Renormgroup origin and analysis of Split Higgsino scenario
V. A. Beylin,∗ V. I. Kuksa,† and G. M. Vereshkov‡
Institute of Physics, Southern Federal University (former
Rostov State University), Rostov-on-Don 344090, Russia
R. S. Pasechnik§
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna 141980, Russia
(Dated: November 4, 2018)
Abstract
We present a renormalization group motivation of scale hierarchies in SUSY SU(5) model. The
Split Higgsino scenario with a high scale of the SUSY breaking is considered in detail. Its mani-
festations in experiments are discussed.
PACS numbers: 12.60.Jv, 95.35.+d, 95.30.Cq
∗Electronic address: vbey@rambler.ru
†Electronic address: kuksa@list.ru
‡Electronic address: gveresh@gmail.com
§Electronic address: rpasech@theor.jinr.ru
http://arxiv.org/abs/0704.0625v1
mailto:vbey@rambler.ru
mailto:kuksa@list.ru
mailto:gveresh@gmail.com
mailto:rpasech@theor.jinr.ru
I. INTRODUCTION
Supersymmetric generalization of the SM permits one to solve some inner problems of
the theory. However, the MSSM is an ambiguous model: despite the defined set of physical
fields, only one characteristic MSSM parameter – electroweak scale MEW – has been fixed in
experiment. The scales of scalar quarks and leptons M0, gaugino M1/2, and Higgsino µ, can
be picked out as arbitrary ones. Expectations of new dynamical effects of the supersymmetry
at the LHC are induced in the MSSM by some specific choice of the SUSY breaking scale
that is not very far from the EW scale, MSUSY ∼ M1/2 ∼ M0 ∼ O (1 TeV). This scale
hierarchy seems “natural”, providing regularization of quadratic divergencies long before
MGUT . Thus, the MSSM as it is motivated by some theoretical and phenomenological
arguments – successful RG evolution, “natural” choice of the renormalization scale and the
reasonable dark matter (DM) description – works well for the specific choice of the SUSY
breaking scale only. However, the “naturalness” as one of the MSSM principles does not
seem to be the obligatory requirement from the QFT point of view. So establishing of the
genuine SUSY scales hierarchy is a problem which can be unambiguously solved beyond the
MSSM.
The arbitrariness in the choice of the scale hierarchy can be sufficiently diminished when
the gauge coupling unification is considered as the most fundamental theoretical basis. With
this consideration as a starting point, we have found that the one-loop RG study, involving
SUSY SU(5) degrees of freedom placed near MGUT , allows one to select some specific classes
of the scale hierarchies. The proton stability is provided simultaneously.
It was shown that states nearMGUT should be taken into account as threshold corrections;
they are crucially important for the selection of scales, giving sufficiently high unification
point. Then the RG consideration at the one-loop level results in two classes of scenarios
with an opposite arrangement of µ and M1/2 scales. At the same time, the analysis does
not fix the characteristic scalar scale M0 due to a specific form of the RG equations which
contain the squark and slepton scales as a ratio Mq̃/Ml̃ only.
For the first class of scenarios the hierarchy |µ| ≫ M1/2 takes place. The second class
is defined by the hierarchy |µ| ≪ M1/2. Due to arbitrariness of M0 it is possible to select
some subscenarios with various M0 arrangements. Among them there can be found some
hierarchies corresponding to Split Symmetry and some ideologically close scenarios which do
not reject fine-tuning and are motivated, on the one hand, by the anthropic principle and,
on the other hand, multi-vacua string landscape arguments [1, 2, 3, 4, 5, 6] (see, however,
comments in [7]).
In particular, the RG analysis results in the hierarchy
M0 ∼ M1/2 ≫ |µ| > MEW , (1.1)
whose spectrum contains two lightest neutralinos degenerated in mass (almost pure Higgsino)
and one of charginos as the states that are the nearest to the electroweak scale. In this Split
Higgsino scenario both M0 and M1/2 are shifted to scales ∼ (107 − 1010) TeV.
In this paper, we consider some features and manifestations of the last scenario only (a
preliminary version of this work was presented in [8], see also [9]).
As it will be shown, direct observation of neutralino (Higgsino) in χ − N scattering is
impossible nowadays due to a very small interaction cross section. As to collider experiments,
the signature of neutralino and chargino production and decays at the LHC crucially depends
on the mass splitting values for these degenerate states [9, 10, 11, 12, 13]. On the one hand,
observation of these hardly detectable effects means that it is possible to get important
information on the high superstate structure, in particular, t̃1,2 and their mixing, from one-
loop calculations of the mass splitting [14, 15, 16]. On the other hand, if only these low-lying
SUSY degrees of freedom are detected in experiments near a TeV scale, the conventional
MSSM spectrum should necessarily be splitted in some manner to produce a high scale for
other superstates. The value predicted for the diffuse gamma flux from the Galactic halo
can be measured by GLAST or ACT in the near future but it is not peculiar to this scenario.
Signals from direct photons do not contain any specific effects too [9].
Thus, the Split Higgsino scenario is an example of SUSY models, in which the lightest
degrees of freedom manifest themselves in some subtle effects as specific decay channels,
correlating with the value of diffuse (or direct) gamma flux. Note, even if collider experiments
will demonstrate the absence of the lowest neutralino signals (due to a high degeneracy of
states, for example, it will certainly testify to some fine-tuning in the SUSY states spectrum),
the SUSY, as a theoretical principle, cannot be rejected. At low O (1 TeV) energies the
SUSY, in a sense, is barely hidden, so it exists at a high scale.
The structure of the paper is as follows. In Section 2, we discuss RG analysis of the SUSY
SU(5) in detail, indicating the most interesting scenarios. In Section 3, the Lagrangian of
the Split Higgsino model and its analysis are presented. There is also an estimation of the
lowest Higgsino mass from the relic abundance data. We discuss expected experimental
manifestations of the scenario in Section 4 and some conclusions are in Section 5.
II. RENORMGROUP ANALYSIS
We consider one-loop RG equations for running couplings and start from their values at
the MZ scale
α−1(MZ) = 127.922± 0.027, αs(MZ) = 0.1200± 0.0028,
sin2 θW (MZ) = 0.23113± 0.00015. (2.1)
In renormgroup equations the following values are used as initial:
(MZ) =
α−1(MZ) cos
2 θW (MZ) = 59.0132∓ (0.0384)sin2 θW ± (0.0124)α,
(MZ) = α
−1(MZ) sin
2 θW (MZ) = 29.5666± (0.0192)sin2 θW ± 0.0062)α, (2.2)
(MZ) = α
s (MZ) = 8.3333± 0.1944.
At the one-loop level the equations for running couplings are well-known
α−1i (Q2) = α
i (Q1) +
, bi =
bij . (2.3)
In the sum all states with masses Mj < Q2/2 at Q2 > Q1 are taken into account.
Namely, we include in the RG analysis the following degrees of freedom: singlet quarks
and their superpartners (DL, D̃L), (DR, D̃R) contained in super-Higgs quintets of SUSY
SU(5); chiral superfields (ΦL, Φ̃L) and (ΨL, Ψ̃L) in adjoint representations of SU(2) and
SU(3), respectively, which survive from super Higgs 24-plet. In the minimal SUSY SU(5)
masses of the states M5 = (MD, MD̃), M24 = (MΨ, MΨ̃, MΦ, MΦ̃) are generated by an
interaction with Higgs condensate at the GUT scale, but the interaction couplings are not
fixed. For the analysis we assume masses M5, M24 to be slightly lower thanMGUT . Note that
heavy states near MGUT are of especial importance for the RG equations. The necessity of
taking into account of such heavy states in the RG analysis as so-called threshold corrections
is well-known [17, 18, 19, 20, 21, 22].
For one-loop running couplings at the scale q2 = (2MGUT )
2 we have
(2MGUT ) = α
(MZ)−
ln 2 +
−7 lnMGUT +
lnMD +
lnMD̃
lnMq̃ +
lnMl̃ +
lnMH +
lnMt +
(2MGUT ) = α
(MZ)−
ln 2 +
−3 lnMGUT +
lnMq̃ +
lnMl̃ +
lnMW̃ +
lnMH +
lnMt −
(2MGUT ) = α
(MZ) +
ln 2 +
(− lnMGUT + 2 lnMΨ̃ + lnMΨ
lnMD +
lnMD̃ + 2 lnMq̃ + 2 lnMg̃ +
lnMt −
(2.4)
Here M0 = (Mq̃, Ml̃) are masses of scalar quarks and leptons averaged over chiralities and
generations; Mt is t-quark mass; other parameters were introduced above. In (2.4) it is
supposed that the lightest Higgs boson mass mh is near to MEW and other Higgs bosons
H, A, H± are placed at some high MH scale. For extra heavy states the only condition is
M24, M5 < MGUT and we suppose that this inequality is fulfilled with accuracy within 1 - 2
orders of magnitude. As to SUSY degrees of freedom M0, M1/2, µ, MH , equations (2.4) do
not depend on any specific arrangement of these scales. So when MGUT is the highest scale
in the system, there is no necessity to establish an initial scale hierarchy. The only demand
is that the RG equations should lead to coupling unification at sufficiently high MGUT scale
for the proton stability. Then the set of possible hierarchies of energy scales arises as the
final result of the study. Note also that if masses of singlet superstates and residual Higgs
superfields are equal to MGUT , equations (2.4) return to the standard form automatically.
The above considerations together with experimental restrictions on the SUSY mass
spectrum and general expectation of the lightest Higgs boson scale (it is not very far from
the EW scale) are sufficient for the RG analysis.
As the first step, all couplings were recalculated at the 2MZ scale, all the SM states
contribute to running of couplings despite W±, Z0, Higgs bosons, and t-quark. At the same
time, terms with ln 2 occur which are quantitatively important for calculations. Between the
(2MZ , 2Mt) scales the following states emerge: W
±, Z0 and one Higgs doublet containing
light h-boson and longitudinal degrees of freedom ofW±, Z0. At these stages, Zq̄q vertex was
used for calculations of α−1
(2Mt), starting from α
(2MZ) value. Above 2Mt calculations
were carried out in a standard manner.
Now, equating couplings at MGUT , from (2.4) we get the following expressions:
MGUT = Ak1MZ
, µ = Bk2MZ
, (2.5)
where
k1 = K
−1/12
GUT1 ≡
)1/12(
M ′GUT
k2 = K
K−1GUT2 ≡
)1/4(
)1/4(
)5/2(
M ′′GUT
1/2 ≡ (MW̃Mg̃)
1/2, M ′GUT ≡ (MΨ̃MΦ̃)
1/3(MΨMΦ)
1/6 ≤ MGUT ,
M ′′GUT ≡
7/6(M2DMD̃)
MΦ)4/3
≤ MGUT ,
A = exp
(5α−1
(MZ)− 3α−12 (MZ)− 2α−13 (MZ))−
= (1.57×1.09
0.92) · 1014,
B = exp
(5α−1
(MZ)− 12α−12 (MZ) + 7α−13 (MZ)) +
= (2.0×0.15
6.56) · 103.
Here all dimensionless parameters K with various indices are defined as quantities having
values larger than unity (see also [23]). Note that KGUT1, KGUT2 are not under theoretical
control either in the MSSM or in the SUSY SU(5). So we assume that 1 ≤ KGUT1 ≃
KGUT2 ≤ 10. For the lightest Higgs boson there is an experimental restriction [24]: Mh >
114.4 GeV, as to other (heavy) Higgs bosons the following interval 2 ≤ KHt ≤ 10 is supposed
for the numerical analysis. Values of Kq̃l̃, Kg̃W̃ can be determined from the renormgroup
evolution from MGUT to M0, M1/2. Here we suppose that 1.5 ≤ Kq̃l̃ ≃ Kg̃W̃ ≤ 2.5. Now all
dimensionless parameters are fixed in some intervals and we analyze M ′
1/2 and µ as functions
of MGUT :
1/2(MGUT ) = (Ak1)
GUT , µ(MGUT ) =
(Ak1)3/2M
×M3/2GUT . (2.6)
As it is seen, the initial system of equations for three running couplings can be rewritten
as a system of two equations for the effective gauginoM ′
and µ scales, depending onMGUT .
It is very essential that all other characteristic scales arise in the equations as dimensionless
ratios only. (This method of RG analysis was used in [23] for investigation of SUSY SU(5)
and E6 energy scale hierarchies; see also [20, 22].)
As an important feature, we have to note that the renormgroup study remains a common
scalar scale M0 as an arbitrary one: the scale M0 turns into (2.6) through the ratio Mq̃/Ml̃
only. For numerical predictions the ratio value was estimated as O(1). Certainly, the
splitting of the Mq̃ and Ml̃ scales will affect the hierarchies of other scales. However, to
change these hierarchies crucially, the ratio Mq̃/Ml̃ must be ∼ O(100) or larger, and here we
do not consider this possibility. Also, we suppose that radiative corrections to superscalar
masses do not change the ratio substantially (numerically, these corrections can be as large
as (1 − 5)%, see [14]). Conclude, the scale M0 should be fixed by some extra arguments
independently.
Contributions of heavy states M24, M5 do not affect the RG hierarchy of SUSY scales
radically because their scales are combined into a ratio too. Nevertheless, the ratio value
is important to fix the MGUT scale: the ratio M24/M5 depends on MZ logarithmically, so
numerical coefficients of the ratio can change the MGUT value.
For the analysis we used the known restrictions for the proton lifetime (τp ≥ 1032 yr at
MGUT ≥ 1015 GeV) and for MSUSY that is ∼ M ′1/2 > 100 GeV when MGUT < 3 · 1016 GeV).
We hope also that loop corrections to all masses in the model do not change results of the
RG analysis drastically, it is supposed that for the scenario they contribute no more than
∼ (10− 15)% [14, 15].
1015 1016
Higgsino
   1: K
=1.5 , K
=3 , K
   2: K
=1.5 , K
=5 , K
   3: K
=2.5 , K
=2 , K
   4: K
=2.0 , K
=8 , K
 ,  GeV
FIG. 1: SUSY SU(5) hierarchies of scales permitted by the one-loop RG analysis.
Numerical results following from (2.6) are shown in Fig. 1. It is clearly seen that the
proposed method of the one-loop RG analysis results in the selection of two proper sets
of SUSY scales only (further we will omit differences between M ′
and M1/2). For this
conclusion we assume that the characteristic µ scale is in the O(1 TeV) region, providing
the neutralino mass value that is sufficient for explanation of the DM properties.
The first class of scenarios is defined by the hierarchy
|µ| ≫ M1/2 & MEW .
As it was noted, M0 value does not affect this scale splitting dictated by the one-loop RG
evolution, and can be placed at an arbitrary but physically reasonable scale. So the class
could be divided into some subscenarios with hierarchies
(a) M0 ≫ |µ| ≫ M1/2 & MEW ,
(b) M0 ∼ |µ| ≫ M1/2 & MEW ,
(c) |µ| ≫ M0 ≫ M1/2 & MEW ,
(d) |µ| ≫ M0 ∼ M1/2 & MEW .
The subscenarios (a), (b) and (c) are close to the known Split SUSY (Gaugino) models [1, 2,
3], but the variant (d), keeping most part of the SUSY degrees of freedom (except Higgsino)
near the EW scale, is similar to the MSSM spectrum with “naturalness”. For these variants
we get MGUT = (1− 5) · 1016GeV and gaugino masses can be as small as (0.1− 1.0) TeV.
The second set of scenarios is generated by the hierarchy
M1/2 ≫ |µ| > MEW .
In this set we have the following subscenarios:
(e) M0 & M1/2 ≫ |µ| > MEW ,
(f) M1/2 ≫ M0 ∼ |µ| > MEW ,
(g) M1/2 ≫ M0 ≫ |µ| > MEW ,
(h) M1/2 ≫ |µ| > M0 & MEW .
Except the last variant (h) with too low M0 scale as contradicting known experimental
data, the other scenarios of this class seem reasonable. All of them introduce Higgsino as
the DM carrier. Note that the hierarchy (f) is close to the MSSM spectrum again, despite
gaugino states at a high scale. Scenarios (e) and (g) shift M0 and M1/2 scales into a multi-
TeV region, splitting the spectrum. In these cases, the µ scale is the nearest to the EW
scale and should be near 1TeV, as it follows from the RG analysis. As to the unification
scale, we have the value MGUT = (1− 2) · 1015GeV which makes the proton stable.
Obviously, Split SUSY models with highM0 andM1/2 scales lead to the damping of heavy
states (squarks, sleptons) contributions near the EW scale. Another important consequence
of this type of scenarios is (nearly) degeneration in mass of the lowest neutralino and chargino
states [9, 11, 13, 16, 25, 26] (experimental searches of this degenerated states were carried
out at LEP energies [27] without any evident signals).
Further, we will consider the Split Higgsino scenario with hierarchy
M0 & M1/2 ≫ |µ| > MEW .
Taking MGUT value as & 10
15 GeV, for µ = 1.0 TeV the SUSY breaking scale is MSUSY ∼
M1/2 ∼ (0.1− 2.5) · 108 GeV; when µ = 1.4 TeV we get MSUSY ∼ (0.5− 2.8) · 108 GeV. For
these cases MGUT ∼ (1.0− 1.7) · 1015 GeV. The used values of µ, as it will be seen further,
agree with modern data on relic DM abundance. Intervals of MSUSY values follow mainly
from uncertainties in the parameters KGUT1 and KGUT2. Most evidently, the scenario is
selected when KGUT1, KGUT2 ∼ 5 − 8; it means that the effect of threshold corrections is
essentially important for the discovery of hierarchy. Namely, if masses M24, M5 are equal
to MGUT , it is questionable to select several possible scenarios that are compatible with the
DM data.
These one-loop results should be refined with taking into account of two-loop β-functions
and loop corrections to masses. By now we expect that two-loop RG analysis does not change
the above hierarchies radically as well as mass corrections. Certainly, various numerical
coefficients will change, arrangements of the scales will move somehow, but the relative
spacing between µ and M1/2 scales will not change, as we hope. Nevertheless, the M0 scale
can be splitted by the RG at the two-loop level, so positions of the Mq̃ and Ml̃ scales can be
determined independently. More carefully this question will be analyzed in a forthcoming
paper. Now we will consider the Lagrangian of the Split Higgsino scenario, its structure,
and features.
III. THE SPLIT HIGGSINO SCENARIO
A. The model Lagrangian and its features
When the group of symmetry is fixed, the Lagrangian as a local gauge group invariant
can be written in a standard manner. Nevertheless, its specific form depends on the chosen
field representation. In this paper, we use the formalism of Majorana spinors and define all
physical fields as having positive masses only.
More precisely, in the scenario considered the neutralino mixing parameter is proportional
to the ratio MZ/MSUSY , so it is negligibly small due to a high scale of SUSY breaking.
However, the mixing can be important when the model restrictions are analyzed in the
framework of a specific mass spectrum with a small splitting of the lightest neutralino
masses.
As to neutralino masses, i.e., the scale of µ parameter, diagonalization of the neutralino
mass matrix by an orthogonal matrix usually leads to emerging of neutralino state with
negative mass. This should be taken into account in calculations by the corresponding
definition of the propagator (if negative mass neutralino is in the intermediate state) or in
the neutralino polarization matrix (if the neutralino is in the initial or in the final state).
To evade this inconvenience, we redefine the neutralino field with the negative mass in the
following way: χ → iγ5χ. This operation provides positive neutralino mass keeping standard
calculation rules simultaneously. Moreover, it does not change Majorana type of the field. It
also has been shown that this procedure is equivalent to the diagonalization by the unitary
matrix instead of the orthogonal one, for details see [28].
Hence, in the scenario two lowest neutralino states are Majorana Higgsino-like ones. The
set of scales leads to the strong splitting in the neutralino and chargino spectra, so masses
of these lightest neutralinos χ0
1, 2 and the light chargino χ
emerge near the µ scale. Light
neutralino states are built from the initial fields h1,2 in the limit of pure Higgsino when
mZ/µ → 0 and mZ/MSUSY → 0; here MSUSY ∼ M1 ∼ M2. Masses of the lightest states are
≃ µ− M
Z(1 + sin 2β)
2M1M2
(M1 cos
2 θW +M2 sin
2 θW ),
≃ µ+ M
Z(1− sin 2β)
2M1M2
(M1 cos
2 θW +M2 sin
2 θW ), (3.1)
≃ µ− µ
sin 2β.
It is seen that the spectrum of the lowest states, χ0
1, 2 and χ
, is nearly degenerated. For
the Higgsino DM this fact is known (see, for example, [16, 25, 26, 29]). Two other neutralino
states, χ0
3, 4, and heavy chargino χ
are placed far from the lightest ones, near the high scale
MSUSY .
The initial Lagrangian of Z −h1,2 interactions, including the µ-term, is the following (for
more details see [28])
gZZµ(h̄1Lγ
µh1L + h̄2Rγ
µh2R)− µ(h̄1Rh2L + h̄1Lh2R). (3.2)
Taking into account that h̄aLγ
µhaL = −h̄aRγµhaR, h̄1Rh2L = h̄2Rh1L, from (3.2) in the limit
of pure neutral Higgsino it follows:
L = −
gZZµh̄Dγ
µhD − µh̄DhD, (3.3)
where hD = h1R + h2L is the Dirac neutral field and gZ = g2/ cos θW . When the field hD is
expressed through Majorana fields χ1 and χ2 as
hD ≡ h1R + h2L = (χ1 − iχ2)/
2, (3.4)
for the transition (h1, h2) → (χ1, χ2) we get
h1,2 =
(χ1 ∓ iγ5χ2); χ1 =
(h1 + h2), χ2 =
(h1 − h2). (3.5)
All processes near the EW scale are described by the SM Lagrangian together with extra
Lagrangian of the Higgsino interactions with photons and vector bosons:
∆L ≃ −
eAµ +
2 cos θ
cos 2θZµ
µχc +
2 cos θ
W+µ (iχ̄
+ χ̄0
)γµχc +
W−µ χ̄cγ
µ(iχ0
), (3.6)
where χc denotes the lowest chargino state, χ
. Remember that χ0
1, 2 states are the Majorana
spinors and chargino χc is the Dirac spinor.
In the above Lagrangian the vertices are considered in the limit of zero mixing, because
contributions to the vertices induced by a mixing are of an order of MZ/MSUSY . So correc-
tions induced by mixing are negligibly small and can be omitted. However, the same small
mixing contributions to the particle mass spectrum should be involved into analysis. The
reason is that the decay widths depend on the mass splitting crucially.
B. Split Higgsino as the DM carrier
Supposing that the lightest Higgsino-like neutralino is a carrier of the DM in the Universe,
we evaluate its mass from the DM relic abundance.
In accordance with the known method for the relic abundance calculation (see, for ex-
ample, [30, 31, 32, 33] and [34] with references therein) before freeze-out neutralinos are
in the thermodynamical equilibrium with other components of the cosmological plasma.
Relic is formed by the irreversible annihilation process – from the moment, when the freeze-
out is reached, up to the present day when temperature is approximately equal to abso-
lute zero (this approximation is sufficient for the case). With the standard estimation of
xf = Mχ/Tf ≈ 20−25 the value of Tf can be well above TEW ∼ 100 GeV only ifmχ > 2 TeV.
Now, if the lowest Higgsino masses were well above TEW , the irreversible annihilation pro-
cess could start before electro-weak phase transition (it is considered as the first order phase
transition) in the high-symmetric phase of the cosmological plasma. Then the plasma does
not contain any Higgs condensate, and all standard particles, except Higgsino, are massless
(more exactly, their masses ≪ T ).
In the ordinary low-symmetric case, the neutralino annihilation cross section is calculated
with the Lagrangian (3.6). In relic calculations all possible coannihilation channels were
taken into account (there is no coannihilation with squarks and/or sleptons), namely
χαχ̄β → ZZ, W+W−
χcχ̄c → ZZ, W+W−, f f̄ , γγ, Zh
χαχ̄c → ZW, lνl, qiq̄j , γW, Wh,
(3.7)
where α, β = 1, 2 denote the lowest neutralino states.
In the scenario the effective neutralino annihilation cross section is:
< (σv)ann >=
128 πM2χ
27 + 4(3 + 5t4W )(c
W + s
W )− kc2W +
(c4W + s
2 c4W
(c4W + s
W ) + (c
W − s2W )4
2 c2W
10− 1
+ 2kc2W
(3.8)
where tW = tanΘW , sW = sinΘW , cW = cosΘW and k = M
W . To compare the
calculated value of Ωh2 with an experimental corridor of the relic data [35], we use known
values of the above parameters and extract the following LSP (Higgsino) mass: Mχ =
1.0 − 1.4 TeV for xf = 25 and Mχ = 1.4 − 1.6 TeV for xf = 20. These values do not spoil
the gauge coupling convergence and are in agreement with the results from [2, 3, 9, 29].
Thus, in the model where two lightest neutralinos and one chargino are the closest to the
EW scale having masses O(1 TeV). Further, we will use Mχ = 1.4 TeV for all numerical
estimations as an average value.
To answer whether the neutralino annihilation process can start in the high-symmetric
phase, we calculate the annihilation cross section with the Lagrangian which contains phys-
ical states presented by chiral fermions and gauge fields B,Wa (a = 1, 2, 3). Due to the
absence of the Higgs condensate neutralino and chargino degrees of freedom join into the
fundamental SU(2) representation, i.e., Dirac field χD. All states of this field are dynam-
ically equivalent and correspond to quantum numbers of the restored (unbroken) SU(2)
symmetry.
In t- and s-channels all cross sections of Higgsino annihilation into gauge bosons and
massless fermions were calculated analogously to the QCD calculation technology. The only
difference is that it is necessary to consider all channels with initial and final states which
have an arbitrary color in two dimensions, corresponding to the restored SU(2) symmetry.
Calculating the total neutralino annihilation cross section, from the known corridor of the
relic abundance values we get the lowest Higgsino mass ∼ 1 TeV again. It contradicts the
initial supposition that Mχ > 2 TeV to provide freeze-out temperature higher than the EW
phase transition temperature. So in the framework of orthodox notions (mechanisms of
extra entropy production or superheavy states decays, etc. are not considered) this scenario
does not allow the DM to be created in the high symmetric phase.
IV. SUSY SCALES AND EXPERIMENTAL POSSIBILITIES
A. χ−N scattering and collider signals
It is important to understand how an experimental study of the Split Higgsino scenario
can be realized. Here we discuss some possibilities that can be given by neutralino-nucleon
scattering processes, collider experiments with SUSY particles creation and the data on
photon spectrum from neutralino annihilation. The last one will be considered in the next
subsection.
An experimental observation of the scenario manifestations depends on neutralino mass
splitting parameters ∆Mχ0 = Mχ0
and ∆Mχc = Mχc −Mχ0
crucially. These splittings
are determined by the sum of tree splittings and radiative corrections to masses. Depending
on the M0 spacing and structure of high energy states, loop corrections can be comparable
with tree values of ∆Mχ (∆Mχ0 or ∆Mχc) or even exceed them [12, 14, 15, 16, 26]. As it is
known [16], the hierarchy of t̃1 and t̃2 states and their mixing angle Θt drive the value of the
mass difference when squarks dominate in loops ∆Mχc ∼ ∆Mχ0 ∼ m3t sin(2Θt) · ln(m2t̃1/m
Due to these (large) corrections the mass splittings can be increased up to ∼ 10GeV [16, 26].
Then let us consider two possible variants.
Firstly, let there be large mass splittings ∆Mχ ∼ (1−10) GeV or even larger. When this
is the case, the lowest Higgsino and chargino states can be detected at the LHC, in particular,
due to specific decay channels of χ0
and χc – the corresponding analysis was considered in
some detail in [9, 10, 11, 12], so we do not repeat it here. In the Split Higgsino hierarchy
M0 & M1/2 ≫ µ, squarks (sleptons) slip out of the LHC experiment (together with the high
energy gaugino), keeping specific chargino (neutralino) decays as the observable only.
If, however, M1/2 ≫ M0, there are two additional possibilities. Namely, if M1/2 ≫
M0 ∼ µ, i.e., squarks (sleptons) have masses ∼ (1 − 2) TeV, an occasion arises to observe
their signals at the LHC. (If mq̄ and/or ml̄ are sufficiently close to the lowest neutralino
mass, some coannihilation contribution to the effective neutralino annihilation cross section
is produced.) When M1/2 ≫ M0 ≫ µ there are no observable effects of squarks (sleptons)
at the LHC scale.
Both of these last subscenarios are very peculiar due to a large splitting between M1/2
and M0 – it can produce relatively long-lived superscalars at TeV (or higher) scale. In this
case, specific manifestations of these states both at colliders (long-lived squarks and /or
sleptons, changes in their decay modes hierarchy etc.), and in neutralino-nucleon scattering
(large contribution to SI interaction) should be. In this paper, however, we concentrate only
on the scenario with M1/2 ∼ M0 where superscalars are very heavy.
Neutralino-nucleon interaction in the scenario behaves as a threshold process due to the
formalism accepted – the corresponding term in the Lagrangian is nondiagonal in neutralino
fields: i(g2/2 cos θ)Zµχ̄
. In the case of pure neutralino states, zero order contribu-
tions to χ − N interaction correspond to the spin-independent (SI) inelastic process. This
conclusion results from the interaction Lagrangian – nondiagonal Higgsino current χ0
interacts with Zµ as a vector, rather than an axial vector, it is a consequence of the real
4-dimensional Majorana formalism used (see [28]).
At the tree level the zero order SI cross section for χ0
− N reaction takes the following
threshold form
σSIχ0N =
g4M2N
64π cos4 θWM
EN −∆Mχ0
. (4.1)
In the non-relativistic case EN = Wk(mN/Mχ), where Wk is an average kinetic energy of
the neutralino in the Sun neighborhood, Wk = Mχv
r/2. For Mχ0 ∼ 1 TeV this energy
Wk ∼ 1 MeV. So for ∆Mχ0 as large as (1 − 10)GeV the threshold value for the reaction
energy is unattainable. Then the process is forbidden, and cosmic neutralinos cannot be
detected in the direct terrestrial experiments.
The cross section for the neutralino-nucleon scattering with chargino production (recharge
process) is similar to (4.1) and for large ∆Mχc ≈ 0.5∆Mχ0 this channel is also closed.
Corrections induced by a nonzero mixing and/or loop diagrams cannot make these non-
diagonal reactions visible because part of correction amplitudes is damped in the limit of
pure Higgsino; contributions from squark exchanges are small due to large squark masses
(see also [36, 37]). Furthermore, due to Majorana nature of neutralino, nonzero loop contri-
butions are proportional to the small parameter q2/M2Z ≪ 1. As concerns elastic (diagonal)
channels, they contribute to the χ−N cross section via loops or due to a nonzero mixing,
so their yield is small as well.
Returning to some RG arguments, it seems that when M0 & M1/2 we can expect ∆Mχ
somewhat lower than for the case M1/2 ≫ M0 when the splitting between t̃1 and t̃2 can be
larger. Certainly, the relative splitting and mixing of these states are really unknown.
So let us consider the second case with ∆Mχ < 1GeV and it can be as low as ∼ (100−
300)MeV if mass splittings are mainly determined by tree contributions. As to collider
signature, in this case only photon, neutrino pair or low energy e−e+, µ−µ+, π−π+ pairs can
be created in the final states, but it is hard to select these events from the background (see
also [9, 10, 11]). So in the case of low ∆Mχ, degenerated Higgsino and chargino are indeed
invisible at the LHC.
The Higgsino-nucleon nondiagonal interaction takes place again in this case, contributing
to the SI cross section. Comparing the model predictions for χ − N reaction with experi-
mental restrictions on the SI cross section [38, 39] it is possible to estimate the MSUSY value.
From the inequality
∆Mχ0 = (M
Z/(M1M2)(M1 cos
2(θW ) +M2 sin
2(θW ) < Wk(MN/Mχ),
it follows that the process χ0
N → χ0
N ′ is closed when MSUSY ≤ 8.3 ·109 GeV and MSUSY ≤
1.2 · 1010 GeV for Mχ0
= 1.0 TeV and 1.4 TeV, respectively. These estimations depend
on tan β weakly and are in agreement with the ones given by the RG analysis. So we
conclude that in this scenario the SI inelastic Higgsino-nucleon scattering cannot be observed
experimentally today.
It seems that the inverse inelastic reaction χ0
N → χ0
N ′ is possible, but χ0
states are
unstable, so they decayed a long time ago. In other words, the only inequality τχ ≤ T0
takes place, where T0 is the age of the Universe. An upper estimation for τχ follows from
the width Γ(χ2 → χ1νν̄) (when ∆Mχ0 is reasonably small we consider only the most ”soft”
channel) and we have
∆M5χ. (4.2)
Then the following restriction emerges
MSUSY ≤ M2Z
G2FT0
. (4.3)
With the value T0 = 3.15 · 1017 s we get MSUSY ≤ 4.25 · 109GeV. It is again in accordance
with the RG results and slightly more stringent than the restrictions following from the
threshold inequality ∆Mχ ≥ Wk(MN/Mχ).
Electroweak corrections to the splitting ∆Mχ± can be as small as ∼ 100MeV due to loops
with γ, Z and W [14]. When ∆Mχ± ∼ 100MeV (squark loop contributions are too small),
recharge process χ0
n → χ±
p accompanied by a track of χ± seems as possible due to very
energetic neutralinos, but their cross section is strongly damped and the reaction is entirely
exotic. So the recharge process cannot be detected experimentally too.
Depending on mass splitting the chargino lifetime can be estimated in the following
manner: from the channel χ± → χ0eν̄e we get
τχ± = (30π
3/G2F )∆M
but in the intermediate interval of ∆Mχ± = (0.1 − 1.0) GeV there are also chargino decay
channels with the final µ and π-meson. For these decays the corresponding formulae are
more cumbersome, and we do not write them here. Gathering all contributions, we get
approximately
τχ± ∼ (10−7 − 10−12) s.
Thus, for both the cases – large or small ∆Mχ – the subscenarios of this class do not
produce practically any visible signal in various channels of neutralino-nucleon scattering at
modern measuring tools.
If ∆Mχ values are sufficiently large, there is a chance to discover at the LHC some
decay modes of the lowest neutralino and chargino states. Moreover, if some specific squark
(slepton) effects occur at low TeV scale too, but gaugino manifestations are absent, it may
correspond to the subscenario M1/2 ≫ M0 ∼ µ. The existence of specific neutralino-chargino
decays together with the absence of other SUSY states at the TeV scale can be understood
in the framework of the subscenarios M0 & M1/2 ≫ µ or M1/2 ≫ M0 ≫ µ.
Absence in an experiment of various decay modes manifestations can indicate that the
subscenarios with M0 & M1/2 ≫ µ or M1/2 ≫ M0 ≫ µ are realized, providing the low-
est neutralino and chargino with a well degenerate spectrum that cannot be observed in
experiment.
Note, for all scenarios with a large splitting between M1/2 and M0 a large contribution
to the SI neutralino-nucleon cross section is possible due to the squark exchange, especially
if the superscalar scale is closed to µ. Naturally, a study of collider and χ− N data corre-
lations is necessary, making details of the state spectrum more precise. Data on neutralino
annihilation photon spectrum are also needed to complete an analysis of capabilities of the
scenarios.
B. Diffuse gamma spectrum from the Galactic halo
As it follows from above, in the Split Higgsino scenario obvious manifestations of SUSY
can be in some latent form: direct interaction of neutralino with nuclei is too small and it
is very questionable whether effects of degenerate neutralino and chargino can be detected
at the LHC if ∆Mχ < 10GeV. It seems that a chance to verify this scenario is to study the
photon spectra from neutralino annihilation in the Galactic halo. The process can produce
gamma quants in two ways: direct photon creation through loop diagrams or formation of a
diffuse (continuous) gamma spectrum due to secondary photons created in radiative decays
of mesons.
In this scenario the dominant mode of continuous gamma spectrum creation is Hig-
gsino annihilation into WW and ZZ bosons followed by creation and radiative decays
of light mesons, mainly through the channel π0 → 2γ. For the lowest Higgsino mass
Mχ = 1.4 TeV we calculate the cross section of Higgsino annihilation into WW and ZZ and
get (σv)WW+ZZ ≈ 0.7 · 10−26 cm3s−1. Further, the same spectrum
e−7.76x
where x = Eγ/Mχ, is used for both (W and Z) channels approximately ( [40, 41]).
Then the total diffuse photon flux from halo can be calculated as
Φγ(E0, Em) ≈ 9.3 · 10−13 cm−2 s−1
· (4.4)
< σv >WW+ZZ
10−26 cm3 s−1
· J̄(∆Ω) ·∆Ω,
where E0 is the threshold photon energy for an apparatus, Em is the maximal registered
photon energy, J̄(∆Ω) is averaged over angle value of the integral J(Ψ) which contains
information on the DM distribution in halo [41, 42].
Fixing some value J̄(10−3) ≈ 1.2 · 103 that is typical of the Navarro-Frenk-White profile,
where ∆Ω = 10−3 sr is used (see [42, 43]), we evaluate the total continuous gamma flux
that can be measured at space-based telescopes (EGRET or GLAST) or at ground based
Atmospheric Cherenkov Telescopes (ACT) (HESS, in particular). Then from (4.5) for the
total flux we get
EGRET ≈ 0.17 · 10
−10 cm−2 s−1, E0 = 1GeV, Em = 20GeV,
GLAST ≈ 0.19 · 10
−10 cm−2 s−1, E0 = 1GeV, Em = 300GeV,
HESS ≈ 0.82 · 10
−12 cm−2 s−1, E0 = 60GeV, Em = 1TeV.
(4.5)
As it is seen, the calculated values are beyond experimental possibilities of these telescopes
– at present, only GLAST has some chance to measure the total continuous gamma flux,
because it can detect a gamma flux as small as Φ
GLAST (exp) ≈ 10−10 cm−2 s−1. In the
near future, however, ACT like HESS, for example, will be able to fix the flux due to high
sensitivity level [41] Φ
HESS(exp) ≈ 10−14 cm−2 s−1. Note also that similar results were
derived in [8] for direct photon signals.
A characteristic value of the flux is not characteristic feature of the scenario, nearly
the same values arise in all schemes with highly degenerate Higgsino as the lowest state.
Nevertheless, from comparison with other model predictions (MSSM, mSUGRA, etc., see,
for example [41, 42, 43, 44, 45, 46, 47]) we note that channel of neutralino annihilation
into quarks increases the total flux up to one order of magnitude, so the flux could be well
over the GLAST sensitivity threshold. Then the Split Higgsino scenario prediction for the
diffuse gamma flux can be discriminated from predictions of models where there is a large
contribution of quark and/or Higgs annihilation modes. Nevertheless, some conclusion on
the Split Higgsino scenario realizability can be made only from the whole data analysis,
using χ−N scattering, collider experiments, and photon spectrum data together.
V. CONCLUSIONS
We have pointed out that from the one-loop RG analysis of the SUSY SU(5) theory there
can be extracted a few sets of energy scales which are compatible with conventional ideas
on the DM structure and manifestations. Threshold corrections, induced by heavy states
near MGUT – M24, M5, – are especially important for establishment of the hierarchies. Due
to a specific form of the RG equations the scalar scale M0 remains arbitrary, and it occurs
the variety of scenarios divided in the two classes: M1/2 ≫ µ or M1/2 ≪ µ. Note also that
the refined RG analysis with two-loop β-functions and mass spectrum improved by radiative
corrections can make the energy scale set more precise. Namely, the M0 scale can be split
to establish squark and slepton scales separately, while the energy scale hierarchies as two
global classes should remain.
In this paper, the hierarchy M0 & M1/2 ≫ µ (the Split Higgsino model) was considered
in detail. In this case, the renormalization group approach determines the SUSY breaking
scale as MSUSY ∼ 108 − 109GeV.
Due to the degenerate mass spectrum of the lightest states in the model the coannihilation
channels are essential for the effective annihilation cross section. With the calculated value
of < σeffv > the relic abundance value is provided by the lowest Higgsino mass in the
interval 1.2 − 1.6TeV.
Experimental observation of this scenario effects crucially depends on ∆Mχ0 and ∆Mχc . If
these differences are ∼ 10GeV, products of χ0
and χ± decays can be, in principle, detected
at the LHC, for small splittings these decay modes are invisible. At the same time, if
superscalars are close to the lowest Higgsino scale, their specific manifestations are possible
too. In particular, such squarks at a TeV scale should increase significantly the SI neutralino-
nucleon cross section. Heavy scalars from the hierarchy M0 ≫ µ do not change the tree
level χ−N cross section significantly.
Even if there are signals from neutralino and /or chargino decays, they are hardly de-
tected. Despite these (possible) effects the Split Higgsino model is characterized by χ− N
scattering with the small and therefore yet unregistered SI cross section and some typical
value of a continuous annihilation gamma flux. For small ∆Mχ values the MSUSY scale is
evaluated as . 1010GeV in agreement with the RG results leading to unobserved SI χ−N
scattering.
Conclude, to detect some footsteps of the scenario, it is necessary to analyze correlation
of all collider data, χ−N cross section measurements and value of diffuse gamma flux from
halo (or direct photon spectrum). Only a comparison of all measured characteristics could
give some conclusions on the scenario realization. In a sense, this model presents a class of
“Hidden SUSY” scenarios which do not reject SUSY ideas and at the same time, can explain
(possible) absence of obvious SUSY signals at the LHC.
If, however, the neutralino annihilation induced continuous gamma flux is not detected
at all, simultaneously with the absence of neutralino and chargino decay modes at the LHC,
it will mean that the DM origin cannot be understood in the MSSM framework. Then
to explain the DM origin and properties there should be attracted some other sources, as
gravitino, for example.
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	Introduction
	Renormgroup analysis
	The Split Higgsino scenario
	The model Lagrangian and its features
	Split Higgsino as the DM carrier
	SUSY scales and experimental possibilities
	-N scattering and collider signals
	Diffuse gamma spectrum from the Galactic halo
	Conclusions
	References

  We present a renormalization group motivation of scale hierarchies in SUSY
SU(5) model. The Split Higgsino scanrio with a high scale of the SUSY breaking
is considered in detail. Its manifestations in experiments are discussed.

Introduction
	Renormgroup analysis
	The Split Higgsino scenario
	The model Lagrangian and its features
	Split Higgsino as the DM carrier
	SUSY scales and experimental possibilities
	-N scattering and collider signals
	Diffuse gamma spectrum from the Galactic halo
	Conclusions
	References
