Integrating out heavy scalars with modied EOMs matching computation of dimension-eight SMEFT coecients

2025-05-05 0 0 838.28KB 32 页 10玖币

侵权投诉

Integrating out heavy scalars with modiﬁed EOMs:

matching computation of dimension-eight SMEFT

coeﬃcients

Upalaparna Banerjee, Joydeep Chakrabortty,1Christoph Englert,2Shakeel Ur

Rahaman,1and Michael Spannowsky3

1Indian Institute of Technology Kanpur, Kalyanpur, Kanpur 208016, Uttar Pradesh, India

2School of Physics & Astronomy, University of Glasgow, Glasgow G12 8QQ, U.K.

Institute for Particle Physics Phenomenology, Department of Physics, Durham University, Durham

DH1 3LE, U.K.

E-mail: upalab@iitk.ac.in, joydeep@iitk.ac.in,

christoph.englert@glasgow.ac.uk, shakel@iitk.ac.in,

michael.spannowsky@durham.ac.uk

Abstract: The shift in focus towards searches for physics beyond the Standard Model (SM)

employing model-independent Eﬀective Field Theory (EFT) methods necessitates a rigorous

approach to matching to guarantee the validity of the obtained results and constraints. The

limits on the leading dimension-six EFT eﬀects can be rather inaccurate for LHC searches

that suﬀer from large uncertainties while exploring an extensive energy range. Similarly,

precise measurements can, in principle, test the subleading eﬀects of the operator expansion.

In this work, we present an algorithmic approach to automatise matching computations

for dimension-eight operators for generic scalar extensions with proper implementation of

equations of motion. We devise a step-by-step procedure to obtain the dimension-eight

Wilson coeﬃcients (WCs) in a non-redundant basis to arrive at complete matching results.

We apply this formalism to a range of scalar extensions of the SM and provide tree-level and

loop-suppressed results. Finally, we discuss the relevance of the dimension-eight operators

for a range of phenomenological analyses, particularly focusing on Higgs and electroweak

physics.

arXiv:2210.14761v2 [hep-ph] 5 Feb 2023

Contents

1 Introduction 1

2 Complete matching at dimension-eight 3

2.1 Removing redundancies: Field Redeﬁnition & Higgs Field EOM 5

2.2 Impact of dimension-six structures on dimension-eight coeﬃcients 7

2.3 Removing redundancies at dimension-eight 8

3 Cross-validation of the method 9

4 Example models 12

4.1 Complex Triplet Scalar 13

4.2 General Two Higgs Doublet Model 14

4.3 Complex quartet Scalar (hypercharge Y = 3/2) 16

4.4 Real Singlet Scalar Model 17

4.5 Scalar Leptoquark 18

5 Impact of dimension-eight operators on BSM scenarios 20

5.1 Electroweak precision observables 20

5.2 Higgs signal strength measurements 21

5.3 Vector boson scattering measurements 22

6 Conclusions 22

A Relevant operator structures 24

B Renormalisable SM Lagrangian and EOM 24

1 Introduction

Searches for physics beyond the Standard Model (BSM) chieﬂy performed at the Large

Hadron Collider (LHC) have, so far, not revealed any signiﬁcant deviation from the Standard

Model (SM) predictions. This is puzzling, on the one hand, given the SM’s plethora of known

ﬂaws and shortcomings. On the other hand, these ﬁndings have motivated the application

of model-independent techniques employing Eﬀective Field Theory (EFT) [

] to LHC

data. The EFT approach breaks away from the assumption of concrete model-dependent

correlations, thus opening up the possibility of revealing new (and perhaps non-canonical)

BSM interactions through a holistic approach to data correlation interpretation. The

inherent assumption of such an approach is that there is a signiﬁcant mass gap between the

BSM spectrum and the (process-dependent) characteristic energy scale at which the LHC

– 1 –

operates to “integrate out” BSM states to obtain a low energy eﬀective description that is

determined by the SM’s particle and symmetry content.

Eﬀorts to apply EFT to the multi-scale processes of the LHC environment have

received considerable interest recently, reaching from theory-led proof-of-principle ﬁts to

LHC data [

–

] (with a history of almost a decade) over the adoption of these techniques

by the LHC experiments (e.g. [

] for recent examples), to perturbative improvements

of the formalism [

–

]. In doing so, most attention has been devoted to SMEFT at

dimension-six level [22]

L=LSM +X

Λ2Oi.(1.1)

While EFT is a formidable tool to put correlations at the forefront of BSM searches,

the signiﬁcant energy coverage of the LHC can lead to blurred sensitivity estimates even

in instances when Eq.

(1.1)

is a suﬃciently accurate expansion. When pushing the cut-oﬀ

scale Λ to large values, the experimental sensitivity to deviations from the SM can be too

small to yield perturbatively meaningful or relevant constraints when matched to concrete

UV scenarios (see e.g. [

]). In contrast, dimension-eight contributions can be sizeable when

the new physics cut-oﬀ Λ is comparably low in the case of more signiﬁcant BSM signals at

the LHC. To understand the ramiﬁcations for concrete UV models, it is then important

to (i) have a ﬂexible approach to mapping out the dimension-eight interactions and (ii)

gauge the importance of dimension-eight contributions relative to those of dimension-six to

quantitatively assess the error of the (potential) dimension-eight truncation.

A common bottleneck in constructing EFT interactions is removing redundancies. This

is historically evidenced by the emergence of the so-called Warsaw basis [

]. Equations

of motion are typically considered in eliminating redundant operators. Still, they are

not identical to general (non-linear) ﬁeld redeﬁnitions, which are the actual redundant

parameters of the ﬁeld theory [

–

]. When truncating a given operator dimension, this

can be viewed as a scheme-dependence not too unfamiliar from renormalisable theories,

however, with less controlled side-eﬀects when the new physics scale is comparably low.

Additional operator structures need to be included to elevate classical equations of motion

to ﬁeld re-deﬁnitions [

] to achieve a consistent classiﬁcation at the dimension-eight

level.

In this work, we devise a generic approach to this issue which enables us to provide a

complete framework to match any dimension-eight structure that emerges in the process

of integrating-out a heavy non-SM scalar and obtaining the form of the WCs. Along the

way of systematically re-organising the operators into a non-redundant basis, resembling

the one discussed in Refs. [

], we show that removing the higher-derivative operators

produced at the dimension-six level itself can induce a non-negligible eﬀect on dimension-

eight matching coeﬃcients along with the direct contribution to the same which can be

computed following the familiar methodologies of the Covariant Derivative Expansion

(CDE) [

] of the path integral [

–

], or diagrammatic approach [

]. Finally, it is

worth mentioning that, even though the one-loop eﬀective action at dimension eight is yet

to be formulated, it is possible to receive equally suppressed, loop-induced corrections from

– 2 –

the dimension-six coeﬃcients computed precisely at one-loop. These can present themselves

as the leading order contributions for the WCs, which generally appear at one-loop.

This paper is organised as follows: in Sec. 2, we discuss the implementation of the Higgs

ﬁeld equation of motion and study its equivalence with ﬁeld redeﬁnitions. This gives rise to

the desired dimension-eight operator structures after removing redundancies (Sec. 2.3). Our

approach is tested and validated against available results for the real triplet scalar extension

in Sec. 3. In Sec. 4, the matching coeﬃcients are presented explicitly considering a range

of scalar extensions of the SM. Finally, the signiﬁcance of the dimension-eight operators

is analysed based on observables in a model-dependent manner in Sec. 5. We conclude in

Sec. 6.

2 Complete matching at dimension-eight

We start by studying the structures of the higher-dimensional operators that can arise from

heavy scalar extensions of the SM generically once the heavy ﬁeld (Φ) is integrated out.

The most generalised structure of the renormalised Lagrangian involving heavy scalars can

be written as [32,39]:

LΦ⊃Φ†(P2−m2−U(x)) Φ + (Φ†B(x) + h.c.) + 1

4λΦ(Φ†Φ)2.(2.1)

Here,

(

) and

(

) contain the interactions that are quadratic and linear in Φ, respectively,

and only involve the lighter degrees of freedom. Once Φ is integrated out, we obtain a tower

of operators that can be arranged according to their canonical dimension. It is important to

note that the operators generated in this process might not be independent. Depending on

phenomenological considerations, several sets of operators are deﬁned in the literature. A

set of dimension-six operators was prescribed in Ref. [

]. It was improved by systematically

removing the redundant structures and promoting it to form a complete non-redundant basis

in Ref. [

]

, popularly known as the Warsaw basis. There is another set of operators known

as the Green’s set [

–

], which is over-complete. The operators here are independent

under the Fierz identities and integration by parts but otherwise redundant on account

of equations of motion

. This source of redundancy contributes to higher dimensional

operators. In this paper, we use the Mathematica package

CoDEx

[

] to generate WCs of

the operators in the SILH set [

] up to one-loop, including the relevant redundant

terms. Since we are interested in the corrections to the dimension-eight coeﬃcients resulting

from the dimension-six redundant structures, we recast the SILH operators into Green’s

set-like structures

to single out redundant and non-redundant operators using the following

equations:

QH≡1

2∂µ(H†H)∂µ(H†H) = −1

2(H†H)(H†H),

1A minimal set of four-fermionic operators is also constructed in Ref. [41].

Here we are being ambiguous about the use of the equation of motion or ﬁeld redeﬁnition in removing

the redundancies, see Ref. [24,29] for more details.

We call it “Green’s set -like structures” because we diﬀer in some redundant operator structures as

deﬁned in Ref. [42], see appendix Afor more details.

– 3 –

QT≡1

2H†←→

DµHH†←→

DµH=−2 (DµH†H)(H†DµH)−1

2(H†H)(H†H),

QR≡(H†H)(DµH†DµH) = 1

2(H†H)(H†H)−(H†H)(D2H†H+H†D2H),

QD≡ D2H†D2H=−1

2(Ypq (ψpψq)D2H+ h.c.) −λ0(H†H)(D2H†H+H†D2H),

Q2W≡ −1

2DµWI

µν 2=−g2

32 Y−1

pq (ψpψq)D2H+ h.c.−ig2

4ψpγµτIψp(H†i←→

µH)

+g2

4λ02Y−1

pq Y−1

qp (H†H)3+g2

8(DµH†H)(H†DµH)−g2

16 QR

+g2

32 (1 + 2λ0Y−1

pq Y−1

qp ) (H†H)(D2H†H+H†D2H),

Q2B≡ −1

2∂µBµν 2=−g2

4Y−1

pq (ψpψq)D2H+ h.c.−ig2

Yψpγµψp(H†i←→

DµH)

+g2

YQR+ 2g2

Yλ02Y−1

pq Y−1

qp (H†H)3+g2

2(1 + λ0Y−1

pq Y−1

qp ) (H†H)(D2H†H+H†D2H),

Q2G≡ −1

2DµGa

µν 2=−g2

3Y−1

pq (ψpψq)D2H+ h.c.+4g2

3λ02Y−1

pq Y−1

qp (H†H)3

+2g2

3λ0Y−1

pq Y−1

qp (H†H)(D2H†H+H†D2H),

QW≡igW(H†τI←→

DµH)DνWI

µν =ig2

2(H†i←→

µH)(ψγµτIψ)−g2

8(DµH†H)(H†DµH)

+g2

16 QR−g2

32 (H†H)(D2H†H+H†D2H),

QB≡igY(H†←→

DµH)∂νBµν =ig2

Y(H†←→

DµH)(ψγµψ)−2g2

YQR

−g2

Y(H†H)(D2H†H+H†D2H),

QW W ≡g2

W(H†H)WI

µν WI,µν ,

QBB ≡g2

Y(H†H)Bµν Bµν ,

QW B ≡2gWgY(H†τIH)WI

µν Bµν ,

QGG ≡g2

G(H†H)Ga

µν Ga µν ,(2.2)

where

Ypq

denotes the SM Yukawa coupling matrix,

{p, q} ∈

3) are the ﬂavour indices.

We denote the Wilson coeﬃcients of the SILH operators as

with

labelling the operators

in Eqs.

(2.2)

. Taking into account all the

-involved structures that can appear from a

scalar extension of the SM, one can write:

L=L(4)

SM +e

λ(H†H)2+ζ(6)

1(H†H)3+ζ(6)

2(H†H)(H†H) + ζ(6)

3(DµH†H)(H†DµH)

+ζ(6)

4(H†H)(Bµν Bµν ) + ζ(6)

5(H†H)(WI

µν WIµν ) + ζ(6)

6(H†τIH)(Bµν WIµν )

+ζ(6)

7(H†H)(Ga

µν Gaµν ) + ζ(6)

8,1(H†i←→

DµH)(ψ γµψ) + ζ(6)

8,2(H†i←→

µH)(ψ τIγµψ)

+ξ(6)

1(H†H)(D2H†H+H†D2H)+ξ(6)

2(ψ ψ)D2H+ h.c..(2.3)

We highlight the redundant terms in Eq.

(2.3)

in bold font; they need to be removed.

– 4 –

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

IntegratingoutheavyscalarswithmodiedEOMs:matchingcomputationofdimension-eightSMEFTcoecientsUpalaparnaBanerjee,JoydeepChakrabortty,1ChristophEnglert,2ShakeelUrRahaman,1andMichaelSpannowsky31IndianInstituteofTechnologyKanpur,Kalyanpur,Kanpur208016,UttarPradesh,India2SchoolofPhysics&Astronomy,Univers...

展开>> 收起<<

Integrating out heavy scalars with modied EOMs matching computation of dimension-eight SMEFT coecients.pdf

共32页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Integrating out heavy scalars with modied EOMs matching computation of dimension-eight SMEFT coecients

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: