One-loop Matching of Scotogenic Model onto Standard Model Eective Field Theory up to Dimension 7 Yi Liaoaband Xiao-Dong Mac

2025-05-02 0 0 904.51KB 28 页 10玖币
侵权投诉
One-loop Matching of Scotogenic Model onto Standard
Model Effective Field Theory up to Dimension 7
Yi Liao a,b∗and Xiao-Dong Ma c†
aGuangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum Matter,
South China Normal University, Guangzhou 510006, China
bGuangdong-Hong Kong Joint Laboratory of Quantum Matter,
Southern Nuclear Science Computing Center,
South China Normal University, Guangzhou 510006, China
cTsung-Dao Lee Institute & School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai 200240, China
Abstract
The scotogenic neutrino seesaw model is a minimal extension of the standard model with three Z2-odd
right-handed singlet fermions Nand one Z2-odd Higgs doublet ηthat can accommodate the tiny neutrino
mass and provide a dark matter candidate in a unified picture. Due to lack of experimental signatures for
electroweak scale new physics, it is appealing to assume these new particles are well above the electroweak
scale and take the effective field theory approach to study their effects on low energy observables. In this
work we apply the recently developed functional matching formalism to the one-loop matching of the model
onto the standard model effective field theory up to dimension seven for the case when all new states N
and ηare heavy to be integrated out. This is a realistic example which has no tree-level matching due to
the Z2symmetry. Using the matching results, we analyze their phenomenological implications for several
physical processes, including the lepton number violating effect, the CDF Wmass excess, and the lepton
flavor violating decays like µ→eγ and µ→3e.
∗liaoy@scnu.edu.cn
†maxid@sjtu.edu.cn
1
arXiv:2210.04270v2 [hep-ph] 3 Dec 2022
Contents
1 Introduction 2
2 Basics of Functional Matching 4
3 Review of Scotogenic Model 6
4 One-loop Matching onto the SMEFT 7
4.1 Matching result in a Green basis 8
4.2 Matching result in the standard basis 10
5 Phenomenology 14
5.1 Dim-5 and dim-7 LNV processes 14
5.2 LNC processes due to dim-6 operators 15
6 Conclusion 19
Acknowledgements 19
A Dim-5, dim-6, and dim-7 operator bases in the SMEFT 20
B Calculation of a power-type supertrace with double insertions of X
X
XNη and X
X
XηN 21
C Reduction into the standard basis 23
References 25
1 Introduction
The origin of neutrino mass and the nature of dark matter are two pieces of well-established evidence for physics
beyond the standard model (SM). Among various beyond SM scenarios an attractive route is to correlate these
two seemingly separate problems and solve them within the same theoretical framework. In this respect the
scotogenic model [1] provides a nice example that can explain the origin of neutrino mass and dark matter
(DM) in a single simple framework, with the introduction only of three Z2-odd right-handed singlet fermions
Nand one Z2-odd scalar doublet ηon top of the SM content. The main merit of the model is that neutrinos
gain radiative mass from interactions with particles in the dark sector. The lightest of the latter is stable
due to the presumed exact Z2symmetry, and thus potentially serves as a DM candidate. Since it was first
proposed in 2006, the model has attracted a lot of attentions from various phenomenological aspects including
the lepton flavor violating (LFV) golden modes µ→eγ, 3eand µ-econversion in nuclei [2–7], the scalar DM
scenario [8–11], the fermionic WIMP and FIMP DM cases [12–16], the running effect of neutrino masses [17]
and the neutrino mass matrix textures [18], low scale leptogenesis [16,19,20], the signals at hadron and lepton
colliders [14,21,22] and in gravitational waves [23], and the LFV Zand Higgs boson decays [24], etc.
Nevertheless, there is no definite signature so far for the predicted particles on the observational side. It
then looks natural to assume that the new particles lie well above the SM electroweak scale to be inaccessible
to current high energy colliders. This motivates us to study their indirect effects on low energy observations by
working with an effective field theory (EFT) in which they are integrated out. The standard model effective
field theory (SMEFT) is tailored exactly for this purpose. It is a low energy effective field theory for SM
2
particles below some new physics scale. In this framework the SM interactions appear as the leading term in
a systematic series expansion, and the effects from new physics at a high scale are incorporated as suppressed
high-dimensional operators and modifications to the SM parameters. The most appealing feature of SMEFT is
perhaps its universality. It contains exclusively the SM fields which are governed by the SM gauge symmetries
but is otherwise not constrained. Different high scale new physics will be reflected in Wilson coefficients (WCs)
and their interrelations. In the past years, the bases of complete and independent high dimensional operators
have been built for the SMEFT up to dimension nine (dim-9) [25–33]. To apply the SMEFT to low energy
phenomenology, one evolves it to the electroweak scale with the help of SMEFT renormalization group equations
(RGE), matches it with the low energy effective field theory (LEFT) at the electroweak scale, and further evolves
the latter to the experimental scale via the LEFT RGEs where one finally calculates the physical observables. In
this way, low energy experimental data can be employed complementarily to constrain physics in the ultraviolet
(UV).
An important task in this approach is the matching of a UV theory onto the SMEFT at the UV scale.
Assuming the UV theory is perturbative, the matching is a double expansion, one in the number of loops and
the other in the inverse power of the heavy scale. The tree-level matching can be easily done by solving the
classical equations of motion (EoM) for the heavy fields followed by a low energy expansion to the desired
order. However, in some cases interesting phenomenology (like flavor changing neutral currents) arises as a loop
effect or precision data demands an improved theoretical analysis, so that one-loop matching becomes more
and more relevant. Confronted with this, the recently developed functional matching via the effective action
is a tailor-made method to achieve this goal [34–44]. Unlike the diagrammatic approach, the matching is done
by calculating some functional supertraces without computing Feynman diagrams one by one for a designed
set of amplitudes. Some (semi-)automatic tools have been developed to facilitate this job for the tree-level
matching [45–47] and one-loop matching [43,47–49]. One-loop matching has recently been practiced for several
UV models, such as the three tree-level seesaws [50–54], the Zee model [51], leptoquark models [51,55,56], and
others [57–63].
Considering rich phenomenology of the scotogenic model and null experimental searches for new heavy states,
it is plausible to take the above EFT approach to investigate its low energy effects by treating both Nand η
as heavy states. It turns out that the existence of Z2symmetry implies no tree-level matching to the SMEFT.
Then we investigate its one-loop matching up to dim-7 operators. The direct result of functional evaluation is
organized in the so-called Green basis with a handful of operators whose origin can be relatively easily tracked
from the diagrammatic picture. To recast the result in terms of the standard dim-5 Weinberg operator [25],
the dim-6 Warsaw basis [26] and the dim-7 basis [29] (further improved in [64]), some manipulations have to be
made of the SM equations of motion (EoM), integration by parts (IBP) and the Fierz identities (FI).
The rest of the paper is organized as follows. In section 2we review the basic ingredients of the functional
matching in effective field theory. Section 3is a brief introduction of the scotogenic model as well as our
notational convention. In section 4we consider the one-loop matching for the scotogenic model to the SMEFT,
and show our main result in both Green and standard bases. The phenomenological analysis together with brief
comparisons to the literature is included in section 5. In section 6, we draw our conclusions. Supplementary
materials presented in the appendices include the collection of the SMEFT operator bases up to dimension 7
in appendix A, the calculation of the supertrace for four-lepton operators in appendix B, and the reduction of
operators from the Green basis to the standard basis in appendix C.
3
2 Basics of Functional Matching
For a UV field theory, whether fundamental or effective, with a hierarchical field spectrum, we collectively
denote the heavy and light (scalar, fermion, or vector) fields as Φ and φ, respectively, with a mass hierarchy
mΦmφ. The low energy dynamics for light particles can be calculated either from the UV Lagrangian
LUV(Φ, φ) consisting of both heavy and light fields, or from the EFT Lagrangian LEFT(φ) consisting only of light
fields. In matching calculation LUV(Φ, φ) is supposed to be known while LEFT(φ) is searched for. To reproduce the
low-energy physics of LUV(Φ, φ), LEFT(φ) has to be carefully determined from LUV(Φ, φ) by integrating out Φ and
performing a matching calculation. Conventionally, this matching is done by designing judiciously a complete
set of amplitudes, computing them in both theories and equating them to determine the Wilson coefficients
in LEFT(φ) which depend on the parameters associated with Φ. The main drawback of this diagrammatical
matching is that one has to first determine the correct basis of operators at each dimension for the sought
LEFT(φ) and compute amplitudes twice. The procedure necessarily involves infrared physics of light particles
which however eventually does not enter LEFT(φ) itself, causing unnecessary complications.
A more elegant approach is the functional matching in the path integral formalism [34–44]. The starting
point is the identification of one-particle-irreducible (1PI) generating functionals ΓEFT[φ]=ΓL
UV[φ] at the matching
scale mΦ. Here ΓL
UV[φ] is computed in the UV theory and irreducible only to the light field φwhile ΓEFT[φ] is
computed in the EFT. This identification is made in a double expansion, one in the inverse power expansion
of mΦand the other in the number of loops. At the tree order, this is easy: one solves in the UV theory the
classical EoM for Φ = Φc[φ] in terms of the light field φwhich is a functional, and makes the inverse power
expansion in mΦto turn it into an infinite series of local functions Φ = Φc(φ). Substituting it into LUV(Φ, φ)
yields the answer:
Ltree
EFT (φ) = LUV(Φc(φ), φ).(2.1)
To perform one-loop matching, let us start on the EFT side whose Lagrangian is
LEFT(φ) = Ltree
EFT (φ) + L1-loop
EFT (φ) + ··· ,(2.2)
where the dots stand for higher-loop contributions and L1-loop
EFT (φ) is what we are seeking. The above contributes
to the one-loop 1PI generating functional in two manners,
Γ1-loop
EFT [φ] = ZddxL1-loop
EFT (φ) + i
2STr[ln(H
H
HEFT)],(2.3)
where the second term arises from one-loop diagrams formed with interactions defined in Ltree
EFT (φ). The super-
trace (STr) includes a minus sign for fields that are quantized as a Grassmanian field. In the path integral
formalism, it results from the Gaussian integral with the Hessian matrix being
H
H
HEFT(x, y) = δ2Stree
EFT [φ]
δ¯
φ(x)δφ(y),(2.4)
where Si
a[ϕ] = ZddzLi
a(ϕ(z)) denotes the action in ddimensional spacetime at the iorder in the atheory for
the field ϕ. (A global factor of µd−4has been suppressed for brevity where µis the usual renormalization scale.)
To count correctly independent degrees of freedom, the fields have been arranged in a self-conjugate form up
to a rotation R, i.e., the conjugate pair of fields ¯ϕand ϕis related by ¯ϕ=ϕTRwith |det(R)|= 1 [43,48]. The
supertrace term in Eq. (2.3) contains all infrared physics associated with the light field φ.
A similar 1-loop manipulation can be made for the generating functional ΓL
UV[φ] in the UV theory. An
important difference from the usual case is that it is a generating functional only for and irreducible only to
4
the light field φalthough the UV theory contains the heavy field Φ as well. Therefore when we make Legendre
transform from the generating functional for connected Green’s functions of φto that for 1PI Green’s functions
φ, we have to implement the classical EoM for the Φ field simply because no external source has been introduced
to it. With this point in mind, we have,
ΓL,1-loop
UV [φ] = i
2STr[ln(H
H
HUV)],(2.5)
where
H
H
HUV(x, y) = δ2SUV[ϕ]
δ¯ϕ(x)δϕ(y)Φ=Φc(φ)
.(2.6)
We notice that the Hessian matrix is defined for the whole field space ϕ= (φ, Φ) and the substitution Φ = Φc(φ)
is made only after the functional derivatives have been finished.
The identification at the 1-loop order of Γ1-loop
EFT [φ] = ΓL,1-loop
UV [φ] then implies
ZddxL1-loop
EFT (φ) = i
2STr[ln(H
H
HUV)] −i
2STr[ln(H
H
HEFT)].(2.7)
The recent development in functional matching is based on the following crucial realization [36,38,39,43]. When
the loop integrals in the UV theory is calculated by integration by regions [65,66],
i
2STr[ln(H
H
HUV)] = i
2STr[ln(H
H
HUV)]hard
+i
2STr[ln(H
H
HUV)]soft
,(2.8)
the soft term exactly cancels out the EFT term in Eq. (2.7). The calculation of one-loop matching then boils
down to the calculation of the hard part in the UV term:
ZddxL1-loop
EFT (φ) = i
2STr[ln(H
H
HUV)]hard
,(2.9)
where the subscript hard means that the loop integrands are first Taylor-expanded for q∼mΦmφ, k where
q, k stand for the loop and external momenta respectively, and then evaluated for the whole qspace in d
dimensions.
The second functional derivative is split into two parts, H
H
HUV =K
K
K−X
X
X, where K
K
Kcontains only kinetic and
mass terms and X
X
Xincludes all remaining interactions. The matrix K
K
Kis in a block-diagonal form:
Ki=
P2−m2
ifor spin-0 fields
/
P−mifor spin-1/2 fields
−gµν (P2−m2
i) for spin-1 fields in Feynman gauge
,(2.10)
where Pµ=iDµwith Dµbeing the covariant derivative with respect to background gauge fields whose operators
are under consideration. To proceed further, we make the Taylor expansion,
ln(K
K
K−X
X
X) = ln(K
K
K) +
∞
X
n=0
1
n(K
K
K−1X
X
X)n.(2.11)
Since X
X
Xcontributes at least a mass dimension 1 (3/2) to operators in search when it involves a bosonic
(fermionic) field and K
K
K−1contributes a nonnegative mass dimension upon finishing loop integrals, the expansion
actually terminates for the sought operators with a given mass dimension. The evaluation of supertraces is then
classified into a log-type and a power-type,
ZddxL1-loop
EFT [φc] = i
2STr [ln(K
K
K)] hard −i
2
∞
X
n=1
1
nSTr hK
K
K−1X
X
Xnihard
.(2.12)
5
摘要：

One-loopMatchingofScotogenicModelontoStandardModelE ectiveFieldTheoryuptoDimension7YiLiaoa;b*andXiao-DongMac„aGuangdongProvincialKeyLaboratoryofNuclearScience,InstituteofQuantumMatter,SouthChinaNormalUniversity,Guangzhou510006,ChinabGuangdong-HongKongJointLaboratoryofQuantumMatter,SouthernNuclearSci...

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