Collective coordinates are frequently employed in path integrals to manage divergences caused by fluctuations around saddle points that align with classical symmetries. These coordinates parametrize a manifold of zero modes and more broadly provide judicious coordinates on the space of fields. However, changing from local coordinates around a saddle point to more global collective coordinates is remarkably subtle. The main complication is that the mapping from local coordinates to collective coordinates is generically multivalued. Consequently one is forced to either restrict the domain of path integral in a delicate way, or otherwise correct for the multivaluedness by dividing the path integral by certain intersection numbers. We provide a careful treatment of how to fix collective coordinates while accounting for these intersection numbers, and then demonstrate the importance of the fix for free theories. We also provide a detailed study of the fix for interacting theories and show that the contributions of higher intersections to the path integral can be nonperturbatively suppressed. Using a variety of examples ranging from single-particle quantum mechanics to quantum field theory, we explain and resolve various pitfalls in the implementation of collective coordinates.

United States

Arindam Bhattacharya

Jordan Cotler

Aurélien Dersy

Matthew D. Schwartz

Collective coordinate fix in the path integral

New measurements of the <math><msub><mrow><mi>T</mi></mrow><mrow><mn>20</mn></mrow></msub></math> component of tensor analyzing power in the reaction <math><mi>γ</mi><mi>d</mi><mo>→</mo><mi>p</mi><mi>p</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></math> for photon energies <math><mn>400</mn><mo>−</mo><mn>700</mn></math> MeV are presented. The experiment was carried out on an internal tensor-polarized gas deuterium target of the VEPP-3 electron storage ring in 2023 with the use of tagged quasi-real photons and two-proton coincidence. For determination of <math><msub><mrow><mi>T</mi></mrow><mrow><mn>20</mn></mrow></msub></math>, the asymmetry caused by a change of the sign of the tensor polarization of the deuteron target was measured. The new data are compared with the results of theoretical calculations based on the quasi-free approximation including πN and NN rescattering effects.

Russia

V.V. Gauzshtein

Saudi Arabia

Egypt

E.M. Darwish

A.I. Fix

M.Ya. Kuzin

Belarus

M.I. Levchuk

A.Yu. Loginov

D.M. Nikolenko

I.A. Rachek

Yu.V. Shestakov

D.K. Toporkov

A.V. Yurchenko

B.I. Vasilishin

S.A. Zevakov

G.N. Baranov

A.V. Bogomyagkov

V.M. Borin

D.V. Dorokhova

V.L. Dorokhov

A.N. Zhuravlev

S.E. Karnaev

K.Yu. Karyukina

A.A. Kovalenko

V.N. Kudryavtsev

E.B. Levichev

I.B. Logashenko

T.V. Maltsev

R.Z. Mamutov

I.A. Morozov

I.N. Okunev

P.A. Piminov

L.I. Shekhtman

E.A. Simonov

S.V. Sinyatkin

M.A. Skamarokha

E.V. Starostina

Measurement of the tensor analyzing power For Incoherent Photoproduction on a Deuteron Above the First Resonance Region

Decays of unstable heavy particles usually involve the coherent sum of several amplitudes, like in a multiple slit experiment. Dedicated amplitude analysis techniques have been widely used to resolve these amplitudes for better understanding of the underlying dynamics. In special cases where two spin-1/2 particles and two (pseudo-) scalar particles are present in the process, multiple equivalent solutions are found owing to intrinsic symmetries in the summed probability density function. In this study, the problem of multiple solutions is discussed, and a scheme to overcome this problem is proposed by fixing some free parameters. Toys are generated to validate the strategy. A new approach to align the helicities of initial- and final-state particles in different decay chains is also introduced.

China

Yuanning Gao

Tianze Rong

Zhenwei Yang

Chenjia Zhang

Yanxi Zhang

A scheme to fix multiple solutions in amplitude analyses  

In previous works, we have developed an approach to fix the leading behaviors of the pure AdS $$_3$$  <math> <msub> <mrow></mrow> <mn>3</mn> </msub> </math>   and BTZ black hole from the entanglement entropies of the free CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>   and finite temperature CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>  , respectively. We exclusively use holographic principle only and make no restriction about the bulk geometry, not only the kinematics but also the dynamics. In order to verify the universality and correctness of our method, in this paper, we apply it to the  $$T\bar{T}$$  <math> <mrow> <mi>T</mi> <mover> <mrow> <mi>T</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   deformed CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>  , which breaks the conformal symmetry. In terms of the physical arguments of the  $$T\bar{T}$$  <math> <mrow> <mi>T</mi> <mover> <mrow> <mi>T</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   deformed CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>  , the derived metric is a deformed BTZ black hole. The requirement that the CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>   lives on a conformally flat boundary leads to  $$r_{c}^{2}=\ 6R_{AdS}^{4}/(\pi c\mu )$$  <math> <mrow> <msubsup> <mi>r</mi> <mrow> <mi>c</mi> </mrow> <mn>2</mn> </msubsup> <mo>=</mo> <mspace width="4pt"></mspace> <mn>6</mn> <msubsup> <mi>R</mi> <mrow> <mi>AdS</mi> </mrow> <mn>4</mn> </msubsup> <mo>/</mo> <mrow> <mo>(</mo> <mi>π</mi> <mi>c</mi> <mi>μ</mi> <mo>)</mo> </mrow> </mrow> </math>   naturally, in perfect agreement with previous conjectures in literature. The energy spectum and propagation speed calculated with this deformed BTZ metric are the same as these derived from  $$T\bar{T}$$  <math> <mrow> <mi>T</mi> <mover> <mrow> <mi>T</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   deformed CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>  . We furthermore fix the dual geometry of highly excited states with our approach. The result contains the descriptions for the conical defect and BTZ black hole.

Peng Wang

Houwen Wu

Haitang Yang

Fix the dual geometries of  $$T\bar{T}$$  <math> <mrow> <mi>T</mi> <mover> <mrow> <mi>T</mi> </mrow> <mrow> <mo>¯</mo> </mrow> </mover> </mrow> </math>   deformed CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>   and highly excited states of CFT $$_2$$  <math> <msub> <mrow></mrow> <mn>2</mn> </msub> </math>