We update the analysis of minimal supersymmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></math> grand unification with a universal, constrained-minimal supersymmetric Standard Model (MSSM)-like soft-breaking spectrum. A pair of Yukawa matrices in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mn mathvariant="bold">10</mn><mo stretchy="false">⊕</mo><mover accent="true"><mn mathvariant="bold">126</mn><mo stretchy="true">¯</mo></mover></math> Higgs sector fixes the charged fermion masses and quark mixing, the baryon number-violating dimension-five operators that mediate proton decay, and the right-handed neutrino Majorana masses of the type-I seesaw. The simultaneously imposed constraints include: (i) gauge coupling unification and vacuum stability; (ii) dimension-five proton decay (the dimension-six gauge mode also computed and negligible), the 125-GeV Higgs mass, and the Large Electron-Positron chargino limit; and (iii) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>μ</mi><mo stretchy="false">→</mo><mi>e</mi><mi>γ</mi></math>, the muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>g</mi><mo>−</mo><mn>2</mn></math>, and electric dipole moments. We perform a global scan over the soft-breaking parameters, complemented by a constrained-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>μ</mi></math> cross-check that fixes the electroweakino sector from the grand unified theory (GUT)-scale boundary conditions through radiative electroweak symmetry breaking. Subject to a single effective colored Higgs scale <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>M</mi><msub><mi>H</mi><mi>C</mi></msub></msub></math> for the proton decay normalization (common to unification and decay, though in a general color triplet mass matrix the two can differ) and to a constrained-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>μ</mi></math> treatment, what survives is a narrow minisplit region: heavy, multitens-of-TeV scalars with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mn>0</mn></msub><mo>≳</mo><mn>11.0</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math> for a dressing electroweakino mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>m</mi><mi>χ</mi></msub><mo>=</mo><mn>1</mn><mtext> </mtext><mtext> </mtext><mi>TeV</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>tan</mi><mi>β</mi><mo>≲</mo><msubsup><mn>8.0</mn><mrow><mo>−</mo><mn>2.0</mn></mrow><mrow><mo>+</mo><mn>5.5</mn></mrow></msubsup></math>, the uncertainty being the Higgs mass theory error. We also examine the fate of the lightest supersymmetric particle as a dark matter candidate, in both the universal model and its free-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>μ</mi></math> nonuniversal-Higgs-mass extension. The surviving region is sharply bounded and testable in the coming decade, most directly by Hyper-Kamiokande proton decay and electroweakino searches, with dark matter direct detection and electric dipole moment experiments.

China

Cheng-Wei Chiang

Japan

Takeshi Fukuyama

Updated analysis of minimal supersymmetric <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>S</mi><mi>O</mi><mo stretchy="false">(</mo><mn>10</mn><mo stretchy="false">)</mo></math> with a universal soft spectrum

We construct asymptotically AdS black brane solutions in a theory of gravity with an infinite series of curvature corrections. The action is based on an $\textit{O}$($\textit{d}$, $\textit{d}$) symmetric ansatz which has been argued to describe the classical NSNS sector of string theories. We find that, for this general class of theories, the singularity behind the horizon is not resolved by the curvature corrections. The approach to the singularity is however generically modified, being characterized by different Kasner exponents. We also show that, in the presence of a non-trivial dilaton, a slight generalization of these types of curvature corrections can generate dynamically a negative cosmological constant in the region of small coupling. This provides a mechanism through which asymptotic freedom could emerge in the hypothetical string dual of QCD.

Netherlands

Pedro Marto

Edwan Préau

$\textit{O}$($\textit{d, d}$) symmetric gravity and finite coupling holography

The study of large-$\textit{N}$ cases of the $\textit{O}$($\textit{N}$)-symmetric model at criticality has attracted significant attention in recent years. In particular, a recent high-precision Monte Carlo study [2] reported the most accurate estimates to date for the critical exponents $$\nu $$ , $$\eta $$ , and $$\omega $$ for $$N \ge 4$$ . Within the framework of the renormalization group (RG) – the cornerstone of the modern theory of critical phenomena – powerful computational approaches such as the $$\varepsilon $$ -expansion and non-perturbative RG are available. Since the effective expansion parameter in the $$\varepsilon $$ -expansion, $$\sigma = \frac{3}{N+8}$$ , decreases with increasing N, one expects resummation techniques to become progressively more accurate in the large-N regime. To test this expectation, we apply the entire-hypergeometric resummation algorithm developed by Shalaby et al. to the recently obtained seven-loop divergent $$\varepsilon $$ -series for $$N \ge 4$$ , yielding high-precision estimates for $$\nu $$ , $$\eta $$ , and $$\omega $$ . Our analysis confirms the anticipated improvement in accuracy with increasing N. To assess the significance of these results, we note that at the same seven-loop order the $\textit{O}$(2) case exhibits errors that are an order of magnitude larger than those obtained from experiment, Monte Carlo simulations, and conformal-field-theory analysis. In contrast, for sufficiently large N, the errors in the present work are of the same order of magnitude as those from Monte Carlo and non-perturbative RG methods, demonstrating the strong predictive power of our resummation approach in the large-N regime.

Qatar

Abouzeid Shalaby

Dependence of critical exponents accuracy on the number of fields in the $\textit{O}$($\textit{N}$)-invariant $$\phi ^4$$ model

The low-lying level structure of 80Se was investigated via the 76Ge( 9Be, 2p3n) reaction at <italic>E</italic> <inf>lab</inf>&#8239;&#8776;&#8239;31 MeV using a Compton-suppressed HPGe clover array, and level lifetimes were measured with the Doppler-shift attenuation method. Measured excitation energies and <italic>B</italic>( <italic>E</italic>2) transition strengths are reasonably well reproduced by Interacting Boson Approximation (IBA) calculations. The observed excitation-energy and transition-strength patterns associated with the <math altimg='si68.svg'><mrow><mi>&#963;</mi><mo linebreak='goodbreak'>=</mo><mn>5</mn></mrow></math> and <math altimg='si69.svg'><mrow><mi>&#963;</mi><mo linebreak='goodbreak'>=</mo><mn>3</mn></mrow></math> representations, together with candidate members of the <math altimg='si70.svg'><mrow><mi>&#963;</mi><mo linebreak='goodbreak'>=</mo><mn>1</mn></mrow></math> representation, exhibit pronounced O(6)-like collective behavior in this five-boson system. The combined experimental observables and finite-boson IBA calculations indicate a surprisingly good correspondence with O(6) expectations for these selected collective structures. At the same time, the presence of low-lying <math altimg='si71.svg'><msubsup><mn>0</mn><mn>2</mn><mo>+</mo></msubsup></math> and <math altimg='si72.svg'><msubsup><mn>2</mn><mn>3</mn><mo>+</mo></msubsup></math> states that are not consistently accommodated within the same classification, together with a finite <italic>B</italic>( <italic>E</italic>2; <math altimg='si73.svg'><mrow><msubsup><mn>0</mn><mn>3</mn><mo>+</mo></msubsup><mo>&#8594;</mo><msubsup><mn>2</mn><mn>1</mn><mo>+</mo></msubsup><mrow><mo>)</mo></mrow></mrow></math> strength, demonstrates that the O(6) limit does not provide a complete description of the low-energy spectrum. We therefore interpret the present results as evidence for pronounced O(6)-like collective structures embedded within a more complex nuclear structure of 80Se.

India

R. Biswas

S. Rajbanshi

Habibur Rahaman

Saikat Sen

G. Manna

R. Palit

Abhijit Bisoi

S. Sounda

Sajad Ali

F.S. Babra

R. Banik

Israel

S. Bhattacharya

S. Bhattacharyya

Poland

P. Dey

Md. S.R. Laskar

G. Mukherjee

S. Nandi

Apurba Kumar Nayek

Romania

H. Pai

Rajkumar Santra

T. Trivedi

Emergence of O(6)-like collective structures in 80Se

We revisit the constrained log-subtracted two-point Padé (LSTP) ensemble for thermal $$\mathcal {N}{=}4$$ supersymmetric Yang-Mills (SYM) thermodynamics in four spacetime dimensions after upgrading the weak-coupling truncation from $$\mathcal {O}(\lambda ^2)$$ to the exact $$\mathcal {O}(\lambda ^{5/2})$$ coefficient. We keep the LSTP interpolation ansatz unchanged and shift the weak-side matching points to the regime where the new term is numerically significant. Within the same parameter scan the admissible set collapses from 9 nominal survivors (3 distinct curves) to a single distinct curve, the crossover range shrinks to a unique value, and the pointwise band width drops to zero within numerical resolution. The Hermite-Padé (HP) central curve does not coincide with the unique LSTP survivor, so the updated weak-side matching removes the LSTP scan uncertainty but not the difference between the two routes. The minimal HP extension that also carries the exact coefficient is uniquely determined and develops a spurious pole-zero pair at small coupling. Away from that pair the HP and LSTP curves still differ, so this difference persists at the same weak-coupling order.

Saudi Arabia

Ubaid Tantary

Qianqian Du

Constrained Padé ensembles for thermal $$\mathcal {N}{=}~4$$ SYM with the exact $$\mathcal {O}(\lambda ^{5/2})$$ coefficient

We critically examine the expectation that in a fundamental quantum field theory, dimensionless couplings in the Lagrangian density should all be of order unity. We propose a measure to quantify the adherence of a theory to this: the spread (the ratio of the largest to the smallest of the magnitudes) of such dimensionless couplings, obtaining various closed-form results. If we take independent and identically distributed (IID) couplings to parameterise our uncertainty on the values of the order-unity couplings, the spread can be much larger than one might naively expect. For a theory with 20 IID unit normal couplings, the probability that the spread is greater than 100 is 0.29, for example. Even when the IID couplings have exponentially suppressed tails, the distribution of the spread has fat <italic>power-law</italic> tails whose amplitude grows with the number of independent couplings.

United Kingdom

Ben Allanach

The fundaments of unity: <math altimg='si17.svg'><mrow><mi mathvariant='script'>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math> dimensionless couplings in quantum field theories

Austria

H. Abele

Germany

G. Angloher

B. Arnold

Italy

M. Atzori Corona

Portugal

A. Bento

France

E. Bossio

F. Buchsteiner

J. Burkhart

F. Cappella

M. Cappelli

N. Casali

R. Cerulli

A. Cruciani

G. Del Castello

M. del Gallo Roccagiovine

S. Dorer

A. Erhart

M. Friedl

S. Fichtinger

V. Ghete

M. Giammei

C. Goupy

J. Hakenmüller

D. Hauff

F. Jeanneau

E. Jericha

M. Kaznacheeva

H. Kluck

A. Langenkämper

T. Lasserre

D. Lhuillier

M. Mancuso

R. Martin

B. Mauri

A. Mazzolari

L. McCallin

H. Neyrial

C. Nones

L. Oberauer

L. Peters

F. Petricca

W. Potzel

F. Pröbst

F. Pucci

F. Reindl

M. Romagnoni

J. Rothe

N. Schermer

J. Schieck

S. Schönert

C. Schwertner

L. Scola

G. Soum-Sidikov

L. Stodolsky

A. Schröder

R. Strauss

R. Thalmeier

C. Tomei

L. Valla

M. Vignati

M. Vivier

A. Wallach

P. Wasser

A. Wex

L. Wienke

Publisher Correction: Characterization of the low energy excess using a NUCLEUS $$\text {Al}_{2}\text {O}_{3}$$ detector

We study the decays $$ h\to \ell \overline{\ell}\left(Z\to {\ell}^{\prime }{\overline{\ell}}^{\prime}\right) $$ and $$ h\to \ell {\overline{\nu}}_{\ell }{\nu}_{\ell^{\prime }}{\overline{\ell}}^{\prime } $$ within the SMEFT framework and including effects up to $$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ , where Λ is the new physics scale suppressing higher dimensional operators. To work to this order, we must include the square of dimension-six operators and the interference of dimension-eight operators with the Standard Model. We study angular asymmetries and other differential decay observables and determine which are most sensitive to $$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ effects. While new kinematic structures arising in higher dimensional operators have the potential to induce novel angular dependency, we find this does not occur for $$ h\to \ell \overline{\ell}\left(Z\to {\ell}^{\prime }{\overline{\ell}}^{\prime}\right) $$ . For $$ h\to \ell {\overline{\nu}}_{\ell }{\nu}_{\ell^{\prime }}{\overline{\ell}}^{\prime } $$ , new angular dependencies do arise at $$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ , though they require a fully reconstructible (meaning we can go to the Higgs rest frame) final state. For non-reconstructible final states such as $$ \ell {\overline{\nu}}_{\ell }{\nu}_{\ell^{\prime }}{\overline{\ell}}^{\prime } $$ , we must study Higgs production and decay together with the appropriate observables, which we find obscures the new angular effects.

United States

Mario Flores-Hernández

Adam Martin

Higgs decays to four leptons to $$ \mathcal{O}\left(1/{\Lambda}^4\right) $$ in SMEFT

We extend the results of Lappi et al. [<a href="http://dx.doi.org/10.1140/epjc/s10052-023-12365-2">Eur. Phys. J. C 84, 84 (2024)</a>] to show that it is possible to obtain expressions for the longitudinal, singlet, and gluon structure functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>L</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>S</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math> in deep inelastic scattering directly in terms of the measured functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>F</mi><mn>2</mn></msub><mo>/</mo><mi>ln</mi><mo stretchy="false">(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> modulo nonsinglet corrections expected to be small at very small <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math>. The latter can be treated at low <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>x</mi></math> using existing quark distributions. Our results are presented consistently to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math>, correcting and extending the mixed-order results of Lappi et al..

Iran

G.R. Boroun

Loyal Durand

Phuoc Ha

Direct determination of the structure functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>L</mi></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mi>S</mi></msub></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>G</mi></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mi>F</mi><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>d</mi><msub><mi>F</mi><mn>2</mn></msub><mo>/</mo><mi>d</mi><msup><mi>Q</mi><mn>2</mn></msup></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>O</mi><mo stretchy="false">(</mo><msubsup><mi>α</mi><mi>s</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo></math>

We derive an exact, scheme-independent generating function <math altimg='si51.svg'><mrow><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math> for the massive <italic>O</italic>( <italic>N</italic>)-symmetric <italic>&#981;</italic> 4 two-point observable <math altimg='si52.svg'><mrow><msub><mi>R</mi><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle></msub><mrow><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>,</mo><mi>a</mi><mo>,</mo><msup><mi>m</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>&#8801;</mo><msup><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mrow><mo>(</mo><msup><mi>Q</mi><mn>2</mn></msup><mo>,</mo><mi>a</mi><mo>,</mo><msup><mi>m</mi><mn>2</mn></msup><mo>)</mo></mrow><mo linebreak='goodbreak'>/</mo><msup><mi>Q</mi><mn>2</mn></msup></mrow></math> within Scale-Factorized Quantum Field Theory (SF-QFT). Path-integral factorization at the physical matching scale <math altimg='si5.svg'><mrow><msup><mi>Q</mi><mo>*</mo></msup><mo linebreak='goodbreak'>=</mo><mi>m</mi></mrow></math>, followed by Seeley&#8211;DeWitt heat-kernel integration of the infrared modes, extracts five universal scheme-independent inputs: the two-loop <italic>&#946;</italic>-function coefficients <italic>&#946;</italic> <inf>0</inf>, <italic>&#946;</italic> <inf>1</inf>, the mass anomalous dimension coefficients <italic>d</italic> <inf>1</inf>, <italic>d</italic> <inf>2</inf>, and the two-loop field anomalous dimension coefficient <italic>e</italic> <inf>2</inf>, in the normalization of Refs. [1,2] with <math altimg='si6.svg'><mrow><mi>a</mi><mo>&#8801;</mo><mi>&#955;</mi><mo linebreak='goodbreak'>=</mo><mi>g</mi><mo linebreak='goodbreak'>/</mo><mo>(</mo><mn>2</mn><msup><mi>&#960;</mi><mn>2</mn></msup><mo>)</mo></mrow></math>. The observable <math altimg='si46.svg'><msub><mi>R</mi><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle></msub></math> carries its own renormalization-group anomalous dimension <math altimg='si53.svg'><mrow><msub><mi>&#947;</mi><msub><mi>R</mi><mstyle mathvariant='normal'><mi>&#915;</mi></mstyle></msub></msub><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mi>a</mi><mo linebreak='goodbreak'>+</mo><msub><mi>&#945;</mi><mn>2</mn></msub><msup><mi>a</mi><mn>2</mn></msup><mo linebreak='goodbreak'>+</mo><mo>&#8943;</mo></mrow></math>, with <math altimg='si54.svg'><mrow><msub><mi>&#945;</mi><mn>1</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>d</mi><mn>1</mn></msub></mrow></math> and <math altimg='si55.svg'><mrow><msub><mi>&#945;</mi><mn>2</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>d</mi><mn>2</mn></msub><mo linebreak='goodbreak'>+</mo><mn>2</mn><msub><mi>e</mi><mn>2</mn></msub></mrow></math>; the factor of two on <italic>e</italic> <inf>2</inf> is the two-leg field renormalization of the two-point function. The Principle of Observable Effective Matching (POEM) at <math altimg='si5.svg'><mrow><msup><mi>Q</mi><mo>*</mo></msup><mo linebreak='goodbreak'>=</mo><mi>m</mi></mrow></math> fixes the boundary data <math altimg='si56.svg'><mrow><msub><mi>f</mi><mn>1</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub></mrow></math> and <math altimg='si57.svg'><mrow><msub><mi>f</mi><mn>2</mn></msub><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>2</mn></msub><mo linebreak='goodbreak'>/</mo><mn>2</mn></mrow></math> and yields a universal three-sector Effective Dynamical Renormalization (EDR) recursion whose coefficients <italic>f</italic> <inf><italic>n</italic>,0</inf> define <math altimg='si31.svg'><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub></math> exactly and converge geometrically with ratio &#8239;&#8764;&#8239;0.003 to 0.01. Summing the recursion gives the master ordinary differential equation (ODE) <math altimg='si58.svg'><mrow><msubsup><mi mathvariant='script'>G</mi><mn>0</mn><mo>&#8242;</mo></msubsup><mrow><mo>(</mo><mn>1</mn><mo linebreak='badbreak'>&#8722;</mo><mi>k</mi><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mo>)</mo></mrow><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mo linebreak='goodbreak'>+</mo><mrow><mo>[</mo><msub><mi>&#945;</mi><mn>2</mn></msub><mo linebreak='badbreak'>&#8722;</mo><mrow><mo>(</mo><mn>1</mn><mo linebreak='badbreak'>+</mo><mi>k</mi><mo>)</mo></mrow><msubsup><mi>&#945;</mi><mrow><mn>1</mn></mrow><mn>2</mn></msubsup><mo>]</mo></mrow><mi>z</mi><mo linebreak='goodbreak'>+</mo><mrow><mo>(</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mo linebreak='badbreak'>+</mo><msub><mi>&#945;</mi><mn>2</mn></msub><mi>z</mi><mo>)</mo></mrow><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub></mrow></math>, with <math altimg='si18.svg'><mrow><mi>k</mi><mo>&#8801;</mo><msub><mi>&#946;</mi><mn>1</mn></msub><mo linebreak='goodbreak'>&#8722;</mo><msub><mi>&#946;</mi><mn>0</mn></msub></mrow></math>, whose two branches are distinguished by whether the anomalous-dimension feedback is active: the massless branch ( <math altimg='si30.svg'><mrow><msub><mi>&#945;</mi><mi>i</mi></msub><mo linebreak='goodbreak'>=</mo><mn>0</mn></mrow></math>) integrates to the square-root <italic>F</italic> <inf>0</inf> of Ref. [3], while the massive branch is an Abel equation of the second kind that admits no elementary closed form, the obstruction being controlled by <italic>&#945;</italic> <inf>1</inf> through <math altimg='si59.svg'><mrow><mi>d</mi><mi mathvariant='script'>S</mi><mo linebreak='goodbreak'>/</mo><mi>d</mi><mi>z</mi><mo linebreak='goodbreak'>=</mo><msub><mi>&#945;</mi><mn>1</mn></msub><mi>c</mi><mo linebreak='goodbreak'>/</mo><msup><mi>&#981;</mi><mrow><mo>&#8242;</mo><mn>2</mn></mrow></msup></mrow></math> with <math altimg='si60.svg'><mrow><mi>c</mi><mo linebreak='goodbreak'>=</mo><mi>k</mi><mrow><mo>(</mo><mi>k</mi><mo linebreak='badbreak'>+</mo><mn>1</mn><mo>)</mo></mrow><msubsup><mi>&#945;</mi><mrow><mn>1</mn></mrow><mn>2</mn></msubsup></mrow></math> and a non-constant Chini invariant. The Lambert- <italic>W</italic> structure nonetheless survives in renormalized form, <math altimg='si61.svg'><mrow><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mo linebreak='goodbreak'>=</mo><mo>&#8722;</mo><mn>1</mn><mo linebreak='goodbreak'>+</mo><msup><mi>&#956;</mi><mrow><mo>&#8722;</mo><mn>1</mn></mrow></msup><msub><mi>W</mi><mn>0</mn></msub><mrow><mo>(</mo><mi>&#956;</mi><mspace width='0.16em' /><msup><mi>e</mi><mrow><mstyle mathvariant='normal'><mi>&#934;</mi></mstyle><mo>(</mo><mi>z</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow></mrow></math> with a self-consistently dressed argument <math altimg='si62.svg'><mrow><msup><mstyle mathvariant='normal'><mi>&#934;</mi></mstyle><mo>&#8242;</mo></msup><mo linebreak='goodbreak'>=</mo><msup><mi>&#981;</mi><mo>&#8242;</mo></msup><mo linebreak='goodbreak'>/</mo><mrow><mo>(</mo><mi>k</mi><mo linebreak='badbreak'>+</mo><mn>1</mn><mo>)</mo></mrow><mo linebreak='goodbreak'>&#8722;</mo><msubsup><mi>&#945;</mi><mrow><mn>1</mn></mrow><mn>2</mn></msubsup><mi>z</mi><mo linebreak='goodbreak'>/</mo><mrow><mo>(</mo><mn>1</mn><mo linebreak='badbreak'>+</mo><msub><mi mathvariant='script'>G</mi><mn>0</mn></msub><mo>)</mo></mrow></mrow></math>, reducing to the explicit closed form in the massless limit. Predictions for all <italic>O</italic>( <italic>N</italic>) universality classes follow from the recursion, and a direct comparison with conventional fixed-order perturbation theory shows the renormalization-scale band of the latter contracting toward the single scheme-free SF-QFT value at every loop order.

Canada

Farrukh A. Chishtie

Exact scheme-independent generating function for the massive <italic>O</italic>( <italic>N</italic>) <italic>&#981;</italic> 4 two-point function in scale-factorized quantum field theory