Generating-function method for tensor products

Bégin, Luc; Cummins, Chris; Mathieu, Pierre

doi:10.1063/1.1286511

Cited by 11 publications

(47 citation statements)

References 22 publications

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“…with the gradings ofÊ 0 , · · · ,Ê 3 respectively given by (1, 0, 0, 0), (1, 1, 1, 0), (1,1,0,1) and (1, 0, 1, 1) for the ordering …”

Section: Fusion Rulesmentioning

confidence: 99%

“…the Ω projection described in section 3 of [1]). After the projection, all the variables A i are set equal to 1.…”

Section: The Generating Function For Su(3) Fusion Rulesmentioning

confidence: 99%

“…This uses the technique of 'composition' of generating functions described previously in section 2.3 of [1].…”

Section: Appendix a Independent Verifications Of The Fusion Generatimentioning

confidence: 99%

“…also the introduction of [1]; for a review of conformal field theory, and in particular fusion rules, see [2]). Even in theories with a Lie group symmetry, the so-called Wess-Zumino-Witten (WZW) models, an intrinsic conformal-field theoretical characterisation is unavoidable.…”

Section: Introductionmentioning

confidence: 99%

“…We first recall some results obtained in [1]. The appropriate basis for the description of sp(4) tensor products reads [20]:…”

Section: The Sp(4) Generating Functionmentioning

confidence: 99%

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Generating-function method for fusion rules

Bégin

Cummins

Mathieu

2000

Journal of Mathematical Physics

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This is the second of two articles devoted to an exposition of the generating-function method for computing fusion rules in affine Lie algebras. The present paper focuses on fusion rules, using the machinery developed for tensor products in the companion article. Although the Kac-Walton algorithm provides a method for constructing a fusion generating function from the corresponding tensor-product generating function, we describe a more powerful approach which starts by first defining the set of fusion elementary couplings from a natural extension of the set of tensor-product elementary couplings.A set of inequalities involving the level are derived from this set using Farkas' lemma.These inequalities, taken in conjunction with the inequalities defining the tensor products, define what we call the fusion basis. Given this basis, the machinery of our previous paper may be applied to construct the fusion generating function. New generating functions for sp(4) and su(4), together with a closed form expression for their threshold levels are presented.05/99, revised 04/00 (arXiv:hepth/0005002) 1

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“…with the gradings ofÊ 0 , · · · ,Ê 3 respectively given by (1, 0, 0, 0), (1, 1, 1, 0), (1,1,0,1) and (1, 0, 1, 1) for the ordering …”

Section: Fusion Rulesmentioning

confidence: 99%

“…the Ω projection described in section 3 of [1]). After the projection, all the variables A i are set equal to 1.…”

Section: The Generating Function For Su(3) Fusion Rulesmentioning

confidence: 99%

“…This uses the technique of 'composition' of generating functions described previously in section 2.3 of [1].…”

Section: Appendix a Independent Verifications Of The Fusion Generatimentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…We first recall some results obtained in [1]. The appropriate basis for the description of sp(4) tensor products reads [20]:…”

Section: The Sp(4) Generating Functionmentioning

confidence: 99%

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Generating-function method for fusion rules

Bégin

Cummins

Mathieu

2000

Journal of Mathematical Physics

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Low-derivative operators of the Standard Model effective field theory via Hilbert series methods

Lehman

Martin

2016

J. High Energ. Phys.

129

169

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In this work, we explore an extension of Hilbert series techniques to count operators that include derivatives. For sufficiently low-derivative operators, we conjecture an algorithm that gives the number of invariant operators, properly accounting for redundancies due to the equations of motion and integration by parts. Specifically, the conjectured technique can be applied whenever there is only one Lorentz invariant for a given partitioning of derivatives among the fields. At higher numbers of derivatives, equation of motion redundancies can be removed, but the increased number of Lorentz contractions spoils the subtraction of integration by parts redundancies. While restricted, this technique is sufficient to automatically recreate the complete set of invariant operators of the Standard Model effective field theory for dimensions 6 and 7 (for arbitrary numbers of flavors). At dimension 8, the algorithm does not automatically generate the complete operator set; however, it suffices for all but five classes of operators. For these remaining classes, there is a well defined procedure to manually determine the number of invariants. Assuming our method is correct, we derive a set of 535 dimension-8 N f = 1 operators.

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On the impact of dimension-eight SMEFT operators on Higgs measurements

Hays

Martin

Sanz

et al. 2019

J. High Energ. Phys.

110

131

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Using the production of a Higgs boson in association with a W boson as a test case, we assess the impact of dimension-8 operators within the context of the Standard Model Effective Field Theory. Dimension-8-SM-interference and dimension-6-squared terms appear at the same order in an expansion in 1/Λ, hence dimension-8 effects can be treated as a systematic uncertainty on the new physics inferred from analyses using dimension-6 operators alone. To study the phenomenological consequences of dimension-8 operators, one must first determine the complete set of operators that can contribute to a given process. We accomplish this through a combination of Hilbert series methods, which yield the number of invariants and their field content, and a step-by-step recipe to convert the Hilbert series output into a phenomenologically useful format. The recipe we provide is general and applies to any other process within the dimension ≤ 8 Standard Model Effective Theory. We quantify the effects of dimension-8 by turning on one dimension-6 operator at a time and setting all dimension-8 operator coefficients to the same magnitude. Under this procedure and given the current accuracy on σ(pp → h W + ), we find the effect of dimension-8 operators on the inferred new physics scale to be small, O(few %), with some variation depending on the relative signs of the dimension-8 coefficients and on which dimension-6 operator is considered. The impact of the dimension-8 terms grows as σ(pp → h W + ) is measured more accurately or (more significantly) in high-mass kinematic regions. We provide a FeynRules implementation of our operator set to be used for further more detailed analyses.

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Generating-function method for tensor products

Cited by 11 publications

References 22 publications

Generating-function method for fusion rules

Generating-function method for fusion rules

Low-derivative operators of the Standard Model effective field theory via Hilbert series methods

On the impact of dimension-eight SMEFT operators on Higgs measurements

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