Perturbation Theory of Transformed Quantum Fields

Paul-Hermann, Balduf,

doi:10.1007/s11040-020-09357-z

Cited by 11 publications

(18 citation statements)

References 21 publications

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“…Field diffeomorphisms. Diffeomorphisms of free fields have been studied in detail recently [42,43,2,50]. In this section we list some central results.…”

Section: Example 13 (4-valent Tree Amplitudementioning

confidence: 99%

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Propagator-cancelling scalar fields

Balduf

2021

Preprint

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We examine a large class of scalar quantum field theories where vertices are able to cancel adjacent propagators. These theories are obtained as diffeomorphisms of the field variable of a free field. Their connected correlations functions can be computed efficiently with an algebraic procedure dubbed connected perspective in earlier work. Specifically,(1) We compare the Feynman rules in momentum-space to their position-space counterparts, they agree. This is a a-posteriori justification for the Feynman rules of the connected perspective.(2) We extend the connected perspective to also incorporate counterterm vertices.(3) We examine a specific choice of diffeomorphism that assumes the vertices at different valence proportional to each other. We find that these relations lead to various simplifications.(4) We perturbatively compute the connected 2-point function to all orders for an arbitrary diffeomorphism of a massless field. We thereby give a systematic perturbative derivation of the exponential superpropagator known from the literature.(5) We compute a number of one-and two-loop counterterms. We find that for the specific diffeomorphism, the one-loop counterterms arise from the bare vertices by a non-linear redefinition of the momentum.(6) We show that every diffeomorphism fulfils a set of infinitely many Slavnov-Taylor-like identities which express diffeomorphism-invariance of the S-matrix at loop-level.Finally, we comment on surprising similarities between the structure of propagator cancelling scalar theories and gauge theories .

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“…Field diffeomorphisms. Diffeomorphisms of free fields have been studied in detail recently [42,43,2,50]. In this section we list some central results.…”

Section: Example 13 (4-valent Tree Amplitudementioning

confidence: 99%

“…6.3] or non-local interactions [16]. The invariance of the S-matrix has been demonstrated in the path integral formalism [53,1] and also using graph theory [65,42,43,2,50,49]. The behaviour of correlation functions of the field diffeomorphism has never been addressed in detail to the author's knowledge apart from the statement that they vanish onshell.…”

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confidence: 99%

Propagator-cancelling scalar fields

Balduf

2021

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“…These combinatorial Green functions then satisfy certain functional equations that are combinatorial version of the Dyson-Schwinger equations (DSEs) of the theory. This theory has been well-developed for the usual Green functions [5,14] and has found use in other areas of mathmatics [40,41] and physics [42,43]. Analogous results apply when we are working with cuts.…”

Section: Dses In Core and Quotient Hopf Algebrasmentioning

confidence: 83%

Algebraic Interplay between Renormalization and Monodromy

Kreimer¹,

Yeats²

2021

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We investigate combinatorial and algebraic aspects of the interplay between renormalization and monodromies for Feynman amplitudes. We clarify how extraction of subgraphs from a Feynman graph interacts with putting edges onshell or with contracting them to obtain reduced graphs. Graph by graph this leads to a study of cointeracting bialgebras. One bialgebra comes from extraction of subgraphs and hence is needed for renormalization. The other bialgebra is an incidence bialgebra for edges put either onor offshell. It is hence related to the monodromies of the multivalued function to which a renormalized graph evaluates. Summing over infinite series of graphs, consequences for Green functions are derived using combinatorial Dyson-Schwinger equations. Contents 2.1. Graphs 2.2. Cuts 2.3. Sub-and co-graphs 3. Hopf algebras 3.1. The core Hopf algebra H core 3.2. Quotient Hopf algebras 3.3. The Hopf algebra H pC 3.4. Pre-Cutkosky Necklaces 3.5. Extension to a core coproduct for pairs (Γ, F ) 3.6. Counting spanning trees 4. Coactions 4.1. H pC → H core ⊗ H pC 4.2. Cointeracting bialgebras 4.3. Sector Decomposition 4.4. Two more coactions 5. DSEs in core and quotient Hopf algebras 5.1. Dyson-Schwinger and cut Dyson-Schwinger setup 5.2. Assembly maps vs Hochschild 1-cocycles 5.3. Core, no cuts 5.4. Core, with cuts 5.5. The coaction on Green functions 5.6. Pairs (Γ, F ) or H C 1

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“…with F ∼ O(1/Λ) a polynomial (local) function of the fields and their derivatives multiplied by appropriate powers of 1/Λ such that the mass dimension of F is equal to that of φ in D = 4 spacetime dimensions. See for instance [107] for a detailed discussion on field redefinitions in EFT and [108] for a recent mathematical perspective. To ensure that F carries the correct factors that preserve normalization of bare operators in (2.6) it will be convenient to also include the power of g b Z 2 that gives F the mass dimension of φ in D = 4−2ε.…”

Section: Eft Lagrangian and Renormalizationmentioning

confidence: 99%

Renormalization and non-renormalization of scalar EFTs at higher orders

Cao

Herzog

Melia

et al. 2021

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We renormalize massless scalar effective field theories (EFTs) to higher loop orders and higher orders in the EFT expansion. To facilitate EFT calculations with the R* renormalization method, we construct suitable operator bases using Hilbert series and related ideas in commutative algebra and conformal representation theory, including their novel application to off-shell correlation functions. We obtain new results ranging from full one loop at mass dimension twelve to five loops at mass dimension six. We explore the structure of the anomalous dimension matrix with an emphasis on its zeros, and investigate the effects of conformal and orthonormal operators. For the real scalar, the zeros can be explained by a 'non-renormalization' rule recently derived by Bern et al. For the complex scalar we find two new selection rules for mixing n-and (n − 2)-field operators, with n the maximal number of fields at a fixed mass dimension. The first appears only when the (n − 2)-field operator is conformal primary, and is valid at one loop. The second appears in more generic bases, and is valid at three loops. Finally, we comment on how the Hilbert series we construct may be used to provide a systematic enumeration of a class of evanescent operators that appear at a particular mass dimension in the scalar EFT.

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Perturbation Theory of Transformed Quantum Fields

Cited by 11 publications

References 21 publications

Propagator-cancelling scalar fields

Propagator-cancelling scalar fields

Algebraic Interplay between Renormalization and Monodromy

Renormalization and non-renormalization of scalar EFTs at higher orders

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