Supermassive gauginos in supergravity inflation with high-scale SUSY breaking
Hun<EMAIL_ADDRESS>and Massimo<EMAIL_ADDRESS>
⋆Center for Quantum Spacetime (CQUeST), Sogang University,
35 Baekbeom-ro, Mapo-gu, Seoul 04107, Republic of Korea,
†Yukawa Institute for Theoretical Physics (YITP), Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan and
♯Research Institute of Basic Sciences (RIBS), Incheon National University,
119, Academy-ro, Yeonsu-gu, Incheon 22012, Republic of Korea
‡Center for Cosmology and Particle Physics
Department of Physics, New York University
726 Broadway, New York, NY 10003, USA

A model of supergravity inflation we recently proposed can produce slow roll
inflation and a realistic spectrum of particles even without F-term
supersymmetry breaking. Supersymmetry is broken only by a D-term induced by a
recently discovered new type of Fayet-Iliopoulos (FI) term. Almost all
supersymmetric partners of the standard model fields can get masses as high as
the inflationary Hubble scale. The exception is gauginos, for which the
vanishing of F-terms implies an exact cancellation that keeps their masses
exactly zero. To cure this problem without spoiling the simplicity of our
model we introduce a new term that further enlarges the space of supergravity
effective actions. It is an F-term that, similarly to the new FI term, becomes
singular in the supersymmetric limit. We show that this term can produce large
gaugino masses without altering the spectrum of other states and without
lowering the cutoff of the effecive theory.

###### Contents

  1. 1 Introduction
  2. 2 Gaugino masses in supergravity with new Fayet-Iliopoulos terms
  3. 3 Supermassive gauginos in supergravity inflation with high-scale SUSY breaking
  4. 4 Consistency check on the UV cutoff
  5. 5 Conclusion

##  1 Introduction

In standard supergravity actions gauginos are massive only if the F and/or D
term of matter fields are non-vanishing [1]. This requires nonvanishing vacuum
expectation values (VEV)s of matter fields which may generate theoretical
difficulty. First, they can make a theory less predictive because it often
needs many additional scalars (moduli). Second, the scalar potential necessary
to have both nonvanishing F and D terms and stable moduli is neither simple
nor natural. Finally, the scalar potential required by moduli stabilization
and realistic gaugino masses may be incompatible with the scalar field
dynamics that we wish to study within the theory.

On the other hand, models with vanishing VEVs of matter fields carry their own
set of problems. A serious one is that the lack of experimental evidence for
superpartners [2] requires among other things heavy gauginos, which are
difficult to obtain in supergravity models where the VEVs of matter fields
vanish. It is thus natural to ask the question: can we obtain large gaugino
masses together with vanishing VEVs of matter fields and a simple moduli
stabilization mechanism? In this paper we will answer in the affirmative by
explicitly presenting a model with all the required properties.

An action containing the so-called new Fayet-Iliopoulos (FI) terms [3, 4],
which is specified by a hidden-sector vector multiplet $V$, does not
contribute to the masses of matter fermions and gauginos, except for the
gaugino of the new-FI vector $V$, but it can make matter scalars (or
equivalently sfermions) parametrically heavier than observable-sector states.
While the new FI terms do not give mass to gauginos, other terms can be added
to the standard supergravity action that do it. In this paper we will describe
in details in Section 2 one such term: a new F-term that couples the gauge
vector multiplets to $V$.

Before embarking in a description of our model, we need to mention several
works relevant to the problem of gaugino masses. First, non-universal gaugino
masses can be obtained in non-minimal GUT models [5, 6, 7, 8, 9, 10, 11],
mirage mediation [12, 13, 14, 15, 16, 17, 18], and non-universal gauge kinetic
functions through string compactifications [19]. Other recent developments in
the non-universal gaugino masses were discussed in [20, 21, 22, 23, 24].
Second, gaugino masses can be obtained by anomaly mediation [25, 26, 27] in
the heavy sfermion scenario, which are proportional to the inverse of the
corresponding gauge coupling constant $g$, its beta function $\beta(g^{2})$,
and gravitino mass $m_{3/2}$. In addition, anomaly-mediated gaugino mass can
be derived in the superspace formulation of supergravity with a Wilsonian
effective action [28], using superconformal anomaly [29], and using the
anomaly of the Peccei-Quinn (PQ) symmetry mediated by additional charged
matters under SM gauge symmetries [30, 31, 32, 33] for KSVZ-type QCD axion
[34, 35]. Furthermore, gaugino masses get corrections as large as the
anomalous one if vector-like matter fields with masses smaller than the
gravitino one exist [36, 37, 38]. Let us point out that the common point of
all those models for gaugino masses is that moduli stabilization with non-
vanishing VEVs of matter fields is necessary and deeply associated with
generating the gaugino masses, signaling a possible backreaction between the
dynamics of scalar fields and gaugino masses. On the contrary, our model for
the gaugino-mass generation can decouple moduli stabilization from scalar
field dynamics while keeping vanishing VEVs of all matter fields (except the
Higgs fields).

This paper describes a mechanism for generating arbitrarily large of gaugino
masses in a simple extension of the supergravity model of inflation with new
Fayet-Iliopoulos (FI) terms [3, 4] presented in [39]. In [39] we used the
Kähler form of the new FI term given in [4] to construct a model with slow
roll inflation, and realistic spectrum and interactions. Among other features,
our model contained the complete spectrum of the minimal supersymmetric
standard model (MSSM) together with its interactions and terms necessary to
give large masses to almost all the supersymmetric partners of standard model
(SM) particles. The exception was gaugino masses. Eq. (77) of Ref. [39] shows
that they are proportional to the F term $WG_{J}=K_{J}W+W_{J}$, where $K$ is
the Kähler potential, $W$ is the superpotential and the indices $J$ are either
those of the modulus $T$ or those of the matter scalars $z^{i}$. Ref. [39]
overlooked the simple fact that the sum $K_{J}W+W_{J}$ is precisely an F term
so incorrectly assumed that it was of the same order of its two summands
$K_{J}W$ and $W_{J}$. So it stated that gaugino masses can be made
$\mathcal{O}(H)$, with $H$, the Hubble parameter during slow-roll inflation.
The gaugino masses are instead zero unless there is an F-term supersymmetry
breaking. A nonzero F-term would spoil the most appealing feature of the
model; namely that a parametrically large mass splitting between standard
model fermions and supersymmetric scalar partners can be achieved even with a
pure D-term supersymmetry breaking, where all F terms vanish at the post-
inflationary minimum ($T=1/2,z^{i}\sim 0$). It would also require
significantly more convoluted superpotentials than those given in [39], which
could spoil good properties of the model such as the possibility of slow roll
inflation, and in any case would only produce gaugino masses much smaller than
those of the other superpartners. The solution we present here is different:
we add a new gaugino mass (GM) term that, much like the new FI terms, becomes
singular when supersymmetry is unbroken. It is an F term of conformal weight 3
whose schematic form in the standard superfield notations is

$\int d^{2}\theta
W_{\alpha}(U)W^{\alpha}(U)W_{\beta}(V)W^{\beta}(V)S_{0}^{-3}W(z,T)^{-1},$
(1.1)

where $V$ is the $U(1)$ vector superfield associated to the new FI term of
Ref. [4], $U$ are the vector superfields of the observable sector, $W$ is as
before the superpotential and $S_{0}$ is the conformal compensator. The
detailed construction of the GM term using superconformal tensor calculus is
given in Sec. 2. The explicit form of the gaugino masses and an analysis of
the range of masses allowed by the GM term, together with their relation to
the cutoff and to the Hubble scale in inflation is given in Sec. 3. Section 4
contains a detailed analysis of the effect of the GM term on the cutoff of the
effective field theory. Since this new term contains inverse powers of F
terms, it could in principle lower the cutoff below the inflationary Hubble
scale $H$. This would make the effective theory useless to describe inflation.
Section 3 finds the range of the free parameters contained in the GM term that
keeps the effective theory cutoff $\Lambda$ larger than $H$.

##  2 Gaugino masses in supergravity with new Fayet-Iliopoulos terms

In this section, we describe a mechanism for generating arbitrarily large of
gaugino masses in supergravity with new Fayet-Iliopoulos (FI) terms. While the
gaugino of the vector multiplet for the new FI term gets a heavy mass from the
new FI term, the gauginos of the SM gauge vector multiplets will get non-
vanishing masses, as heavy as the Hubble scale via the new GM F term (1.1) .

As in Ref. [39] we work in $M_{pl}=1$ units and introduce matter chiral
multiplets $Z^{i}$, the chiral compensator $S_{0}$, a real multiplet $V$, and
another real multiplet $(V)_{D}$, whose lowest component is the auxiliary D
term of the real multiplet $V$. The components of the various superconformal
multiplets are given as follows.

Chiral matter multiplets

$\displaystyle
Z^{i}=(z^{i},-i\sqrt{2}P_{L}\chi^{i},-2F^{i},0,+i\mathcal{D}_{\mu}z^{i},0,0)=\\{z^{i},P_{L}\chi^{i},F^{i}\\},$
(2.1)
$\displaystyle\bar{Z}^{\bar{i}}=(\bar{z}^{\bar{i}},+i\sqrt{2}P_{R}\chi^{\bar{i}},0,-2\bar{F}^{\bar{i}},-i\mathcal{D}_{\mu}\bar{z}^{\bar{i}},0,0)=\\{\bar{z}^{\bar{i}},P_{R}\chi^{\bar{i}},\bar{F}^{\bar{i}}\\}.$
(2.2)

Conformal compensator multiplet

$\displaystyle
S_{0}=(s_{0},-i\sqrt{2}P_{L}\chi^{0},-2F_{0},0,+i\mathcal{D}_{\mu}s_{0},0,0)=\\{s_{0},P_{L}\chi^{0},F_{0}\\},$
(2.3)
$\displaystyle\bar{S}_{0}=(\bar{s}_{0},+i\sqrt{2}P_{R}\chi^{0},0,-2\bar{F}_{0},-i\mathcal{D}_{\mu}\bar{s}_{0},0,0)=\\{\bar{s}_{0},P_{R}\chi^{0},\bar{F}_{0}\\}.$
(2.4)

Field strength multiplet for the new FI vector $V$

$\displaystyle V$ $\displaystyle=$
$\displaystyle\\{0,0,0,0,v_{\mu},\lambda_{V},D_{V}\\}~{}\textrm{in the Wess-
Zumino gauge}$ (2.5) $\displaystyle(\bar{\lambda}P_{L}\lambda)_{V}$
$\displaystyle=$
$\displaystyle(\bar{\lambda}_{V}P_{L}\lambda_{V},-i\sqrt{2}P_{L}\Lambda_{V},2D_{-}^{2},0,+i\mathcal{D}_{\mu}(\bar{\lambda}_{V}P_{L}\lambda_{V}),0,0)$
(2.6) $\displaystyle=$
$\displaystyle\\{\bar{\lambda}_{V}P_{L}\lambda_{V},P_{L}\Lambda_{V},-D_{-}^{2}\\},$
$\displaystyle(\bar{\lambda}P_{R}\lambda)_{V}$ $\displaystyle=$
$\displaystyle(\bar{\lambda}_{V}P_{R}\lambda_{V},+i\sqrt{2}P_{R}\Lambda_{V},0,2D_{+}^{2},-i\mathcal{D}_{\mu}(\bar{\lambda}_{V}P_{R}\lambda_{V}),0,0)$
(2.7) $\displaystyle=$
$\displaystyle\\{\bar{\lambda}_{V}P_{R}\lambda_{V},P_{R}\Lambda_{V},-D_{+}^{2}\\},$

where

$\displaystyle
P_{L}\Lambda_{V}\equiv\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}+iD_{V})\lambda_{V},\qquad
P_{R}\Lambda_{V}\equiv\sqrt{2}P_{R}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}-iD_{V})\lambda_{V},$
(2.8) $\displaystyle D_{V-}^{2}\equiv
D_{V}^{2}-\hat{F}_{V}^{-}\cdot\hat{F}_{V}^{-}-2\bar{\lambda}_{V}P_{L}\cancel{\mathcal{D}}\lambda_{V},\qquad
D_{V+}^{2}\equiv
D_{V}^{2}-\hat{F}_{V}^{+}\cdot\hat{F}_{V}^{+}-2\bar{\lambda}_{V}P_{R}\cancel{\mathcal{D}}\lambda_{V},$
(2.9)
$\displaystyle\mathcal{D}_{\mu}\lambda_{V}\equiv\bigg{(}\partial_{\mu}-\frac{3}{2}b_{\mu}+\frac{1}{4}w_{\mu}^{ab}\gamma_{ab}-\frac{3}{2}i\gamma_{*}\mathcal{A}_{\mu}\bigg{)}\lambda_{V}-\bigg{(}\frac{1}{4}\gamma^{ab}\hat{F}_{Vab}+\frac{1}{2}i\gamma_{*}D_{V}\bigg{)}\psi_{\mu}$
(2.10) $\displaystyle\hat{F}_{Vab}\equiv
F_{Vab}+e_{a}^{~{}\mu}e_{b}^{~{}\nu}\bar{\psi}_{[\mu}\gamma_{\nu]}\lambda_{V},\qquad
F_{Vab}\equiv e_{a}^{~{}\mu}e_{b}^{~{}\nu}(2\partial_{[\mu}v_{\nu]}),$ (2.11)
$\displaystyle\hat{F}^{\pm}_{V\mu\nu}\equiv\frac{1}{2}(\hat{F}_{V\mu\nu}\pm\tilde{\hat{F}}_{V\mu\nu}),\qquad\tilde{\hat{F}}_{V\mu\nu}\equiv-\frac{1}{2}i\epsilon_{\mu\nu\rho\sigma}\hat{F}_{V}^{\rho\sigma}.$
(2.12)

Field strength multiplet for the SM gauge vector $U$

$\displaystyle U$ $\displaystyle=$
$\displaystyle\\{0,0,0,0,u_{\mu},\lambda_{U},D_{U}\\}~{}\textrm{in the Wess-
Zumino gauge}$ (2.13) $\displaystyle(\bar{\lambda}P_{L}\lambda)_{U}$
$\displaystyle=$
$\displaystyle(\bar{\lambda}_{U}P_{L}\lambda_{U},-i\sqrt{2}P_{L}\Lambda_{U},2D_{U-}^{2},0,+i\mathcal{D}_{\mu}(\bar{\lambda}_{U}P_{L}\lambda_{U}),0,0)$
(2.14) $\displaystyle=$
$\displaystyle\\{\bar{\lambda}_{U}P_{L}\lambda_{U},P_{L}\Lambda_{U},-D_{U-}^{2}\\},$
$\displaystyle(\bar{\lambda}P_{R}\lambda)_{U}$ $\displaystyle=$
$\displaystyle(\bar{\lambda}_{U}P_{R}\lambda_{U},+i\sqrt{2}P_{R}\Lambda_{U},0,2D_{U+}^{2},-i\mathcal{D}_{\mu}(\bar{\lambda}_{U}P_{R}\lambda_{U}),0,0)$
(2.15) $\displaystyle=$
$\displaystyle\\{\bar{\lambda}_{U}P_{R}\lambda_{U},P_{R}\Lambda_{U},-D_{U+}^{2}\\},$

where

$\displaystyle
P_{L}\Lambda_{U}\equiv\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}+iD_{U})\lambda_{U},\qquad
P_{R}\Lambda_{U}\equiv\sqrt{2}P_{R}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}-iD_{U})\lambda_{U},$
(2.16) $\displaystyle D_{U-}^{2}\equiv
D_{U}^{2}-\hat{F}_{U}^{-}\cdot\hat{F}_{U}^{-}-2\bar{\lambda}_{U}P_{L}\cancel{\mathcal{D}}\lambda_{U},\qquad
D_{U+}^{2}\equiv
D_{U}^{2}-\hat{F}_{U}^{+}\cdot\hat{F}_{U}^{+}-2\bar{\lambda}_{U}P_{R}\cancel{\mathcal{D}}\lambda_{U},$
(2.17)
$\displaystyle\mathcal{D}_{\mu}\lambda_{U}\equiv\bigg{(}\partial_{\mu}-\frac{3}{2}b_{\mu}+\frac{1}{4}w_{\mu}^{ab}\gamma_{ab}-\frac{3}{2}i\gamma_{*}\mathcal{A}_{\mu}\bigg{)}\lambda_{U}-\bigg{(}\frac{1}{4}\gamma^{ab}\hat{F}_{Uab}+\frac{1}{2}i\gamma_{*}D_{U}\bigg{)}\psi_{\mu}$
(2.18) $\displaystyle\hat{F}_{Uab}\equiv
F_{Uab}+e_{a}^{~{}\mu}e_{b}^{~{}\nu}\bar{\psi}_{[\mu}\gamma_{\nu]}\lambda_{U},\qquad
F_{Uab}\equiv e_{a}^{~{}\mu}e_{b}^{~{}\nu}(2\partial_{[\mu}u_{\nu]}),$ (2.19)
$\displaystyle\hat{F}^{\pm}_{U\mu\nu}\equiv\frac{1}{2}(\hat{F}_{U\mu\nu}\pm\tilde{\hat{F}}_{U\mu\nu}),\qquad\tilde{\hat{F}}_{U\mu\nu}\equiv-\frac{1}{2}i\epsilon_{\mu\nu\rho\sigma}\hat{F}_{U}^{\rho\sigma}.$
(2.20)

The GM F-term Lagrangian, written in the notations of superconformal tensor
calculus of [1], is

$\displaystyle\mathcal{L}_{GM}=[(\bar{\lambda}P_{L}\lambda)_{U}(\bar{\lambda}P_{L}\lambda)_{V}S_{0}^{-3}W(Z)^{-1}]_{F}+c.c.,$
(2.21)

where $W(Z)$ is the superpotential of our theory, and
$(\bar{\lambda}P_{L}\lambda)_{U}$ and $(\bar{\lambda}P_{L}\lambda)_{V}$ are
the composite multiplets corresponding to each of the two vectors. It is an
F-term density of conformal weight 3 and chiral weight 3, so it is invariant
under supersymmetry. The superpotential multiplet is given by

$\displaystyle
W(Z)=\Big{\\{}W(z),W_{i}P_{L}\chi^{i},W_{i}F^{i}-\frac{1}{2}W_{ij}\bar{\chi}^{i}P_{L}\chi^{j}\Big{\\}}.$
(2.22)

To compute the Lagrangian (2.21), we can use the following results of
superconformal tensor calculus

$\displaystyle W^{-1}(Z)$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{1}{W},-\frac{W_{i}}{W^{2}}P_{L}\chi^{i},-\frac{W_{i}F^{i}}{W^{2}}+\Big{(}\frac{W_{ij}}{2W^{2}}+\frac{W_{i}W_{j}}{W^{3}}\Big{)}\bar{\chi}^{i}P_{L}\chi^{j}\Big{\\}},$
(2.23) $\displaystyle S_{0}^{-3}$ $\displaystyle=$
$\displaystyle\Big{\\{}\frac{1}{s_{0}^{3}},-\frac{3}{s_{0}^{4}}P_{L}\chi^{0},-\frac{3F^{0}}{s_{0}^{4}}\Big{\\}}.$
(2.24)

Multiplying the supermultiplets we get

$\displaystyle S_{0}^{-3}W^{-1}(Z)$
$\displaystyle=\Big{\\{}\frac{1}{s_{0}^{3}W},-\frac{3}{s_{0}^{4}W}P_{L}\chi^{0}-\frac{W_{i}}{s_{0}^{3}W^{2}}P_{L}\chi^{i},$
$\displaystyle\qquad-\Big{(}\frac{3F^{0}}{s_{0}^{4}W}+\frac{W_{i}F^{i}}{s_{0}^{3}W^{2}}\Big{)}+\Big{(}\frac{W_{ij}}{2s_{0}^{3}W^{2}}+\frac{W_{i}W_{j}}{s_{0}^{3}W^{3}}\Big{)}\bar{\chi}^{i}P_{L}\chi^{j}-\frac{3W_{i}}{s_{0}^{4}W^{2}}\bar{\chi}^{i}P_{L}\chi^{0}\Big{\\}}.$
(2.25)

The multiplication of the field strength mutliplets of $V$ and $U$ gives

$\displaystyle(\bar{\lambda}P_{L}\lambda)_{U}(\bar{\lambda}P_{L}\lambda)_{V}$
$\displaystyle=\\{(\bar{\lambda}_{U}P_{L}\lambda_{U})(\bar{\lambda}_{V}P_{L}\lambda_{V}),$
$\displaystyle\qquad(\bar{\lambda}_{U}P_{L}\lambda_{U})\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}+iD_{V})\lambda_{V}+(\bar{\lambda}_{V}P_{L}\lambda_{V})\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}+iD_{U})\lambda_{U},$
$\displaystyle\qquad-(\bar{\lambda}_{U}P_{L}\lambda_{U})(D_{V}^{2}-\hat{F}_{V}^{-}\cdot\hat{F}_{V}^{-}-2\bar{\lambda}_{V}P_{L}\cancel{\mathcal{D}}\lambda_{V})-(\bar{\lambda}_{V}P_{L}\lambda_{V})(D_{U}^{2}-\hat{F}_{U}^{-}\cdot\hat{F}_{U}^{-}-2\bar{\lambda}_{U}P_{L}\cancel{\mathcal{D}}\lambda_{U})$
$\displaystyle\qquad-2\bar{\lambda}_{U}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}+iD_{U})P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}+iD_{V})\lambda_{V}\\}.$
(2.26)

Since the Lagrangian (2.21) is the F term of the two composite multiplets
(2.25) and (2.26), we find

$\displaystyle\mathcal{L}_{\textrm{GM}}$ $\displaystyle=$
$\displaystyle\bigg{[}-\frac{1}{s_{0}^{3}W}(\bar{\lambda}_{U}P_{L}\lambda_{U})(D_{V}^{2}-\hat{F}_{V}^{-}\cdot\hat{F}_{V}^{-}-2\bar{\lambda}_{V}P_{L}\cancel{\mathcal{D}}\lambda_{V})$
(2.27)
$\displaystyle-\frac{1}{s_{0}^{3}W}(\bar{\lambda}_{V}P_{L}\lambda_{V})(D_{U}^{2}-\hat{F}_{U}^{-}\cdot\hat{F}_{U}^{-}-2\bar{\lambda}_{U}P_{L}\cancel{\mathcal{D}}\lambda_{U})$
$\displaystyle-2\frac{1}{s_{0}^{3}W}\bar{\lambda}_{U}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}+iD_{U})P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}+iD_{V})\lambda_{V}$
$\displaystyle-\Big{(}\frac{3F^{0}}{s_{0}^{4}W}+\frac{W_{i}F^{i}}{s_{0}^{3}W^{2}}\Big{)}(\bar{\lambda}_{U}P_{L}\lambda_{U})(\bar{\lambda}_{V}P_{L}\lambda_{V})$
$\displaystyle+\Big{(}\frac{W_{ij}}{2s_{0}^{3}W^{2}}+\frac{W_{i}W_{j}}{s_{0}^{3}W^{3}}\Big{)}\bar{\chi}^{i}P_{L}\chi^{j}(\bar{\lambda}_{U}P_{L}\lambda_{U})(\bar{\lambda}_{V}P_{L}\lambda_{V})$
$\displaystyle-\frac{3W_{i}}{s_{0}^{4}W^{2}}(\bar{\chi}^{i}P_{L}\chi^{0})(\bar{\lambda}_{U}P_{L}\lambda_{U})(\bar{\lambda}_{V}P_{L}\lambda_{V})$
$\displaystyle+\Big{(}\frac{3}{s_{0}^{4}W}\bar{\chi}^{0}P_{L}+\frac{W_{i}}{s_{0}^{3}W^{2}}\bar{\chi}^{i}P_{L}\Big{)}(\bar{\lambda}_{U}P_{L}\lambda_{U})\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}+iD_{V})\lambda_{V}$
$\displaystyle+\Big{(}\frac{3}{s_{0}^{4}W}\bar{\chi}^{0}P_{L}+\frac{W_{i}}{s_{0}^{3}W^{2}}\bar{\chi}^{i}P_{L}\Big{)}(\bar{\lambda}_{V}P_{L}\lambda_{V})\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}+iD_{U})\lambda_{U}\bigg{]}+c.c.$

The GM term produces another contribution to the solutions of the equations of
motion for the auxiliary field of the compensator $S_{0}$, $F^{0}$, and for
the compensators $F^{j}$ of the physical chiral multiplets. We write it inside
large brackets

$\displaystyle F^{0}$ $\displaystyle=$ $\displaystyle
e^{2K/3}\bar{W}(\bar{z})-\frac{1}{3}e^{K/6}(\bar{\lambda}P_{R}\lambda)_{V}+\Biggl{[}s_{0}^{*-4}\bar{W}^{-1}(\bar{z})e^{K/3}(\bar{\lambda}P_{R}\lambda)_{U}(\bar{\lambda}P_{R}\lambda)_{V}\Biggr{]},$
$\displaystyle F^{i}$ $\displaystyle=$
$\displaystyle-3e^{K/2}G^{i\bar{j}}(\bar{W}_{\bar{j}}+K_{\bar{j}}\bar{W})-G^{i\bar{j}}\Big{(}9\frac{\mathcal{U}_{\bar{j}}}{\mathcal{U}}+3K_{\bar{j}}\Big{)}(\bar{\lambda}P_{R}\lambda)_{V}$
(2.28)
$\displaystyle+\Biggl{[}9s_{0}^{*-3}\bar{W}_{\bar{j}}(\bar{z})\bar{W}^{-2}(\bar{z})(\bar{\lambda}P_{R}\lambda)_{U}(\bar{\lambda}P_{R}\lambda)_{V}\Biggr{]}.$

Here $K$ is the Kähler potential and the second term in Eq. (2.28) comes from
the new FI terms introduced in Ref. [39]. Notice that the fermionic component
of the composite multiplet defining the Lagrangian (2.21) contains couplings
with three or five fermions only, so the GM term (2.21) does not alter the
original goldstino contribution to the fermion masses when the gravitino
becomes massive. In other words, the GM term contribution to the goldstino
mass term produces only higher-order interactions, not mass terms. Therefore,
all the mass matrices except those of the gaugini remain the same as those of
Ref. [39].

##  3 Supermassive gauginos in supergravity inflation with high-scale SUSY
breaking

The contribution of (2.27) to gaugino masses is contained in the terms

$\displaystyle\mathcal{L}_{\textrm{2 fermions}}$ $\displaystyle\supset$
$\displaystyle(\bar{\lambda}P_{L}\lambda)_{V}s_{0}^{-3}W(z)^{-1}(-D_{U-}^{2})+(\bar{\lambda}P_{L}\lambda)_{U}s_{0}^{-3}W(z)^{-1}(-D_{V-}^{2})$
(3.1)
$\displaystyle+2D_{U}D_{V}s_{0}^{-3}W(z)^{-1}\bar{\lambda}_{U}P_{L}\lambda_{V}+2D_{U}D_{V}s_{0}^{-3}W(z)^{-1}\bar{\lambda}_{V}P_{L}\lambda_{U}$

where $z,s_{0}$ are complex scalars, and $D_{U/V-}^{2}\equiv
D_{U/V}^{2}-\hat{F}_{U/V}^{-}\cdot\hat{F}_{U/V}^{-}-2(\bar{\lambda}P_{L}\cancel{\mathcal{D}}\lambda)_{U/V}$.
Notice that there are off diagonal quadratic terms in the gaugino fermions.
The other terms in (2.27) merely produce matter-fermion couplings and gaugino
interactions. In general, since the Lagrangian (3.1) contains off diagonal
quadratic terms, we need to diagonalize the gaugino mass matrix

$\displaystyle
M_{\textrm{Gaugino}}=\frac{1}{W(z)s_{0}^{3}}\begin{pmatrix}-D_{U}^{2}&2D_{U}D_{V}\\\
2D_{V}D_{U}&-D_{V}^{2}.\end{pmatrix}$ (3.2)

However, the D terms of the SM vector multiplet for $SU(3)_{c}$ vanish, while
those for $U(1)_{Y}$ and $SU(2)_{L}$ are given by
$D_{U}\sim\mathcal{O}(g_{a}(v_{u}^{2}-v_{d}^{2}))$, where $g_{a}$’s are the SM
($U(1)_{Y},SU(2)_{L}$) gauge coupling constant and $v_{u,d}$ are the vacuum
expectation values of the two Higgs fields, $H_{u}$ and $H_{d}$, present in
the MSSM. These D terms are much smaller than those of the new FI term vector
$V$; therefore, we can neglect them and write the Lagrangian as follows

$\displaystyle\mathcal{L}_{\textrm{Gaugino-
mass}}\supset-\frac{D_{V}^{2}}{W(z)s_{0}^{3}}(\bar{\lambda}P_{L}\lambda)_{U}=-\frac{D_{V}^{2}}{W(z)e^{K/2}}(\bar{\lambda}P_{L}\lambda)_{U}=-\frac{D_{V}^{2}}{m_{3/2}}(\bar{\lambda}P_{L}\lambda)_{U}$
$\displaystyle\implies M_{\textrm{gaugino}}=\frac{D_{V}^{2}}{m_{3/2}}\sim H.$
(3.3)

Here we used the superconformal gauge-fixing condition on the compensator
$s_{0}=e^{K/6}$, and the gravitino mass relation $m_{3/2}=W(z)e^{K/2}$. Notice
that the gaugino masses of the three SM gauge groups can be as large as the
inflationary Hubble scale $H$, both during and after inflation, as long as the
gravitino mass remains of $\mathcal{O}(H)$ during and after slow-roll
inflation.

$\displaystyle D_{V}\sim\xi=M_{S}^{2}\sim H,\qquad m_{3/2}\sim H.$ (3.4)

Here $\xi$ is the mass scale of the inflationary potential and $M_{S}$ is the
supersymmetry breaking scale, both given in $M_{pl}=1$ units.

More precisely, the gravitino mass $m_{3/2}$ along the inflationary trajectory
is given by

$\displaystyle m_{3/2}$ $\displaystyle=$ $\displaystyle
We^{K/2}=\frac{A(e^{-aT}-c)}{(T+\bar{T})^{3/2}}=\frac{A(e^{-aX/2}-c)}{X^{3/2}}$
(3.5) $\displaystyle=$ $\displaystyle
X^{-3/2}\sqrt{3}H(a^{-1}e^{-a(X-1)/2}-a^{-1}-1/3),$

where we use the same superpotential $W=A(e^{-aT}-c)$ as in Ref. [39],
$K=-3\ln[T+\bar{T}]$, $X=T+\bar{T}$, $T$ is real along the inflationary
trajectory where VEVs of the matter scalars are vanishing, and

$\displaystyle A=a^{-1}e^{a/2}\sqrt{3(M_{I}^{4}-\Lambda)}\approx
a^{-1}e^{a/2}\sqrt{3}H,\quad c=(1+a/3)e^{-a/2},\quad M_{I}^{4}\approx H^{2},$
(3.6)

with $a$ a so-far-unspecified real-number parameter. Our effective theory
describes physics only up to a finite cutoff in energy, $\Lambda_{cut}$. The
gravitino can be described by this effective theory only if its mass is below
the cutoff (and large enough to be unobservable). To satisfy both constraints
we restrict the range of the gravitino mass to

$\displaystyle\alpha H\lesssim|m_{3/2}|<\Lambda_{cut},$ (3.7)

where $0<\alpha<1$ is another yet-to-be-constrained real parameter. Extra
scalar degrees of freedom are required to be heavier than the Hubble scale in
order to be integrated out in our single-field inflationary model while
fermionic degrees of freedom do not have to be heavier than the Hubble scale,
but they still have to but lighter than the cutoff.

For a very small number $a$, we have

$\displaystyle m_{3/2}$ $\displaystyle\approx$ $\displaystyle
X^{-3/2}\sqrt{3}H\Big{(}a^{-1}(1-a(X-1)/2)-a^{-1}-1/3\Big{)}$ (3.8)
$\displaystyle=$
$\displaystyle-X^{-3/2}\sqrt{3}H\Big{(}\frac{X-1}{2}+\frac{1}{3}\Big{)}=X^{-3/2}\sqrt{3}H\frac{(1-3X)}{6},$

so that the allowed range is

$\displaystyle\alpha
H\lesssim\frac{\sqrt{3}}{6}X^{-3/2}|1-3X|H<\Lambda_{cut}.$ (3.9)

The inflationary trajectory starts from around $X\sim 100$ at the beginning of
the slow roll epoch and ends at the vacuum $X=1$, where the inflaton gets
frozen. We are thus interested in the range $1/3<X$. Then, constraint (3.7) is

$\displaystyle\alpha
H\lesssim\frac{\sqrt{3}}{6}X^{-3/2}(3X-1)H<\Lambda_{cut}.$ (3.10)

To investigate this constraint, we define a function

$\displaystyle F(X,\alpha)\equiv\frac{\sqrt{3}}{6}X^{-3/2}(3X-1)-\alpha,$
(3.11)

leading to

$\displaystyle 0\lesssim F(X,\alpha)<\frac{\Lambda_{cut}}{H}-\alpha.$ (3.12)

We therefore obtain

$\displaystyle X$ $\displaystyle\lesssim$
$\displaystyle\frac{1}{4\alpha^{2}}+\frac{3-8\alpha^{2}}{4\alpha^{2}\sqrt[3]{3}(9-36\alpha^{2}+24\alpha^{4}+8\alpha^{3}\sqrt{-3+9\alpha^{2}})^{1/3}}$
(3.13)
$\displaystyle+\frac{(9-36\alpha^{2}+24\alpha^{4}+8\alpha^{3}\sqrt{-3+9\alpha^{2}})^{1/3}}{4\alpha^{2}\sqrt[3]{9}},$

which we computed using Mathematica. To ensure a sufficient number of
e-foldings in the slow-roll inflation epoch, we set the right side of Eq.
(3.13) at $X=100$. This leads to

$\displaystyle\alpha\approx 0.086,$ (3.14)

which can also be obtained numerically using Mathematica. In Fig. 1 we plot
$F(X,\alpha)$ vs $X$ for various values of $\alpha$ to illustrate this bound.

Figure 1: The behaviors of $F(X,\alpha)$ in the direction of $X$ for different
$\alpha$’s. We can see that the maximum value of $X$ decreases as $\alpha$
increases. In order to have around 100 e-foldings during slow-roll inflation,
$\alpha\sim 0.086$.

With this value of $\alpha$, the gravitino mass can remain around the Hubble
scale over the whole inflationary trajectory. Re-introducing the reduced
Planck mass $M_{pl}~{}(\sim 2.4\times 10^{18}~{}\textrm{GeV)}$ and setting
$H\sim 10^{-5}M_{pl}$, the gravitino mass bound is given by

$\displaystyle 0.086H=8.6\times 10^{-7}M_{pl}\lesssim
m_{3/2}<\Lambda_{cut}\sim\sqrt{HM_{pl}}\sim 10^{-2.5}M_{pl}$ (3.15)

or equivalently in GeV

$\displaystyle 10^{12}~{}\textrm{GeV}\lesssim
m_{3/2}<10^{15.5}~{}\textrm{GeV}.$ (3.16)

Since the gaugino mass is given by $M_{Gaugino}\sim\frac{H^{2}}{m_{3/2}}$, we
find that

$\displaystyle
10^{12}~{}\textrm{GeV}\lesssim\frac{H^{2}}{M_{Gaugino}}\sim\frac{10^{8}~{}\textrm{GeV}^{2}}{M_{Gaugino}}<10^{15.5}~{}\textrm{GeV},$
(3.17)

which immediately produces the final result for the gaugino-mass bound

$\displaystyle 10^{10.5}~{}\textrm{GeV}<M_{Gaugino}\lesssim
10^{14}~{}\textrm{GeV}<\Lambda_{cut}=M_{S}\sim 10^{15.5}~{}\textrm{GeV}.$
(3.18)

Note that the gaugino masses can be very large while still being within the
above range.

Next, let us see whether the range we consider is consistent with the inflaton
dynamics during slow-roll inflation and reheating. Our scalar potential has
the following “Starobinsky” form

$\displaystyle
V=H^{2}-\frac{1}{X^{2}}\bigg{(}-2aA^{2}e^{-aX}+2acA^{2}e^{-aX/2}\bigg{)}+\frac{1}{3X}a^{2}A^{2}e^{-aX},$
(3.19)

where $A^{2}\approx 3a^{-2}e^{a}M_{S}^{4}=3a^{-2}e^{a}H^{2}$ and
$c=(1+a/3)e^{-a/2}$. The redefinition that makes the kinetic term of the
inflaton canonically normalized is $X=e^{\sqrt{2/3}\phi}$. We re-write $V$

$\displaystyle
V=H^{2}\bigg{(}1-\frac{1}{X^{2}}\bigg{(}-2ae^{-aX}+2a(1+a/3)e^{-a/2}e^{-aX/2}\bigg{)}3a^{-2}e^{a}+\frac{1}{X}e^{-a(X-1)}\bigg{)}$
(3.20)

and note that its minimum is located at $X=1$. For the scalar potential at
large $X$ during inflation we have $V\sim H^{2}M_{pl}^{2}$. On the contrary,
around the point $X=1/3$ (which is NOT a minimum), we have

$\displaystyle
V=H^{2}\bigg{(}1+\frac{-54e^{a/3}-18ae^{a/3}+54e^{2a/3}+3ae^{2a/3}}{a}\bigg{)}.$
(3.21)

If we assume that $a\ll 1$, after a short calculation and after expanding the
exponential terms up to linear order in $a$ we obtain

$\displaystyle V\approx 4H^{2}(1-a)\sim 4H^{2}.$ (3.22)

This implies that during the reheating era after inflation, the inflaton can
fluctuate around the vacuum at $X=1$ without reaching the point $X=1/3$, since
$V(X=1/3)\sim 4H^{2}M_{pl}^{2}$ is greater than $V(X\gg 1)\sim
H^{2}M_{pl}^{2}$, which is the potential at the beginning of slow-roll
inflation (and the Hubble friction dissipates energy). Thus, it is consistent
to restrict the range of the inflaton’s motion to $1/3<X\lesssim 100$ along
the inflationary trajectory –whose minimum at $X=1$, as shown in Fig. 2. In
particular, within this range, the upper bound in Eq. (3.10) is always
satisfied, because we have $H\sim\mathcal{O}(10^{-5})$ while, in $M_{pl}=1$
units, $\Lambda_{cut}\sim\mathcal{O}(10^{-2.5})$.

Figure 2: The scalar potentials of Starobinsky inflation for different values
of $a$ in Eq. (3.20), whose minimum is located at $X=1$. Note that during
inflation and reheating the potential energy is much smaller than at $X=1/3$.

##  4 Consistency check on the UV cutoff

To study the scale at which nonrenormalizable interactions appear we follow
the method used in [40, 41, 42, 39]. We analyze nonrenormalizable terms that
come from either the new-FI D term or the new GM F term. We can generalize the
strategy introduced in Ref. [39] in the following way. Let us write the
Lagrangian density of the new FI and GM terms as

$\displaystyle\mathcal{L}={1\over 2}D^{2}-\Xi D+\sum_{n\geq 0,m\geq
0}D^{-n}M_{Pl}^{-m}O^{2n+m+4}_{\textrm{newFI}}+\mathcal{L}_{GM}$ (4.1)

where $\Xi$ is the new-FI-term constant and $O^{2n+m+4}_{\textrm{newFI}}$ are
nonrenormalizable local operators of scaling dimension $2n+m+4$. All terms
with $m>0$ vanish in the global supersymmetry limit $M_{pl}\rightarrow\infty$
and $D\sim\Xi$ fixed. This shows that the strongest constraint on the UV
cutoff comes either from the term with $m=0$ or from $\mathcal{L}_{GM}$. The
$m=0$ terms gives

$\displaystyle\Lambda_{cut}^{2}\lesssim D\sim\Xi\sim HM_{pl}\sim
M_{S}^{2}\implies\Lambda_{cut}\lesssim\sqrt{HM_{pl}},;$ (4.2)

this is the same bound obtained in [39]. Moreover, in the limit of
$M_{pl}\rightarrow\infty$ and in the superconformal gauge
($s_{0}=\bar{s}_{0}=M_{pl}e^{K/6M_{pl}^{2}}$), the $\mathcal{L}_{GM}$ term in
the Lagrangian reduces to

$\displaystyle\mathcal{L}_{\textrm{GM}}$
$\displaystyle=-\frac{1}{W}(\bar{\lambda}_{U}P_{L}\lambda_{U})(D_{V}^{2}-\hat{F}_{V}^{-}\cdot\hat{F}_{V}^{-}-2\bar{\lambda}_{V}P_{L}\cancel{\mathcal{D}}\lambda_{V})$
$\displaystyle-\frac{1}{W}(\bar{\lambda}_{V}P_{L}\lambda_{V})(D_{U}^{2}-\hat{F}_{U}^{-}\cdot\hat{F}_{U}^{-}-2\bar{\lambda}_{U}P_{L}\cancel{\mathcal{D}}\lambda_{U})$
$\displaystyle-2\frac{1}{W}\bar{\lambda}_{U}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}+iD_{U})P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}+iD_{V})\lambda_{V}$
$\displaystyle-\Big{(}\frac{W_{i}F^{i}}{W^{2}}\Big{)}(\bar{\lambda}_{U}P_{L}\lambda_{U})(\bar{\lambda}_{V}P_{L}\lambda_{V})$
$\displaystyle+\Big{(}\frac{W_{ij}}{2W^{2}}+\frac{W_{i}W_{j}}{W^{3}}\Big{)}\bar{\chi}^{i}P_{L}\chi^{j}(\bar{\lambda}_{U}P_{L}\lambda_{U})(\bar{\lambda}_{V}P_{L}\lambda_{V})$
$\displaystyle+\Big{(}\frac{W_{i}}{W^{2}}\bar{\chi}^{i}P_{L}\Big{)}(\bar{\lambda}_{U}P_{L}\lambda_{U})\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{V}+iD_{V})\lambda_{V}$
$\displaystyle+\Big{(}\frac{W_{i}}{W^{2}}\bar{\chi}^{i}P_{L}\Big{)}(\bar{\lambda}_{V}P_{L}\lambda_{V})\sqrt{2}P_{L}(-\frac{1}{2}\gamma\cdot\hat{F}_{U}+iD_{U})\lambda_{U}.$
(4.3)

This equation includes both renormalizable and nonrenormalizable terms. From
the latter we find new constraints on the cutoff scale $\Lambda_{cut}$. They
are found to be

$\displaystyle\small{1}⃝~{}\frac{1}{W}\lesssim\Lambda_{cut}^{-3},~{}\small{2}⃝~{}\frac{D_{V}}{W}\lesssim\Lambda_{cut}^{-1},~{}\small{3}⃝~{}\frac{W_{i}F^{i}}{W^{2}}\lesssim\Lambda_{cut}^{-2},$
$\displaystyle\small{4}⃝~{}\Big{(}\frac{W_{ij}}{2W^{2}}+\frac{W_{i}W_{j}}{W^{3}}\Big{)}\lesssim\Lambda_{cut}^{-5},~{}\small{5}⃝~{}\frac{W_{i}}{W^{2}}\lesssim\Lambda_{cut}^{-4},~{}\small{6}⃝~{}\frac{W_{i}D_{V}}{W^{2}}\lesssim\Lambda_{cut}^{-2}.$
(4.4)

Let us examine each of them separately. The constraint \small{1}⃝ reduces to

$\displaystyle\small{1}⃝~{}\Lambda_{cut}\lesssim W^{1/3}.$ (4.5)

From \small{1}⃝ and \small{2}⃝, we find

$\displaystyle\small{2}⃝~{}D_{V}\lesssim\Lambda_{cut}^{2}.$ (4.6)

From \small{5}⃝ and \small{3}⃝, we find

$\displaystyle\small{3}⃝~{}F^{i}\lesssim\Lambda_{cut}^{2}.$ (4.7)

From \small{1}⃝ and \small{4}⃝, we find

$\displaystyle\small{4}⃝~{}\frac{W_{ij}}{2}+\frac{W_{i}W_{j}}{W}\lesssim\Lambda_{cut}\implies\frac{W_{ij}}{2}\lesssim\Lambda_{cut}\quad\mathrm{and}\quad\frac{W_{i}W_{j}}{W}\lesssim\Lambda_{cut}.$
(4.8)

From \small{1}⃝ and \small{5}⃝, we find

$\displaystyle\small{5}⃝~{}W_{i}\lesssim\Lambda_{cut}^{2}.$ (4.9)

Using \small{5}⃝ and \small{6}⃝, we find that the contraint \small{6}⃝ in Eq.
(4.4) reduces to the constraint \small{2}⃝ in (4.6). Notice that Eqs. (4.2)
and (4.6) bound the value of the cutoff scale as follows

$\displaystyle D_{V}\sim\Lambda_{cut}^{2}\sim M_{S}^{2}\sim
HM_{pl}\implies\Lambda_{cut}\sim M_{S}\sim\sqrt{HM_{pl}}.$ (4.10)

This is essential for the bounds coming from the new FI term and the gaugino
mass to be compatible with each other. With this result, the constraints
\small{1}⃝, \small{3}⃝, \small{4}⃝, and \small{5}⃝ show that

$\displaystyle M_{S}^{3}\lesssim W,\quad F^{i}\lesssim M_{S}^{2},\quad
W_{ij}\lesssim M_{S},\quad W_{i}\lesssim M_{S}^{2},$ (4.11)

or equivalently

$\displaystyle HM_{pl}\sqrt{HM_{pl}}\lesssim W,\quad F^{i}\lesssim
HM_{pl},\quad W_{ij}\lesssim\sqrt{HM_{pl}},\quad W_{i}\lesssim HM_{pl},$
(4.12)

which can be satisfied for the values $F^{i}\lesssim HM_{pl}$ and
$W\lesssim\mathcal{O}(HM_{pl}^{2})$ that we used in our supergravity model of
inflation with the new FI terms. Summing all up we find that the constraints
in Eqs. (4.5) through Eq. (4.9), which arise from the gaugino mass Lagrangian
(2.21), can be satisfied while also satisfying Eq. (4.2).

Let’s study in detail the constraints following from Eq. (4.11). We begin with
looking at the first one by writing the superpotential considered in Ref. [39]

$\displaystyle W(T,z^{i})=\underbrace{W_{0}+Ae^{-aT}}_{\textrm{hidden
sector}}+\underbrace{W_{MSSM}(z^{i})}_{\textrm{observable sector}},$ (4.13)

where $W_{0}=-cA$ and $c=(1+a/3)e^{-a/2}$. From Ref. [39], we see that
$M_{S}^{4}=\Lambda+\frac{a^{2}A^{2}e^{-a}}{3}\approx(\frac{aAe^{-a/2}}{\sqrt{3}})^{2}$.
At the minimum $(T=1/2,z\sim 0)$, the value of the superpotential is
$|W|_{vac}=\frac{a}{3}e^{-a/2}A=M_{S}^{2}/\sqrt{3}$. Therefore, the constraint
$M_{S}^{3}\lesssim W$ is satisfied at the post-inflation vacuum as long as
$M_{S}<M_{pl}=1$. For large $T$ along the inflationary plateau, the
superpotential reduces to
$|W_{0}|=Ae^{-a/2}+aAe^{-a/2}/3=\frac{\sqrt{3}M_{S}^{2}}{a}+M_{S}^{2}/\sqrt{3}$,
which automatically satisfies the constraint due to the previous condition for
the vacuum. The second condition in (4.11) holds because the scalar part of
$F^{i}$ in Eq. (2.28) is given by
$F^{i}\sim-3e^{K/2}G^{i\bar{j}}(\bar{W}_{\bar{j}}+K_{\bar{j}}\bar{W})\sim 0$
along the inflationary trajectory (i.e. $z^{i}\approx 0$). The third and
fourth ones can be easily satisfied due to the fact that the derivatives of
the superpotential with respect to the matter scalars are either of order of a
low scale, much smaller than the supersymmetry breaking one, or vanish along
the inflationary trajectory.

##  5 Conclusion

We presented a simple, predictive model with moduli stabilization without
backreaction on the scalar field dynamics and parametrically large gaugino
masses. We saw that the gaugino masses in our model can be given a value close
to the Hubble scale

$\displaystyle
M_{\textrm{gaugino}}=\frac{\left<D_{V}\right>^{2}}{m_{3/2}M_{pl}^{2}}=\frac{M_{S}^{4}}{m_{3/2}M_{pl}^{2}}\sim\mathcal{O}(H)=\mathcal{O}(10^{-5}M_{pl}),$
(5.1)

where $\left<D_{V}\right>=\xi=M_{S}^{2}=HM_{pl}$, $m_{3/2}\sim H$, and $M_{S}$
is the SUSY-breaking mass scale, which is of order $10^{-2.5}M_{pl}$. This
means that gauginos can be integrated out at energies below the Hubble scale,
and thus are to all effects undetectable. Note that the gaugino masses (5.1)
do not depend on the VEVs of matter fields –which either vanish at the post-
inflation vacua or give negligibly small contributions to the masses.

As a further study, it would be interesting to seek string realization of the
gaugino mass term (2.21) proposed here. The hint for this may come from the
four-dimensional gaugino-mass term in Ref. [43], which is obtained by
dimensional reduction of a fermionic space-filling D$p$-brane action carrying
both RR and NSNS fluxes.

### Acknowledgments

H.J. is supported in part by Incheon National University RIBS Grant in 2023,
and in part by Basic Science Research Program through the National Research
Foundation of Korea (NRF) funded by the Ministry of Education through the
Center for Quantum Spacetime (CQUeST) of Sogang University
(NRF-2023R1A2C2005360). H.J. would like to thank YITP at Kyoto University for
its kind hospitality. M.P. is supported in part by NSF grant PHY-2210349 and
by the Leverhulme Trust through a Leverhulme Visiting Professorship at
Imperial College, London.

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Multi | 53 | 52 | 105
Uni | 51 | 44 | 95
$\beta_{1}^{(2)}(t)$ | 104 | 96 | 200

Dim 3 | Multi | Uni | $\beta_{0}^{(3)}(t)$
---|---|---|---
Multi | 36 | 39 | 75
Uni | 40 | 85 | 125
$\beta_{1}^{(3)}(t)$ | 76 | 124 | 200

Table 4: Crosstables for the number of simulation runs, where the univariate
(Uni) or the multivariate (Multi) estimation approach result in lower rrMSE
values for the coefficient effect functions $\beta_{0}(t)$ and $\beta_{1}(t)$
over the three dimensions in the data. The rows correspond to $\beta_{0}(t)$
with the last column showing the marginal frequencies and analogous for the
columns and last row for $\beta_{1}(t)$.

5(a) Impact of sampling regime: Estimation based on true eigenfunctions | $\gamma_{0}$ | $\gamma_{1}$
---|---|---
Sampling | Bias | rMSE | FC | Bias | rMSE | FC
Sparse | -0.003 | 0.051 | 0.960 | -0.000 | 0.073 | 0.930
Regular | -0.002 | 0.030 | 0.965 | -0.000 | 0.043 | 0.960
Irregular | 0.000 | 0.024 | 0.960 | -0.001 | 0.033 | 0.940

5(b) Impact of MFPC basis: Estimation on sparse setting with MFPCs estimated from different settings | $\gamma_{0}$ | $\gamma_{1}$
---|---|---
Sampling | Bias | rMSE | FC | Bias | rMSE | FC
True | -0.003 | 0.051 | 0.960 | -0.000 | 0.073 | 0.930
Sparse | 0.108 | 0.122 | 0.405 | -0.064 | 0.097 | 0.825
Regular | 0.131 | 0.220 | 0.495 | -0.070 | 0.124 | 0.795
Irregular | 0.093 | 0.110 | 0.520 | -0.057 | 0.093 | 0.870

5(c) Impact of MFPC truncation: Estimation with MFPCs truncated at $98\%$ based on different scalar products | $\gamma_{0}$ | $\gamma_{1}$
---|---|---
Sampling | Bias | rMSE | FC | Bias | rMSE | FC
No Truncation | 0.108 | 0.122 | 0.405 | -0.064 | 0.097 | 0.825
Equal Weights | 0.145 | 0.177 | 0.300 | -0.078 | 0.177 | 0.775
Weighted | 0.232 | 0.292 | 0.160 | -0.105 | 0.156 | 0.730

5(d) Univariate vs. multivariate: Estimation on sparse setting with true eigenfunctions | $\gamma_{0}$ | $\gamma_{1}$
---|---|---
Sampling | Bias | rMSE | FC | Bias | rMSE | FC
Multivariate | -0.003 | 0.051 | 0.960 | -0.000 | 0.073 | 0.930
Univariate | 0.005 | 0.052 | 0.955 | -0.003 | 0.073 | 0.935

Table 5: Evaluation of the estimation of the coefficients $\gamma_{0}$ and
$\gamma_{1}$ in the additive predictor $\eta_{i2}^{(3)}$ of the scale
parameter of dimension three for the different simulation scenarios. Figure
15: Estimated values of $\gamma_{0}$ and $\gamma_{1}$ of the additive
predictor of the scale parameter of dimension three for the different examined
scenarios. The red line corresponds to the true values.

##  Appendix B Application

###  B.1 Data Preprocessing

Figure 16 provides an example of a detector failure in the Berlin traffic
data. The site “Müllerstraße” is not fully observed on the time interval and
features an unrealistic jump in the outcome $vCar$ between 2017 and 2021. As
other outcomes are not affected, only this section is removed from the final
data set.

###  B.2 Estimation of Univariate Generalized FPCs

We follow the outlined approach by Zhou et al. [2023] to estimate the
univariate generalized FPCs and adapt it to the application at hand. In the
first step, the data is grouped into $s=1,...,24$ overlapping bins with one
bin containing the measurement data $t_{ij}\in[s-2,s+2]$. Note that we use
data over the ends of the domains, e.g. the first bin contains all
observations between 11 p.m. and 3 a.m., as the traffic data are of a cyclic
nature. For each of these bins, four univariate BAMLSS regression models are
estimated with the distributional assumption analoguous to the multivariate
functional data. These models ignore the functional nature of the data and
estimate site-specific and site-year-specific (scalar) random intercepts while
accounting for the covariate effects. This results in additive predictors for
the respective subset of data as

$\displaystyle\eta_{ij1}^{s,k^{\prime}}(t)=\beta_{0}^{s,k^{\prime}}+f^{s,k^{\prime}}(year_{j})+lanes1_{i}\beta_{1}^{s,k^{\prime}}+lanes3_{i}\beta_{2}^{s,k^{\prime}}+b_{1i}^{s,k^{\prime}}+b_{0ij}^{s,k^{\prime}},$
$\displaystyle\eta_{ij1}^{s,k^{\prime\prime}}(t)=\beta_{0}^{s,k^{\prime\prime}}+f^{s,k^{\prime}}(year_{j})+limit30_{i}\beta_{1}^{s,k^{\prime\prime}}+limit60_{i}\beta_{1}^{s,k^{\prime\prime}}+b_{1i}^{s,k^{\prime}}+b_{0ij}^{s,k^{\prime}}$
$\displaystyle t=s-2,...,s+2,\quad k^{\prime}\in\\{qCar,qTruck\\},\quad
k^{\prime\prime}\in\\{vCar,vTruck\\},$

where the random intercepts are denoted as $b_{1i}^{s,k}$ and $b_{0ij}^{s,k}$.
The models are estimated with 1000 MCMC samples after a burn-in of 200 using
seven cubic P-spline basis functions with second order difference penalty for
the nonlinear effect of $year$. Estimating one such univariate regression
model takes five to ten minutes on a Linux system with $12$ cores at $3.5$ GHz
and $32$ GB memory, which amounts to 11 hours total without parallelization.

The estimated random intercepts $\hat{b}_{1i}^{s,k}$ and
$\hat{b}_{0ij}^{s,k},k\in\\{qCar,qTruck,vCar,vTruck\\}$ are then used in the
fast multilevel FPCA method proposed by Cui et al. [2023] to obtain smooth
estimates of the univariate eigenfunctions $\phi_{1m}^{(k)}(t)$ and
$\phi_{0m}^{(k)}$ of the site-specific and site-year-specific processes,
taking only a few seconds. For the smooth estimation of eigenfunctions, we
choose a cyclic cubic P-spline basis with second order difference penalty and
ten knots, as we find a larger number of basis functions to yield
undersmoothed results [as described similarly by Leroux et al., 2023]. The
number of eigenfunctions $M_{1}^{k}$ and $M_{0}^{k}$ estimated by the fast
FPCA approach is determined based on the cumulative percentage of variance
explained with a threshold of 0.99. For the site-specific random process,
$\left(M_{1}^{qCar},M_{1}^{qTruck},M_{1}^{vCar},M_{1}^{vTruck}\right)=(3,6,3,3)$,
and for the site-year-specific random process
$\left(M_{0}^{qCar},M_{0}^{qTruck},M_{0}^{vCar},M_{0}^{vTruck}\right)=(6,5,6,6)$
FPCs are estimated with the estimates given in Figure 17.

In order to avoid bias introduced by the binning step, we re-estimate the
univariate scores in separate univariate generalized functional additive mixed
models with additive predictors

$\displaystyle\eta_{ij1}^{(k^{\prime})}\left(t\mid\left\\{\hat{\phi}_{1m}^{(k^{\prime})}(t)\right\\}_{m=1}^{M_{1}^{k^{\prime}}},\left\\{\hat{\phi}_{0m}^{(k^{\prime})}(t)\right\\}_{m=1}^{M_{0}^{k^{\prime}}}\right)=$
$\displaystyle\beta_{0}^{(k^{\prime})}(t)+f^{(k^{\prime})}(year_{j},t)$
$\displaystyle+lanes1_{i}\beta_{1}^{(k^{\prime})}(t)+lanes3_{i}\beta_{2}^{(k^{\prime})}(t)$
$\displaystyle+\sum_{m=1}^{M_{1}^{k^{\prime}}}\xi_{1im}^{(k^{\prime})}\hat{\phi}_{1m}^{(k^{\prime})}(t)+\sum_{m=1}^{M_{0}^{k^{\prime}}}\xi_{0ijm}^{(k^{\prime})}\hat{\phi}_{0m}^{(k^{\prime})}(t)$
$\displaystyle\eta_{ij1}^{(k^{\prime\prime})}\left(t\mid\left\\{\hat{\phi}_{1m}^{(k^{\prime\prime})}(t)\right\\}_{m=1}^{M_{1}^{k^{\prime\prime}}},\left\\{\hat{\phi}_{0m}^{(k^{\prime\prime})}(t)\right\\}_{m=1}^{M_{0}^{k^{\prime\prime}}}\right)=$
$\displaystyle\beta_{0}^{(k^{\prime\prime})}(t)+f^{(k^{\prime\prime})}(year_{j},t)+$
$\displaystyle+limit30_{i}\beta_{1}^{(k^{\prime\prime})}(t)+limit60_{i}\beta_{2}^{(k^{\prime\prime})}(t)$
$\displaystyle+\sum_{m=1}^{M_{1}^{k^{\prime\prime}}}\xi_{im}^{(k^{\prime\prime})}\hat{\phi}_{1m}^{(k^{\prime\prime})}(t)+\sum_{m=1}^{M_{0}^{k^{\prime\prime}}}\xi_{ijm}^{(k^{\prime\prime})}\hat{\phi}_{0m}^{(k^{\prime\prime})}(t),$

for $k^{\prime}\in\\{qCar,qTruck\\}$ and
$k^{\prime\prime}\in\\{vCar,vTruck\\}$, conditioning on the estimated
univariate FPCs $\hat{\phi}_{1m}^{(k)}(t)$ and
$\hat{\phi}_{0m}^{(k)}(t),k\in\\{qCar,qTruck,vCar,vTruck\\}$. The functional
intercepts $\beta_{0}^{(k)}(t)$ and the categorical functional effects
$\beta_{1}^{(k)}(t)$ and $\beta_{2}^{(k)}(t)$ are represented using $14$ cubic
P-splines while the bivariate interaction terms $f^{(k)}(year_{j},t)$ are
modeled using anisotropic tensor product splines of $7$ (for $year$) and $14$
(for $t$) marginal cubic P-splines allowing different variance parameters for
the different margins. These four univariate models are again estimated in the
BAMLSS framework with 1,000 MCMC samples after a burn-in of 1,000 and a
thinning of 5 in order to get more stable estimates of the univariate scores.
The computation of the univariate generalized functional additive mixed models
takes 12 to 18 hours per outcome. We then use the posterior mean of the scores
as estimates of the univariate scores and Table 6 reports the estimated
univariate eigenvalues from this univariate generalized FPCA approach.

###  B.3 Additional Results

Table 7 reports the estimated multivariate eigenvalues based on a weighted
scalar product. Figure 18 presents the areas where pointwise $95\%$ credible
intervals of the bivariate interaction effect $f^{(k)}(year_{j},t)$ do not
include zero. Areas that do not include zero are highlighted by their effect
(either positive or negative) compared to the global functional intercept.
Figure 19 shows the estimated coefficient effect functions
$\hat{\beta}_{1}^{(k^{\prime\prime})}(t)$ and
$\hat{\beta}_{2}^{(k^{\prime\prime})}(t),k^{\prime\prime}\in\\{vCar,vTruck\\}$
for sites with a 30 km/h or 60 km/h (or higher) speed limit, respectively,
depicting the difference compared to a site with a 50 km/h speed limit on the
linear predictor scale, all other things being equal. The dashed lines show
the pointwise $95\%$ credible intervals. Figure 20 constructs a 90% and a 95%
credible region for the joint estimate of
$\left(\hat{\beta}_{1}^{(vCar)}(t),\hat{\beta}_{1}^{(vTruck)}(t)\right)^{\top}$
at time point $t=12$ based on a two-dimensional kernel density estimate on the
1000 MCMC samples. This can be used to test the null hypothesis
$H_{0}:\hat{\beta}_{1}^{(vCar)}(12)=\hat{\beta}_{1}^{(vTruck)}(12)=0$ by
checking whether the origin $(0,0)^{\top}$ (marked in red) is included in the
$1-\alpha$ credible region. Figure 21 shows the impact of the five leading
estimated MFPCs on the mean for the site-year-specific (level 2) random
process. The black line corresponds to the estimated mean of a site with two
lanes and 50 km/h speed limit in 2020 with plus (red +) or minus (blue -)
$2\sqrt{\hat{\nu}_{2m}},m=1,2,3$ times the estimated MFPC. Table 8 reports the
relative contributions of the estimated posterior score variances to the total
(posterior unweighted) variation for the site-specific random process (level
1, $\nu_{1,m},m=1,...,9$) and the site-year-specific random process (level 2,
$\nu_{0,m},m=1,...,$) based on an MFPCA with scalar product with unequal
weights. Figure 22 shows the fitted location parameter together with the
scalar observations for three different years at the traffic detector site
“Goerzallee”. Figure 23 presents outcome specific worm plots [van Buuren and
Fredriks, 2001] of normalized (randomized) quantile residuals [Dunn and Smyth,
1996].

Figure 16: Example of detector failure for site “Müllerstraße” in the Berlin
traffic data. It is apparent that the time-series for outcome $vCar$ is
incomplete and shows an unrealistic jump for the observations from 2017 to
2021.

Figure 17: Estimated univariate FPCs for the different outcomes with level 1
denoting the estimates $\hat{\phi}_{1m}^{(k)}(t),m=1,...,M_{1}^{k}$
corresponding to the site-specific random process. Level 2 denotes the
estimates $\hat{\phi}_{m}^{(k)}(t),m=1,...,M^{k}$ for the site-year-specific
random process. The estimates are available at the bin centers and are
linearly interpolated only for this representation.

Level 1
---
Absolute | $qCar$ | $qTruck$ | $vCar$ | $vTruck$
$\upsilon_{11}^{k}$ | 0.227 | 0.393 | 0.041 | 0.056
$\upsilon_{12}^{k}$ | 0.019 | 0.073 | 0.002 | 0.010
$\upsilon_{13}^{k}$ | 0.013 | 0.016 | 0.001 | 0.001
$\upsilon_{14}^{k}$ | – | 0.010 | – | –
$\upsilon_{15}^{k}$ | – | 0.009 | – | –
$\upsilon_{16}^{k}$ | – | 0.004 | – | –
$\sum_{M^{k}}\upsilon_{1m}^{k}$ | 0.259 | 0.504 | 0.045 | 0.067
Level 2
---
Absolute | $qCar$ | $qTruck$ | $vCar$ | $vTruck$
$\upsilon_{01}^{k}$ | 0.0079 | 0.1344 | 0.0070 | 0.0112
$\upsilon_{02}^{k}$ | 0.0022 | 0.0190 | 0.0013 | 0.0019
$\upsilon_{03}^{k}$ | 0.0010 | 0.0145 | 0.0008 | 0.0013
$\upsilon_{04}^{k}$ | 0.0007 | 0.0041 | 0.0003 | 0.0005
$\upsilon_{05}^{k}$ | 0.0006 | 0.0020 | 0.0002 | 0.0004
$\upsilon_{06}^{k}$ | 0.0003 | – | 0.0001 | 0.0002
$\sum_{M^{k}}\upsilon_{0m}^{k}$ | 0.0127 | 0.1739 | 0.0097 | 0.0155

Level 1
---
Relative | $qCar$ | $qTruck$ | $vCar$ | $vTruck$
$\upsilon_{11}^{k}$ | 0.877 | 0.779 | 0.924 | 0.838
$\upsilon_{12}^{k}$ | 0.074 | 0.145 | 0.043 | 0.143
$\upsilon_{13}^{k}$ | 0.049 | 0.031 | 0.033 | 0.019
$\upsilon_{14}^{k}$ | – | 0.019 | – | –
$\upsilon_{15}^{k}$ | – | 0.017 | – | –
$\upsilon_{16}^{k}$ | – | 0.008 | – | –
$\sum_{M^{k}}\upsilon_{m}^{k}$ | 0.296 | 0.576 | 0.051 | 0.076
Level 2
---
Relative | $qCar$ | $qTruck$ | $vCar$ | $vTruck$
$\upsilon_{01}^{k}$ | 0.624 | 0.773 | 0.727 | 0.723
$\upsilon_{02}^{k}$ | 0.176 | 0.109 | 0.132 | 0.125
$\upsilon_{03}^{k}$ | 0.078 | 0.083 | 0.084 | 0.083
$\upsilon_{04}^{k}$ | 0.057 | 0.024 | 0.027 | 0.035
$\upsilon_{05}^{k}$ | 0.044 | 0.011 | 0.020 | 0.023
$\upsilon_{06}^{k}$ | 0.020 | – | 0.011 | 0.012
$\sum_{M^{k}}\upsilon_{m}^{k}$ | 0.060 | 0.821 | 0.046 | 0.073

Table 6: Estimated univariate eigenvalues for the site-specific random process
(level 1) and the site-year-specific random process (level 2). The tables on
the left report the absolute values with the last row containing the sum of
the estimated eigenvalues per dimension. The tables on the right report the
corresponding percent of explained variation within the dimension and the last
row contains the percent of explained variation per dimension over the
multivariate functional process.

Level 1
---
| Absolute | Relative | Cumulative
$\nu_{1,1}$ | 15382.305 | 0.429 | 0.429
$\nu_{1,2}$ | 9432.398 | 0.263 | 0.692
$\nu_{1,3}$ | 5163.201 | 0.144 | 0.837
$\nu_{1,4}$ | 1820.428 | 0.051 | 0.887
$\nu_{1,5}$ | 1525.451 | 0.043 | 0.930
$\nu_{1,6}$ | 838.268 | 0.023 | 0.953
$\nu_{1,7}$ | 498.967 | 0.014 | 0.967
$\nu_{1,8}$ | 430.993 | 0.012 | 0.979
$\nu_{1,9}$ | 270.716 | 0.008 | 0.987
$\nu_{1,10}$ | 187.071 | 0.005 | 0.992
$\nu_{1,11}$ | 129.847 | 0.004 | 0.996
$\nu_{1,12}$ | 60.462 | 0.002 | 0.997
$\nu_{1,13}$ | 42.722 | 0.001 | 0.999
$\nu_{1,14}$ | 26.987 | 0.001 | 0.999
$\nu_{1,15}$ | 25.547 | 0.001 | 1.000

Level 2
---
| Absolute | Relative | Cumulative
$\nu_{0,1}$ | 28455.212 | 0.272 | 0.272
$\nu_{0,2}$ | 22257.918 | 0.212 | 0.484
$\nu_{0,3}$ | 11770.427 | 0.112 | 0.596
$\nu_{0,4}$ | 10453.869 | 0.100 | 0.696
$\nu_{0,5}$ | 7540.123 | 0.072 | 0.768
$\nu_{0,6}$ | 5301.006 | 0.051 | 0.819
$\nu_{0,7}$ | 4994.258 | 0.048 | 0.866
$\nu_{0,8}$ | 3057.358 | 0.029 | 0.896
$\nu_{0,9}$ | 2603.484 | 0.025 | 0.920
$\nu_{0,10}$ | 1846.931 | 0.018 | 0.938
$\nu_{0,11}$ | 1350.284 | 0.013 | 0.951
$\nu_{0,12}$ | 1269.752 | 0.012 | 0.963
$\nu_{0,13}$ | 963.401 | 0.009 | 0.972
$\nu_{0,14}$ | 893.100 | 0.009 | 0.981
$\nu_{0,15}$ | 756.275 | 0.007 | 0.988
$\nu_{0,16}$ | 384.172 | 0.004 | 0.992
$\nu_{0,17}$ | 352.843 | 0.003 | 0.995
$\nu_{0,18}$ | 317.335 | 0.003 | 0.998
$\nu_{0,19}$ | 169.490 | 0.002 | 1.000
$\nu_{0,20}$ | 30.391 | 0.000 | 1.000
$\nu_{0,21}$ | 8.652 | 0.000 | 1.000
$\nu_{0,22}$ | 0.038 | 0.000 | 1.000
$\nu_{0,23}$ | 0.031 | 0.000 | 1.000

Table 7: Estimated multivariate eigenvalues for the site-specific random
process (level 1, $\nu_{1,m},m=1,...,15$) and the site-year-specific random
process (level 2, $\nu_{0,m},m=1,...,23$) for an MFPCA with weighted scalar
product. Figure 18: Areas where the pointwise $95\%$ credible intervals of the
estimated bivariate interaction effect $f^{(k)}(year_{j},t)$ do not include 0.
Areas where the estimated effect is larger than 0 are colored in red, effects
lower than 0 in blue. Figure 19: Estimated coefficient effect functions
$\hat{\beta}_{1}^{(k^{\prime\prime})}(t)$ and
$\hat{\beta}_{2}^{(k^{\prime\prime})}(t),k^{\prime\prime}\in\\{vCar,vTruck\\}$
for sites with a 30 km/h or 60 km/h (or higher) speed limit, respectively, on
the linear predictor scale. The dashed lines show the pointwise $95\%$
credible intervals. Figure 20: Estimated coefficient effects of
$\hat{\beta}_{1}^{(vCar)}(t)$ and $\hat{\beta}_{1}^{(vTruck)}(t)$ at time
point $t=12$ based on the MCMC samples. Empirical credible regions are
included as contour lines with the origin marked in red.

Figure 21: Leading five estimated MFPCs based on a weighted scalar product for
the site-year-specific (level 2) random process. The black line corresponds to
the estimated mean of a site with two lanes and 50 km/h speed limit in 2020
with plus (red +) or minus (blue -) $2\sqrt{\hat{\nu}_{2m}},m=1,2,3$ times the
estimated MFPC.

Level 1
---
| Relative
$\hat{\nu}_{1,1}$ | 0.032
$\hat{\nu}_{1,2}$ | 0.404
$\hat{\nu}_{1,3}$ | 0.385
$\hat{\nu}_{1,4}$ | 0.040
$\hat{\nu}_{1,5}$ | 0.067
$\hat{\nu}_{1,6}$ | 0.019
$\hat{\nu}_{1,7}$ | 0.008
$\hat{\nu}_{1,8}$ | 0.025
$\hat{\nu}_{1,9}$ | 0.020

Level 2
---
| Relative
$\hat{\nu}_{0,1}$ | 0.084
$\hat{\nu}_{0,2}$ | 0.068
$\hat{\nu}_{0,3}$ | 0.059
$\hat{\nu}_{0,4}$ | 0.058
$\hat{\nu}_{0,5}$ | 0.431
$\hat{\nu}_{0,6}$ | 0.017
$\hat{\nu}_{0,7}$ | 0.018
$\hat{\nu}_{0,8}$ | 0.074
$\hat{\nu}_{0,9}$ | 0.017
$\hat{\nu}_{0,10}$ | 0.047
$\hat{\nu}_{0,11}$ | 0.010
$\hat{\nu}_{0,12}$ | 0.030
$\hat{\nu}_{0,13}$ | 0.031
$\hat{\nu}_{0,14}$ | 0.056

Table 8: Relative contributions of the estimated posterior score variances to
the total (posterior unweighted) variation for the site-specific random
process (level 1, $\nu_{1,m},m=1,...,9$) and the site-year-specific random
process (level 2, $\nu_{0,m},m=1,...,14$) based on an MFPCA with scalar
product with unequal weights. Figure 22: Estimated mean function (line) and
scalar observations (points) for site “Goerzallee” for three different years.
Figure 23: Dimension specific worm plots corresponding to de-trended QQ-plots
of normalized (randomized) quantile residuals.