Consistency of Dalgarno–Lewis perturbation theory for confining two-bodys ystems: second-order corrections in the Cornell potential

Consistency of Dalgarno–Lewis perturbation theory for confining two-bodys ystems: second-order corrections in the Cornell potential | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Consistency of Dalgarno–Lewis perturbation theory for confining two-bodys ystems: second-order corrections in the Cornell potential Krishna Kingkar Pathak This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8655369/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 05 Apr, 2026 Read the published version in Few-Body Systems → Version 1 posted 7 You are reading this latest preprint version Abstract Confining two-body systems provide a useful setting for examining the internal consistency ofperturbative treatments of bound-state wavefunctions. In this work, we investigate theapplicability of Dalgarno--Lewis perturbation theory beyond first order within aCornell-type potential,\( V(r)=br-\tfrac{4\alpha_s}{3r}+c \),treating the linear confinement term as the parent Hamiltonian and incorporating the Coulombicinteraction perturbatively so as to retain analytic control over the solutions. Going beyond standard first-order implementations, we carry out a systematic second-orderDalgarno--Lewis correction to the Airy-function ground state and analyze its impact onwavefunction moments. We show that observables involving higher spatial moments are notstabilized at first order, indicating a breakdown of first-order Dalgarno--Lewis theory for suchquantities in confining two-body systems. Inclusion of second-order contributions is thereforerequired for internal consistency. As an explicit application, we evaluate effective overlap functions, their slopes and curvatures,as well as geometric and electromagnetic radii for representative heavy--light mesons($D$, $D_s$, $B$, and $B_s$). While the analysis is model dependent and does not rely on strictheavy-quark universality, the resulting magnitudes and systematic trends are compatible withthose obtained in relativistic quark models, QCD sum-rule approaches, and continuumDyson--Schwinger/Bethe--Salpeter studies. The present results provide semi-analytic benchmarksthat clarify the role of higher-order wavefunction corrections in few-body bound-state problems. PACS numbers: 12.39.Pn, 12.39.Jh, 14.40.Lb, 14.40.Nd heavy–light mesons hadron structure quark potential models Cornell potential Dalgarno–Lewis perturbation theory Isgur–Wise function charge radius Full Text Additional Declarations No competing interests reported. Supplementary Files SIFBS.pdf Cite Share Download PDF Status: Published Journal Publication published 05 Apr, 2026 Read the published version in Few-Body Systems → Version 1 posted Editorial decision: Revision requested 02 Mar, 2026 Reviews received at journal 02 Mar, 2026 Reviewers agreed at journal 28 Jan, 2026 Reviewers invited by journal 27 Jan, 2026 Editor assigned by journal 22 Jan, 2026 Submission checks completed at journal 22 Jan, 2026 First submitted to journal 21 Jan, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8655369","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":581838739,"identity":"b1f3230a-21cd-4d34-bac4-3b3b77c039e9","order_by":0,"name":"Krishna Kingkar Pathak","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2ElEQVRIiWNgGAWjYBACCR7GBgaGAwwM/CBeQgEpWiQbQFoMiNICIoFaDA6AGMRokew53Pbwyxm7fOPzqxM/PDBgkOcXO4BfizRvY7uxzI1ky2033m6WADrMcObsBPxa5PgZ26QlPjAbmN04uwGkJcHgNnFa6g2MZ5zd/IMoLUCHtUl+uHHYwIC/dxtxtkj2HGyTZjhz3EDiBu82iwQDCcJ+kTiT/kzyx7FqA/7+s5tv/qiwkeeXJqAFBJjBcSMBVilBWDkIMP4AkfwHiFM9CkbBKBgFIw8AABq/RV+gR3AbAAAAAElFTkSuQmCC","orcid":"","institution":"Arya Vidyapeeth College","correspondingAuthor":true,"prefix":"","firstName":"Krishna","middleName":"Kingkar","lastName":"Pathak","suffix":""}],"badges":[],"createdAt":"2026-01-21 05:25:12","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8655369/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8655369/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00601-026-02041-y","type":"published","date":"2026-04-05T15:58:35+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":106343750,"identity":"58baee6d-04fd-4197-bf2e-2262d8798706","added_by":"auto","created_at":"2026-04-07 16:08:48","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":262148,"visible":true,"origin":"","legend":"","description":"","filename":"FBSmain.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8655369/v1_covered_374ff99c-9981-424c-9475-fcc2615d2d80.pdf"},{"id":101380394,"identity":"64bc6d60-666e-4897-a28b-7bb30b06b819","added_by":"auto","created_at":"2026-01-29 06:05:50","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":206548,"visible":true,"origin":"","legend":"","description":"","filename":"SIFBS.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8655369/v1/5981ea04a05241fba375acce.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Consistency of Dalgarno–Lewis perturbation theory for confining two-bodys ystems: second-order corrections in the Cornell potential","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"few-body-systems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"fbsy","sideBox":"Learn more about [Few-Body Systems](http://link.springer.com/journal/601)","snPcode":"601","submissionUrl":"https://submission.nature.com/new-submission/601/3","title":"Few-Body Systems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"heavy–light mesons, hadron structure, quark potential models, Cornell potential, Dalgarno–Lewis perturbation theory, Isgur–Wise function, charge radius","lastPublishedDoi":"10.21203/rs.3.rs-8655369/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8655369/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eConfining two-body systems provide a useful setting for examining the internal consistency ofperturbative treatments of bound-state wavefunctions. In this work, we investigate theapplicability of Dalgarno--Lewis perturbation theory beyond first order within aCornell-type potential,\\( V(r)=br-\\tfrac{4\\alpha_s}{3r}+c \\),treating the linear confinement term as the parent Hamiltonian and incorporating the Coulombicinteraction perturbatively so as to retain analytic control over the solutions. Going beyond standard first-order implementations, we carry out a systematic second-orderDalgarno--Lewis correction to the Airy-function ground state and analyze its impact onwavefunction moments. We show that observables involving higher spatial moments are notstabilized at first order, indicating a breakdown of first-order Dalgarno--Lewis theory for suchquantities in confining two-body systems. Inclusion of second-order contributions is thereforerequired for internal consistency. As an explicit application, we evaluate effective overlap functions, their slopes and curvatures,as well as geometric and electromagnetic radii for representative heavy--light mesons($D$, $D_s$, $B$, and $B_s$). While the analysis is model dependent and does not rely on strictheavy-quark universality, the resulting magnitudes and systematic trends are compatible withthose obtained in relativistic quark models, QCD sum-rule approaches, and continuumDyson--Schwinger/Bethe--Salpeter studies. The present results provide semi-analytic benchmarksthat clarify the role of higher-order wavefunction corrections in few-body bound-state problems.\u003c/p\u003e\n\u003cp\u003ePACS numbers: 12.39.Pn, 12.39.Jh, 14.40.Lb, 14.40.Nd\u003c/p\u003e","manuscriptTitle":"Consistency of Dalgarno–Lewis perturbation theory for confining two-bodys ystems: second-order corrections in the Cornell potential","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-29 06:05:45","doi":"10.21203/rs.3.rs-8655369/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-03-02T08:32:22+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-03-02T07:00:43+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"97322548459111114511781962285413221542","date":"2026-01-28T13:32:28+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-27T13:15:37+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-22T11:40:35+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-01-22T11:40:15+00:00","index":"","fulltext":""},{"type":"submitted","content":"Few-Body Systems","date":"2026-01-21T05:04:43+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"few-body-systems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"fbsy","sideBox":"Learn more about [Few-Body Systems](http://link.springer.com/journal/601)","snPcode":"601","submissionUrl":"https://submission.nature.com/new-submission/601/3","title":"Few-Body Systems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"f77f004f-e1c4-4559-8be8-4ffbec2ca638","owner":[],"postedDate":"January 29th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2026-04-07T16:04:39+00:00","versionOfRecord":{"articleIdentity":"rs-8655369","link":"https://doi.org/10.1007/s00601-026-02041-y","journal":{"identity":"few-body-systems","isVorOnly":false,"title":"Few-Body Systems"},"publishedOn":"2026-04-05 15:58:35","publishedOnDateReadable":"April 5th, 2026"},"versionCreatedAt":"2026-01-29 06:05:45","video":"","vorDoi":"10.1007/s00601-026-02041-y","vorDoiUrl":"https://doi.org/10.1007/s00601-026-02041-y","workflowStages":[]},"version":"v1","identity":"rs-8655369","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8655369","identity":"rs-8655369","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}