocal potential defined in the center of mass frame with Nambu-Bethe-Salpeter wave functions in a laboratory frame. Based on the relation, we propose the time-dependent method to extract potentials, often used in practice for the HALQCD method in the center of mass frame. For numerical simulations in the <math><mi>I</mi><mo>=</mo><mn>2</mn></math> <math><mi>π</mi><mi>π</mi></math> system, we employ (<math><mrow><mn>2</mn><mo>+</mo><mn>1</mn></mrow></math>)-flavor gauge configurations on a <math><mrow><msup><mrow><mn>32</mn></mrow><mrow><mn>3</mn></mrow></msup><mo>×</mo><mn>64</mn></mrow></math> lattice at the lattice spacing <math><mrow><mi>a</mi><mo>≈</mo><mn>0.0907</mn><mtext> </mtext><mtext> </mtext><mi>fm</mi></mrow></math> and <math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>π</mi></mrow></msub><mo>≈</mo><mn>700</mn><mtext> </mtext><mtext> </mtext><mi>MeV</mi></mrow></math>. Both effective leading order (LO) potentials and corresponding phase shifts obtained in laboratory frames agree with those obtained in the center-of-mass frame by the conventional HAL QCD method within somewhat larger statistical errors. In addition, we observe a consistency in scattering phase shifts between ours and results by the finite-volume method as well. The HAL QCD method with nonzero total momenta, established in this study, brings more flexibility to the HAL QCD method, which enables us to handle systems having the same quantum numbers with a vacuum or to access energy regions prohibited in the center of mass frame.

Yutaro Akahoshi

Interaction potentials for two-particle states with nonzero total momenta in lattice QCD

We re-analyse the bulk reconstruction for a scalar field in Lorentzian AdS spacetime, both for the case of even and odd dimensions, for an extended range of conformal dimensions where the original HKLL reconstruction has to be modified. We also discuss the use of space-like Green’s functions in the bulk reconstruction. We demonstrate that in the extended range also the singular part of the Green’s function, omitted in the original papers, has be included. The results are particularly simple and physically interesting for integer conformal dimensions below the range considered in the original HKLL papers.

Extension of the HKLL bulk reconstruction for small ∆

Abstract We study decuplet baryons from meson–baryon interactions in lattice quantum chromodynamics (QCD), in particular, Δ and Ω baryons from P-wave I = 3/2 Nπ and I = 0  interactions, respectively. Interaction potentials are calculated in the HAL QCD method using 3-quark-type source operators at m$_{π}$ ≈ 410 MeV and m$_{K}$ ≈ 635 MeV, where Δ as well as Ω baryons are stable. We use the conventional stochastic estimate of all-to-all propagators combined with the all-mode averaging to reduce statistical fluctuations. We have found that the  system has a weaker attraction than the Nπ system while the binding energy from the threshold is larger for Ω than Δ. This suggests that an inequality  comes mainly from a smaller spatial size of a  bound state due to a larger reduced mass, rather than its interaction. Root-mean-square distances of bound states in both systems are small, indicating that Δ and Ω are tightly bound states and thus can be regarded qualitatively as composite states of three quarks. Results of binding energies agree with those obtained from temporal two-point functions within large systematic errors, which arise dominantly from the lattice artifact at short distances.

Kotaro Murakami

Kenji Sasaki(HAL QCDCollaboration)

Lattice quantum chromodynamics (QCD) studies on decuplet baryons as meson–baryon bound states in the HAL QCD method

Abstract We develop a framework for the reconstruction of the bulk theory dual to conformal field theory without any assumption by means of a flow equation. To this end we investigate a minimal extension of the free-flow equation and find that at a special parametrization the conformal transformation for a normalized smeared operator exactly becomes the isometry of anti-de Sitter space (AdS). By employing this special flow equation for O(N) vector models, we explicitly show that the AdS geometry as well as the scalar field satisfying the GKP–Witten relation concurrently emerge in this framework.