{"title":"Exact results for scaling dimensions of neutral operators in scalar conformal field theories","authors":"Oleg Antipin, Jahmall Bersini, Francesco Sannino","doi":"10.1103/physrevd.111.l041701","DOIUrl":null,"url":null,"abstract":"We determine the scaling dimension Δ</a:mi>n</a:mi></a:msub></a:math> for the class of composite operators <d:math xmlns:d=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><d:msup><d:mi>ϕ</d:mi><d:mi>n</d:mi></d:msup></d:math> in the <f:math xmlns:f=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><f:mi>λ</f:mi><f:msup><f:mi>ϕ</f:mi><f:mn>4</f:mn></f:msup></f:math> theory in <h:math xmlns:h=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><h:mi>d</h:mi><h:mo>=</h:mo><h:mn>4</h:mn><h:mo>−</h:mo><h:mi>ε</h:mi></h:math> taking the double scaling limit <j:math xmlns:j=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><j:mi>n</j:mi><j:mo stretchy=\"false\">→</j:mo><j:mi>∞</j:mi></j:math> and <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><m:mi>λ</m:mi><m:mo stretchy=\"false\">→</m:mo><m:mn>0</m:mn></m:math> with fixed <p:math xmlns:p=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><p:mi>λ</p:mi><p:mi>n</p:mi></p:math> via a semiclassical approach. Our results resum the leading power of <r:math xmlns:r=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><r:mi>n</r:mi></r:math> at any loop order. In the small <t:math xmlns:t=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><t:mi>λ</t:mi><t:mi>n</t:mi></t:math> regime we reproduce the known diagrammatic results and predict the infinite series of higher-order terms. For intermediate values of <v:math xmlns:v=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><v:mi>λ</v:mi><v:mi>n</v:mi></v:math> we find that <x:math xmlns:x=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><x:msub><x:mi mathvariant=\"normal\">Δ</x:mi><x:mi>n</x:mi></x:msub><x:mo>/</x:mo><x:mi>n</x:mi></x:math> increases monotonically approaching a (</ab:mo>λ</ab:mi>n</ab:mi>)</ab:mo>1</ab:mn>/</ab:mo>3</ab:mn></ab:mrow></ab:msup></ab:math> behavior in the <eb:math xmlns:eb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><eb:mi>λ</eb:mi><eb:mi>n</eb:mi><eb:mo stretchy=\"false\">→</eb:mo><eb:mi>∞</eb:mi></eb:math> limit. We further generalize our results to neutral operators in the <hb:math xmlns:hb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><hb:msup><hb:mi>ϕ</hb:mi><hb:mn>4</hb:mn></hb:msup></hb:math> in <jb:math xmlns:jb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><jb:mi>d</jb:mi><jb:mo>=</jb:mo><jb:mn>4</jb:mn><jb:mo>−</jb:mo><jb:mi>ε</jb:mi></jb:math>, <lb:math xmlns:lb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><lb:msup><lb:mi>ϕ</lb:mi><lb:mn>3</lb:mn></lb:msup></lb:math> in <nb:math xmlns:nb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><nb:mi>d</nb:mi><nb:mo>=</nb:mo><nb:mn>6</nb:mn><nb:mo>−</nb:mo><nb:mi>ε</nb:mi></nb:math>, and <pb:math xmlns:pb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><pb:msup><pb:mi>ϕ</pb:mi><pb:mn>6</pb:mn></pb:msup></pb:math> in <rb:math xmlns:rb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><rb:mi>d</rb:mi><rb:mo>=</rb:mo><rb:mn>3</rb:mn><rb:mo>−</rb:mo><rb:mi>ε</rb:mi></rb:math> theories with <tb:math xmlns:tb=\"http://www.w3.org/1998/Math/MathML\" display=\"inline\"><tb:mi>O</tb:mi><tb:mo stretchy=\"false\">(</tb:mo><tb:mi>N</tb:mi><tb:mo stretchy=\"false\">)</tb:mo></tb:math> symmetry. <jats:supplementary-material> <jats:copyright-statement>Published by the American Physical Society</jats:copyright-statement> <jats:copyright-year>2025</jats:copyright-year> </jats:permissions> </jats:supplementary-material>","PeriodicalId":20167,"journal":{"name":"Physical Review D","volume":"66 1","pages":""},"PeriodicalIF":5.0000,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review D","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevd.111.l041701","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
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Abstract
We determine the scaling dimension Δn for the class of composite operators ϕn in the λϕ4 theory in d=4−ε taking the double scaling limit n→∞ and λ→0 with fixed λn via a semiclassical approach. Our results resum the leading power of n at any loop order. In the small λn regime we reproduce the known diagrammatic results and predict the infinite series of higher-order terms. For intermediate values of λn we find that Δn/n increases monotonically approaching a (λn)1/3 behavior in the λn→∞ limit. We further generalize our results to neutral operators in the ϕ4 in d=4−ε, ϕ3 in d=6−ε, and ϕ6 in d=3−ε theories with O(N) symmetry. Published by the American Physical Society2025
期刊介绍：
Physical Review D (PRD) is a leading journal in elementary particle physics, field theory, gravitation, and cosmology and is one of the top-cited journals in high-energy physics.
PRD covers experimental and theoretical results in all aspects of particle physics, field theory, gravitation and cosmology, including:
Particle physics experiments,
Electroweak interactions,
Strong interactions,
Lattice field theories, lattice QCD,
Beyond the standard model physics,
Phenomenological aspects of field theory, general methods,
Gravity, cosmology, cosmic rays,
Astrophysics and astroparticle physics,
General relativity,
Formal aspects of field theory, field theory in curved space,
String theory, quantum gravity, gauge/gravity duality.
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