Several possible excesses around $ 95 $ GeV hint at an additional light scalar beyond the Standard Model. We examine the capability of the Circular Electron-Positron Collider (CEPC) to test this hypothesis in the Higgs-strahlung channel $ e^{+}e^{-} \to ZS $ with $ Z \to\mu^{+}\mu^{-} $ and $ S \to\tau^{+}\tau^{-} $ . A full detector simulation shows that the optimal center-of-mass energy for studying the 95 GeV light scalar is 210 GeV. A deep neural network classifier reduces the luminosity required for discovery by half. At $ L = 20\; \mathrm{ab}^{-1} $ , the $ 5\sigma $ sensitivity of the CEPC to the signal strength $ \mu_{\tau\tau}^{ZS} $ reaches 0.016 and 0.020 for $ \sqrt{s} = 210 $ GeV and 240 GeV, respectively. The corresponding thresholds for a 5% precision measurement are $ \mu_{\tau\tau}^{ZS} \gt 0.10 $ and $ \gt 0.12 $ . At $ \sqrt{s}=210 $ GeV (240 GeV), the $ 5\sigma $ coverage of all N2HDM-Flipped samples with $ \chi^2_{h_{95}} \lt 7.82 $ requires $ L=800\ \mathrm{fb}^{-1} $ (1.22 $ \mathrm{ab}^{-1} $ ). These results establish a 210 GeV run, augmented by machine-learning selection, as the most efficient strategy to confirm or refute the 95 GeV excess at future lepton colliders.

China

Yabo Dong

Manqi Ruan

Kun Wang

Haijun Yang

Jingya Zhu

Testing a 95 GeV Scalar at the CEPC with Machine Learning * This work was supported by the National Natural Science Foundation of China under Grant No. 12275066 and by the startup research funds of Henan University. The work of K. Wang was also supported by the Open Project of the Shanghai Key Laboratory of Particle Physics and Cosmology under Grant No. 22DZ2229013-3. The work of M. Ruan was also supported by the National Key Program for S&T Research and Development under Contract No. 2024YFA1610603. The work of H. Yang was also supported by the National Natural Science Foundation of China under Grant No. W2441004.

An analytical and numerical systematic study of the neutrino mass matrix with one texture zero is presented in a basis where the charged leptons are diagonal. Under the assumption that neutrinos are Dirac particles, the analysis is conducted in detail for the normal and inverted hierarchy mass spectra. Our study is performed without any approximations, first analytically and then numerically, using current neutrino oscillation data. The analysis constrains the parameter space in such a way that, among the six possible one-texture-zero patterns, only four are favored in the normal hierarchy and one in the inverted hierarchy by current oscillation data at the $3 \sigma$ level. Phenomenological implications for the lepton CP-violating phase and neutrino masses are also explored.

Colombia

Richard H. Benavides

Yessica Lenis

John D. Gómez

William A. Ponce

One texture zero for Dirac neutrinos in a diagonal charged lepton basis * Richard H. Benavides acknowledges additional financial support from Minciencias (CD82315 CT ICETEX 2021-1080)

One-loop contributions for the decay process $ H^{\pm} \rightarrow W^{\pm}Z $ within the Two-Higgs-Doublet Model (THDM) are computed in the general $ {\cal{R}}_{\xi} $ gauge, and its phenomenological applications at future muon–TeV colliders are investigated. Analytic results are confirmed by several consistency tests, including those of $\textit{ξ}$-independence, renormalization-scale stability, and the ultraviolet finiteness of the one-loop amplitude. We first perform an updated parameter scan of the Type-X THDM in phenomenological studies. The production of charged Higgs boson pairs at future muon–TeV colliders is then investigated through two processes: $ \mu^+\mu^- \rightarrow H^+H^- \rightarrow W^{\pm}W^{\mp}Zh $ and $ \mu^+\mu^- \rightarrow \gamma\gamma \rightarrow H^+H^- \rightarrow W^{\pm}W^{\mp}Zh $ . Both signal events and their significance are evaluated considering the corresponding Standard Model backgrounds. We find that the signal significances can exceed $ 5\sigma $ at several benchmark points in the viable parameter space of the Type-X THDM.

Vietnam

Dzung Tri Tran

Quang Hoang-Minh Pham

Khoa Ngo-Thanh Ho

Khiem Hong Phan

One-loop expressions for ${H^\pm \rightarrow W^\pm Z} $ and their implications at muon-TeV colliders * This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) (103.01-2023.16)

We construct, for the first time, new static non-extremal five-dimensional black hole solutions (without or with squashed horizons) endowing with four different electric charge parameters in the $\textit{D}$ = 5, $$ \mathcal{N} $$ = 2 supergravity coupled to $\textit{three}$ vector multiplets with a specific pre-potential $$ \mathcal{V} $$ = $\textit{STU}$ – $\textit{W}$ $\textit{U}$ ≡ 1. When the fourth charge parameter disappears, the solution simplify reduces to the three-charge static black hole solution previously presented in ref. [1], which belongs to the solution to the $\textit{D}$ = 5, $$ \mathcal{N} $$ = 2 supergravity coupled to $\textit{two}$ vector multiplets (also notably known as the $\textit{STU}$ model). We parameterize the model in such a simple fashion that not only can one easily recover the static three-charge solution but also it is very convenient to study their thermodynamical properties of the obtained black hole solutions in the case without a squashing horizon. We then show that the thermodynamical quantities perfectly obey both the differential first law and integral Smarr formula of thermodynamics. Finally, we also extend to present its generalizations with squashed horizons or including a nonzero cosmological constant.

Di Wu

Shuang-Qing Wu

Four-charge static non-extremal black holes in the five-dimensional $$ \mathcal{N} $$ = 2, $\textit{STU}$ − $\textit{W}$

In this paper, we investigate thermodynamical aspects of a novel static non-extremal black hole solution in the five-dimensional anti-de Sitter (AdS) space carrying four independent electric charge parameters and with a negative cosmological constant, within the framework of $$D=5$$ , $$\mathcal {N}= 2$$ gauged supergravity coupled to three vector multiplets specified by the pre-potential $$\mathcal {V}= STU -W^2U \equiv 1$$ . A central outcome of our work is the demonstration that the thermodynamics of this solution consistently satisfies both the differential and integral forms of the mass formulae of black hole thermodynamics. When the fourth charge is set to zero, our configuration reduces to the known three-charge static non-extremal AdS $$_5$$ black hole within the gauged $\textit{STU}$ model, and its thermodynamic behavior connects smoothly to that limit. Finally, we also provide a brief discussion about topological classifications of its thermodynamics.

Thermodynamics and topological classifications of static non-extremal four-charge AdS black hole in the five-dimensional $$\mathcal {N}= 2$$ , $$STU-W^2U$$ gauged supergravity

We propose an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>AdS</mi><mo>×</mo><mi mathvariant="normal">S</mi></mrow></math> factorization formula at the level of the generating function for correlators with arbitrary Kaluza-Klein configurations and implement it in the supergravity limit of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></mrow></math> supersymmetric Yang-Mills theory. By incorporating this mechanism into Mellin space bootstrap, together with an observed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mn>2</mn></msub></math> symmetry under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>AdS</mi><mo stretchy="false">↔</mo><mi mathvariant="normal">S</mi></mrow></math>, we manage to simultaneously work out unified formulas both for all five-point half-BPS correlators and for all four-point correlators with one superdescendant. This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>AdS</mi><mo>×</mo><mi mathvariant="normal">S</mi></mrow></math> bootstrap method is directly applicable to generic multipoint computation at tree level.

Portugal

Bruno Fernandes

Vasco Gonçalves

Zhongjie Huang

Yichao Tang

United Kingdom

Joao Vilas Boas

Ellis Ye Yuan

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>AdS</mi><mo stretchy="false">×</mo><mi mathvariant="normal">S</mi></mrow></math> Mellin Bootstrap, Hidden 10d Symmetry and Five-Point Kaluza-Klein Functions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi mathvariant="script">N</mi><mo>=</mo><mn>4</mn></mrow></math> Supersymmetric Yang-Mills Theory

We report decay-rate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetries of the singly-Cabibbo-suppressed decays <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>h</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>h</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><mi>p</mi><msup><mrow><mi>h</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>h</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></mrow></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>h</mi><mo>=</mo><mi>K</mi></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>π</mi></mrow></math>, measured using <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn>428</mn><mtext> </mtext><mtext> </mtext><msup><mi>fb</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> collisions collected by the Belle II experiment at the SuperKEKB collider. The results, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>3.7</mn><mo>±</mo><mn>6.6</mn><mo>±</mo><mn>0.6</mn><mo stretchy="false">)</mo><mo>%</mo></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>9.5</mn><mo>±</mo><mn>6.8</mn><mo>±</mo><mn>0.5</mn><mo stretchy="false">)</mo><mo>%</mo></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><mi>p</mi><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>3.9</mn><mo>±</mo><mn>1.7</mn><mo>±</mo><mn>0.7</mn><mo stretchy="false">)</mo><mo>%</mo></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mspace linebreak="goodbreak"></mspace><mrow><mi>p</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0.3</mn><mo>±</mo><mn>1.0</mn><mo>±</mo><mn>0.2</mn><mo stretchy="false">)</mo><mo>%</mo></mrow></math>, where the first uncertainties are statistical and the second systematic, agree with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> symmetry. From these results we derive the sums <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>+</mo><mspace linebreak="goodbreak"></mspace><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><mi>p</mi><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>13.4</mn><mo>±</mo><mn>7.0</mn><mo>±</mo><mn>0.9</mn><mo stretchy="false">)</mo><mo>%</mo></mrow></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>K</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>C</mi><mi>P</mi></mrow></msub><mo stretchy="false">(</mo><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mrow><mi>p</mi><msup><mrow><mi>π</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>π</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>4.0</mn><mo>±</mo><mspace linebreak="goodbreak"></mspace><mn>6.6</mn><mo>±</mo><mn>0.7</mn><mo stretchy="false">)</mo><mo>%</mo></mrow></math>, which are consistent with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>U</mi></math>-spin symmetry prediction of zero. These are the first measurements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> asymmetries for individual hadronic three-body charmed-baryon decays.

M. Abumusabh

I. Adachi

H. Ahmed

Y. Ahn

H. Aihara

N. Akopov

S. Alghamdi

M. Alhakami

N. Althubiti

K. Amos

N. Anh Ky

D.M. Asner

H. Atmacan

R. Ayad

V. Babu

N.K. Baghel

S. Bahinipati

P. Bambade

Sw. Banerjee

M. Bartl

J. Baudot

A. Beaubien

J. Becker

J.V. Bennett

V. Bertacchi

M. Bertemes

E. Bertholet

S. Bettarini

V. Bhardwaj

D. Biswas

A. Bobrov

D. Bodrov

G. Bonvicini

J. Borah

A. Boschetti

M. Bračko

P. Branchini

R.A. Briere

T.E. Browder

A. Budano

S. Bussino

Q. Campagna

M. Campajola

G. Casarosa

C. Cecchi

P. Chang

P. Cheema

L. Chen

B.G. Cheon

C. Cheshta

H. Chetri

K. Chilikin

J. Chin

K. Chirapatpimol

H.E. Cho

K. Cho

S.J. Cho

S.K. Choi

S. Choudhury

S. Chutia

J.A. Colorado-Caicedo

I. Consigny

L. Corona

J.X. Cui

S. Das

S.A. De La Motte

G. De Nardo

G. De Pietro

R. de Sangro

M. Destefanis

A. Di Canto

Z. Doležal

I. Domínguez Jiménez

T.V. Dong

X. Dong

M. Dorigo

G. Dujany

P. Ecker

D. Epifanov

J. Eppelt

R. Farkas

P. Feichtinger

T. Ferber

T. Fillinger

G. Finocchiaro

F. Forti

B.G. Fulsom

A. Gabrielli

E. Ganiev

R. Garg

G. Gaudino

V. Gaur

V. Gautam

A. Gaz

A. Gellrich

D. Ghosh

H. Ghumaryan

R. Giordano

A. Giri

P. Gironella Gironell

Search for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>C</mi><mi>P</mi></math> violation in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Ξ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><msup><mrow><mi mathvariant="normal">Σ</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>h</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>h</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi mathvariant="normal">Λ</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>+</mo></mrow></msubsup><mo stretchy="false">→</mo><mi>p</mi><msup><mrow><mi>h</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>h</mi></mrow><mrow><mo>−</mo></mrow></msup></mrow></math> at Belle II

We use analytic (super-)conformal bootstrap methods to derive explicit expressions for the structure constants of $$ \mathcal{N} $$ = 1 Liouville CFT in the ‘timelike’ regime of the superconformal central charge. The obtained expressions take the form of inverses of the appropriate spacelike counterparts, which we explain concretely by elucidating the analytic properties of the corresponding shift relations in the NS- and R-sectors for the normalization-independent bootstrap data on the sphere. In a particular normalization, the timelike structure constants are shown to agree with the OPE coefficients of $$ \mathcal{N} $$ = 1 Minimal Models when specified at degenerate values of the momenta, exactly as in the non-supersymmetric case. We discuss possible applications of our results, with emphasis on the construction of the $$ \mathcal{N} $$ = 1 supersymmetric analog of the Virasoro Minimal String.

United States

Beatrix Mühlmann

Vladimir Narovlansky

France

Ioannis Tsiares

On the three-point functions in timelike $$ \mathcal{N} $$ = 1 Liouville CFT

How do we describe non-trivial bulk measurements relative to an observer (i.e. relationally) when both the observer and the system it probes may/may not evolve in time? How can we interpret this holographically; particularly for zero-energy BPS states in supersymmetric theories? We address these questions, in the $$ \mathcal{N} $$ = 2 double-scaled SYK model and its putative bulk dual by: (i) formulating a holographic procedure in the language of quantum reference frames to gravitationally dress bulk observables to “clocks” parametrized by both boundary time and R-charge; and (ii) proposing a $\textit{new measure of Krylov complexity}$ with R-charge in the boundary theory that probes zero-energy BPS states. Holographically, this proposal reproduces a relational bulk observable, a BPS wormhole length. We contrast this to the Krylov complexity for Hartle-Hawking states with non-trivial time flow. The latter reproduces the same observable as for the bosonic DSSYK in the semiclassical limit, while its quantum fluctuations can capture supersymmetric corrections depending on the specific initial state.

Japan

Sergio Aguilar-Gutierrez

Evolution with(out) time: relational holography & BPS complexity growth in $$ \mathcal{N} $$ = 2 double-scaled SYK

In this work, we study the long-range interactions in nongravitational field theories and their behavior in the deep infrared. To model such effects, we consider a nonlocal scalar theory obtained by adding a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><msup><mo>□</mo><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>ϕ</mi></math> term to the local action. Using the functional renormalization group, we analyze its infrared fixed-point structure. Within the local potential approximation (LPA), we show that nonlocality modifies phase-transition patterns and can induce symmetry breaking. Extending the LPA beyond polynomial truncations, we examine the convexity property of the effective potential as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>k</mi><mo stretchy="false">→</mo><mn>0</mn></math> and find that the flow becomes singular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>λ</mi><mn>2</mn></msup><mo>&gt;</mo><mn>0</mn></math> before reaching the deep infrared. Including the scale-dependent wave function renormalization, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>Z</mi><mi>k</mi></msub></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msup><mi>LPA</mi><mo>′</mo></msup></math>), we find that the infrared-stable fixed point is the nonlocal Gaussian fixed point. We then generalize the model to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>ϕ</mi><msup><mo>□</mo><mrow><mi>σ</mi><mo>/</mo><mn>2</mn></mrow></msup><mi>ϕ</mi></math> and analyze how the infrared properties depend on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi></math>. With appropriate scaling choices, we show that the infrared behavior remains unchanged up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi><mo>=</mo><mi>d</mi><mo>/</mo><mn>2</mn></math> and follows Sak’s prediction up to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi><mo>=</mo><mn>2</mn></math>. Finally, we study higher-derivative cases within the LPA, focusing on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>σ</mi><mo>=</mo><mn>4</mn></math>, which corresponds to isotropic Lifshitz criticality, and obtain results consistent with earlier work.