Hidden Spin-3/2 Field in the Standard Model
Durmuş Demir Canan Karahan Beste Korutlu Ozan Sargın










t1This work is supported in part by the TUBITAK grant 115F212.



e1e-mail: demir@physics.iztech.edu.tr
e2e-mail: cananduzturk@iyte.edu.tr
e3e-mail: beste.korutlu@tubitak.gov.tr
e4e-mail: ozansargin@iyte.edu.tr


Department of Physics, İzmir Institute of Technology, Urla, İzmir, 35430 TURKEY 
           TÜBİTAK National Metrology Institute, Gebze, Kocaeli, 41470 TURKEY 
           







Hidden Spin-3/2 Field in the Standard Modelt1

    Durmuş Demire1,addr1
        Canan  Karahane2,addr1
        Beste Korutlue3,addr2
        Ozan Sargıne4,addr1

    Received: December 30, 2023/ Accepted: December 30, 2023
==================================================================================================================




Here we show that a massive spin-3/2 field can hide in the
SM spectrum in a way revealing itself only virtually. We study 
collider signatures and loop effects of this field, 
and determine its role in Higgs inflation and 
its potential as Dark Matter. We show that this spin-3/2
field has a rich linear collider phenomenology and motivates
consideration of a neutrino-Higgs collider. We also show
that study of Higgs inflation, dark matter and dark energy 
can reveal more about the neutrino and dark sector.

12.60.-i 95.35.+d





§ INTRODUCTION

The Standard Model (SM) of strong and electroweak interactions, spectrally
completed by the discovery of its Higgs boson at the LHC <cit.>, 
seems to be the model of the physics at the Fermi energies. It does so because various experiments
have revealed so far no new particles beyond the SM spectrum. There is, however, at least the dark matter (DM), which requires new particles beyond the SM. Physically, therefore, we must use every 
opportunity to understand where those new particles can hide, if any.

In the present work we study a massive spin-3/2 field hidden in the SM spectrum. This higher-spin field, described by the Rarita-Schwinger equations <cit.>, has to obey certain constraints to have correct degrees of freedom when it is on the physical shell. At the renormalizable level, it can couple to the SM matter via only the neutrino portal (the composite SM singlet formed by the lepton doublet and the Higgs field). This interaction is such that it vanishes when the spin-3/2 field is on shell. In Sec. 2 below we give the model and basic constraints on the spin-3/2 field. 

In Sec. 3 we study collider signatures of the spin-3/2 field. We study there ν_L h →ν_L h and e^-e^+→ W^+W^-  scatterings in detail. We give analytical computations and numerical predictions. We propose there a neutrino-Higgs collider and emphasize importance of the linear collider in probing the spin-3/2 field.

In Sec. 4 we turn to loop effects of the spin-3/2 field. We find that the spin-3/2 field adds logarithmic and quartic UV-sensitivities atop the logarithmic and quadratic ones in the SM. We convert  power-law UV-dependent terms into curvature terms as a result of the incorporation of gravity into the SM. Here we use the results of <cit.> which shows that gravity can be incorporated into the SM properly and naturally (i) if the requisite curved geometry is structured by interpreting the UV cutoff as a constant value assigned to the spacetime curvature, and (ii) if the SM is extended by a secluded new physics (NP) that does not have to interact with the SM. This mechanism eliminates big hierarchy problem by metamorphosing the quadratic UV part of the Higgs boson mass turns into Higgs-curvature coupling. 

In Sec. 5 we discuss possibility of Higgs inflation via the large Higgs non-minimal coupling induced by the spin-3/2 field. We find that Higgs inflation is possible in a wide range of parameters provided that the secluded NP sector is crowded enough. 

In Sec. 6 we discuss the DM. We show therein that the spin-3/2 field is a viable DM candidate. We also show that the singlet fields in the NP can form a non-interacting DM component. 

In Sec. 7 we conclude. There, we give a brief list of problems that can be studied as furthering of the material presented this work. 



§ A LIGHT SPIN-3/2 FIELD

Introduced for the first time by
Rarita and Schwinger <cit.>, ψ_μ propagates with

    S^αβ(p) = i/p - MΠ^αβ(p),

to carry one spin-3/2 and two spin-1/2 components through the
projector <cit.>


    Π^αβ = -η^αβ +
    γ^αγ^β/3+
    (γ^αp^β -
    γ^βp^α)/3M+2
    p^αp^β/3 M^2,


that exhibits both spinor and vector characteristics.

It is necessary to impose <cit.>


    p^μψ_μ(p)⌋_p^2=M^2=0,


and


    γ^μψ_μ(p)⌋_p^2=M^2=0,


to eliminate the two spin-1/2 components to make ψ_μ
satisfy the Dirac equation


    (p - M)ψ_μ=0


as expected of an on-shell fermion. The constraints (<ref>) and (<ref>) imply that p^μψ_μ(p) and γ^μψ_μ(p) both vanish on the physical shell p^2=M^2. The latter is illustrated in Fig. <ref> taking ψ_μ on-shell.

Characteristic of singlet fermions, the ψ_μ, at the renormalizable level, makes contact with the SM via 

    ℒ^(int)_3/2 = c^i_3/2L^i H γ^μψ_μ + h.c.

in which 

    L^i = ([ ν_ℓ L;   ℓ_L ])_i

is the lepton doublet (i=1,2,3), and 

    H = 1/√(2)([ v + h + i φ^0;       √(2)φ^- ])

is the Higgs doublet with vacuum expectation value v≈ 246  GeV, Higgs boson h, and Goldstone bosons  φ^-, φ^0 and φ^+ (forming the longitudinal components of W^-, Z and W^+ bosons, respectively). 

In general, neutrinos are sensitive probes of singlet fermions. They can get masses through, for instance, the Yukawa interaction (<ref>), which leads to the Majorana mass matrix 

    (m_ν)^i j_3/2∝ c^i_3/2v^2/M c^⋆ j_3/2

after integrating out ψ_μ. This mass matrix can, however, not lead to the experimentally known neutrino mixings <cit.>. This means that flavor structures necessitate additional singlet fermions. Of such are the right-handed neutrinos  ν_R^k of mass M_k (k=1,2,3,…), which interact with the SM through 

    ℒ^(int)_R = c_R^i kL̅^i H ν_R^k + h.c.

to generate the neutrino Majorana masses 

    (m_ν)^i j_R∝ c_R^i kv^2/M_k c_R^⋆ k j

of more general flavor structure. This mass matrix must have enough degrees of freedom  to fit to the data <cit.>.








Here we make a pivotal assumption. We assume that ψ_μ and ν_R^k can weigh as low as a TeV, and that c^i_3/2 and some of c_R^i k can be 𝒪(1). We, however, require that contributions to neutrino masses from 
ψ_μ and ν_R add up to reproduce with experimental result

    (m_ν)^i j_3/2 + (m_ν)^i j_R≈ (m_ν)^i j_exp

via cancellations among different terms.  We therefore take 

    c_3/2≲𝒪(1) ,  M≳ TeV

and investigate the physics of  ψ_μ. This cancellation requirement does not have to cause any excessive fine-tuning simply because ψ_μ and ν_R^k  can have appropriate symmetries that correlate their couplings. One possible symmetry would be rotation of γ^μψ_μ and ν_R^k into each other. We defer study of possible symmetries to another work in progress <cit.>. The right-handed sector, which can involve many ν_R^k fields, is interesting by itself but hereon we focus on ψ_μ and take, for simplicity, c^i_3/2 real and family-universal (c^i_3/2=c_3/2 for ∀ i).





§ SPIN-3/2 FIELD AT COLLIDERS

It is only when it is off-shell that ψ_μ can reveal itself through the interaction (<ref>). This means that its effects are restricted to modifications in scattering rates of the SM particles. To this end, as follows from (<ref>), it participates in 


  * ν_L h →ν_L h (and also ν_Lν_L→ h h)

  * e^+ e^- → W^+_L W^-_L (and also ν_Lν_L→ Z_L Z_L)

at the tree level. They are analyzed below in detail.




 §.§ ν_L h →ν_L h Scattering

Shown in Fig. <ref> are the two box diagrams which enable ν_L h →ν_L h scattering in the SM.  Added to this loop-suppressed SM piece is the ψ_μ piece depicted in Fig. <ref>. The two contributions add up to give the cross section 

    dσ(ν_L h →ν_L h)/dt= 1/16π𝒯_ν h(s,t)/(s-m_h^2)^2

in which the squared matrix element 



    𝒯_ν h(s,t)    =    9(c_3/2/3 M)^4((s-m_h^2)^2 + st) - 16(c_3/2/3 M)^2(
    2(s-m_h^2)^2 +(2s -m_h^2)t) 𝕃+ 2(
    s-m_h^2)(s + t-m_h^2) 𝕃^2


involves the loop factor

    𝕃=(g_W^2+g_Y^2)^2 M_Z^2 m_h^2 I(M_Z)/192 π^2 + g_W^4 M_W^2 m_h^2 I(M_W)/96 π^2

in which g_W (g_Y) is the isospin (hypercharge) gauge coupling, and



    I(μ)=∫_0^1dx∫_0^1-xdy∫_0^1-x-ydz ((s-m_h^2)(x+y+z-1) y - txz + m_h^2 y (y-1) + μ^2 (x + y + z))^-2


is the box function. In Fig. <ref>, we plot the total cross section σ(ν_L h →ν_L h)  as a function of the neutrino-Higgs center-of-mass energy for different M values. The first important thing about the plot is that there is no resonance formation around √(s)=M. This confirms the fact that ψ_μ, under the constraint (<ref>), cannot come to physical shell with the couplings in (<ref>). In consequence, the main search strategy for ψ_μ is to look for deviations from the SM rates rather than resonance shapes. The second important thing about the plot is that, in general, as revealed by (<ref>), larger the M smaller the ψ_μ contribution. The cross section starts around 10^-7  pb, and falls rapidly with √(s). (The SM piece, as a loop effect, is too tiny to be observable: σ(ν_L h →ν_L h)≲ 10^-17  pb). It is necessary to have some 10^4/fb integrated luminosity (100 times the target luminosity at the LHC) to observe few events in a year. This means that ν_L ν_L → h h scattering can probe ψ_μ at only high luminosity but with a completely new scattering scheme. 
 







Fig. <ref> shows that neutrino-Higgs scattering can be a promising channel to probe ψ_μ (at high-luminosity, high-energy machines). The requisite experimental setup would involve crossing of Higgs factories with accelerator neutrinos. The setup, schematically depicted in  Fig. <ref>, can be viewed as incorporating  future Higgs (CEPC<cit.>, FCC-ee <cit.> and ILC <cit.>) and neutrino <cit.> factories. If ever realized, it could be a rather clean experiment with negligible SM background. This hypothetical “neutrino-Higgs collider”, depicted in Fig. <ref>, must have, as suggested by Fig. <ref>, some 10^4/fb integrated luminosity to be able to probe a TeV-scale ψ_μ. In general, need to high luminosities is a disadvantage of this channel.  (Feasibility study, technical design and possible realization of a “neutrino-Higgs collider” falls outside the scope of the present work.)







 §.§ e^+ e^- → W_L^+ W_L^- Scattering

It is clear that ψ_μ directly couples to the Goldstone bosons φ^+,-,0 via (<ref>). The Goldstones, though eaten up by the W and Z bosons in acquiring their masses, reveal themselves at high energies. In fact, the Goldstone equivalence theorem <cit.> states that scatterings at energy E involving longitudinal W^±_L bosons are equal to scatterings that involve φ^± up to terms 𝒪(M_W^2/E^2). This theorem, with similar equivalence for the longitudinal Z boson, provides a different way of probing ψ_μ. In this regard, depicted in  Fig. <ref> is ψ_μ contribution to
e^+ e^- → W_L^+ W_L^- scattering in light of the Goldstone equivalence. The SM amplitude is  given in <cit.>. The total differential cross section

    dσ(e^+ e^- → W^+_L W^-_L)/dt= 1/16π s^2𝒯_W_L W_L(s,t)

involves the squared matrix element 



    𝒯_W_L W_L(s,t)   =   (g_W^2/s-M_Z^2(-1+M_Z^2/4 M_W^2 +M_Z^2-M_W^2/s) +g_W^2/s-4 M_Z^2(1+M_W^2/t) +c^2_3/2/3 M^2)^2(-2 s M_W^2 -2 (t-M_W^2)^2) 
       +   c^4_3/2 s/18 M^2(4 + t/t-M^2)^2


Plotted in Fig. <ref> is σ(e^+ e^- → W^+_L W^-_L) as a function of the e^+ e^- center-of-mass energy for different values of M. The cross section, which falls with √(s) without exhibiting a resonance shape, is seen to be large enough to be measurable at the ILC <cit.>. In general, larger the M smaller the cross section but even 1/fb luminosity is sufficient for probing ψ_μ for a wide range of mass values.



Collider searches for ψ_μ, as illustrated by  ν_L h →ν_L h and e^-e^+→ W^+W^- scatterings, can access spin-3/2 fields of several TeV mass. For instance, the ILC, depending on its precision, can confirm or exclude a ψ_μ of even 5 TeV mass with an integrated luminosity around 1/fb. Depending on possibility and feasibility of a neutrino-neutrino collider (mainly accelerator neutrinos), it may be possible to study also ν_L ν_L → h h and ν_L ν_L → Z_L Z_L scatterings, which are expected to have similar sensitivities to M. 



§ SPIN-3/2 FIELD IN LOOPS

As an inherently off-shell field, ψ_μ is expected to reveal itself mainly in loops. Its one possible loop effect would be generation of neutrino masses but chirality forbids it. Despite the couplings in (<ref>), therefore, neutrino masses do not get any contribution from ψ_μ-h loop.

One other loop effect of ψ_μ would be radiative corrections to the Higgs boson mass. This is not forbidden by any symmetry. The relevant Feynman diagram is depicted in Fig. <ref>. It adds to the Higgs boson squared-mass a logarithmic piece

    (δ m_h^2)_log = c_3/2^2/12π^2M^2log G_F M^2

relative to the logarithmic piece log G_F Λ^2 in the SM, and a quartic piece


    (δ m_h^2)_4 = c_3/2^2/ 48 π^2Λ^4/M^2


which have the potential to override the experimental result <cit.> depending on how large the UV cutoff Λ is compared to the Fermi scale G_F^-1/2 = 293  GeV. 

The logarithmic contribution in (<ref>), which originates from the η^αβ part of (<ref>), gives rise to the little hierarchy problem in that larger the M stronger the destabilization of the SM Higgs sector. Leaving aside the possibility of cancellations with similar contributions from the right-handed neutrinos ν_R^k in (<ref>), the little hierarchy problem can be prevented if M (more precisely M/c_3/2) lies in the TeV domain. 

The quartic contribution in (<ref>), which originates from the longitudinal p^α p^β term in (<ref>), gives cause to the notorious big hierarchy problem in that larger the Λ larger the destabilization of the SM Higgs sector. This power-law UV sensitivity exists already in the SM

    (δ m_h^2)_2   =   3 Λ^2/16 π^2 |⟨ H ⟩|^2( m_h^2 + 2 M_W^2 + M_Z^2 - 4 m_t^2)

at the quadratic level <cit.> and
violates the LHC bounds unless Λ≲ 550 GeV. This bound obviously contradicts with the
LHC experiments since the latter continue to confirm the
SM at multi TeV energies. This experimental fact makes
it obligatory to find a natural UV completion to the SM. 






One possibility is to require  (δ m_h^2)_4 to cancel out  (δ m_h^2)_2. This requirement involves a severe fine-tuning (as with a scalar field
<cit.>, Stueckelberg vector <cit.> and
spacetime curvature <cit.>) and cannot form a viable
stabilization mechanism.

Another possibility would be to switch, for instance, to dimensional
regularization scheme, wherein the quartic and quadratic
UV-dependencies are known to disappear. This, however, is not a
solution. The reason is that the SM, as a quantum field theory of
the strong and electroweak interactions, needs gravity to be
incorporated as the forth known force. And the fundamental scale of
gravity, M_Pl, inevitably sets an ineliminable physical UV
cutoff (rendering Λ physical). This cutoff forces quantum field theories to exist in between
physical UV and IR scales. The SM plus ψ_μ (plus right-handed neutrinos), for instance,
ranges from G_F^-1/2 at the IR up to Λ at the UV such
that both scales are physical (not to be confused with the formal
momentum cutoffs employed in the cutoff regularization).

To stabilize the SM, it is necessary to metamorphose the
destabilizing UV effects. This necessitates a physical agent. The
most obvious candidate is gravity. That is to say, the
UV-naturalness problems can be a clue to how quantized matter must
gravitate. Indeed, quantized matter in classical curved geometry
suffers from inconsistencies. The situation can be improved by
considering long-wavelength matter by integrating out high-frequency
modes. This means that the theory to be carried into curved geometry
for incorporating gravity is not the full action but the effective
action (see the discussions in <cit.> and <cit.>). Thus, starting with
the SM effective action in flat spacetime with well-known
logarithmic, quartic and quadratic UV-sensitivities, gravity can be
incorporated in a way ensuring UV-naturalness. More precisely,
gravity gets incorporated properly and naturally  (i) if the
requisite curved geometry is structured by interpreting Λ^2
as a constant value assigned to the spacetime curvature, and (ii) if the SM is extended by new physics (NP) that does not have
to interact with the SM. The ψ_μ can well be an NP field.  
Incorporating gravity by identifying Λ^2 g_μν with
the Ricci curvature R_μν(g), fundamental scale of 
gravity gets generated as

    M_Pl^2 ≈(n_b-n_f)/2(8 π)^2Λ^2

where n_b (n_f) are the total number of bosons (fermions) in the
SM plus the NP. The ψ_μ increases n_f by 4, right-handed neutrinos by 2. There are
various other fields in the NP, which contribute to n_b  and n_f
to ensure Λ≲ M_Pl. Excepting ψ_μ,  they
do not need to interact with the SM fields. Induction of M_Pl
ensures that the quadratic UV-contributions to vacuum energy are
canalized not to the cosmological constant but to the gravitational
constant (see  <cit.> arriving at this result in a
different context). This suppresses the cosmological constant down
to the neutrino mass scale.

The quartic UV contributions in (<ref>) and the quadrat­ic
contributions in (<ref>) (suppressing contributions from the right-handed 
neutrinos ν_R^k) change their roles with the inclusion
of gravity. Indeed, corrections to the Higgs mass term [(δ m_h^2)_4+(δ m_h^2)_2] H^† H  turns
into

    [3(m_h^2 + 2 M_W^2 + M_Z^2 - 4 m_t^2)/(8π)^2|⟨ H ⟩|^2+c_3/2^2/12(n_b-n_f)M_Pl^2/M^2] R H^† H

which is nothing but the direct coupling of the Higgs field to the
scalar curvature R. This Higgs-curvature coupling is perfectly
natural; it has no potential to de-stabilize the Higgs sector.
Incorporation of gravity as in <cit.> leads, therefore, to
UV-naturalization of the SM with a nontrivial NP sector
containing ψ_μ as its interacting member. 



§ SPIN-3/2 FIELD AS ENABLER OF HIGGS INFLATION

The non-minimal Higgs-curvature coupling in (<ref>) reminds one at once the possibility of Higgs inlation. Indeed, the Higgs field has been shown in <cit.> to lead to correct inflationary expansion provided that 

    c_3/2^2/12(n_b-n_f)M_Pl^2/M^2≈ 1.7× 10^4

after dropping the small SM contribution in (<ref>). This relation puts constraints on M and Λ depending on how crowded the NP is. 

For a Planckian UV cutoff Λ≈ M_Pl, the Planck scale in (<ref>) requires n_b - n_f≈ 1300, and this leads to M/c_3/2≈ 6.3× 10^13  GeV. This heavy ψ_μ, weighing not far from the see-saw and axion scales, acts as an enabler of Higgs inflation. (Of course, all this makes sense if the ψ_μ contribution in (<ref>) is neutralized by similar contributions from the right-handed neutrinos ν_R^k to alleviate the little 
hierarchy problem.) 

For an intermediate UV cutoff Λ≪ M_Pl, n_b-n_f can be large enough to bring M down to lower scales. In fact, M gets lowered to M∼ TeV for n_b-n_f≃ 10^24, and this sets the UV cutoff Λ∼ 3  TeV. This highly crowded NP illustrates how small M
and Λ can be. Less crowded NP sectors lead to intermediate-scale M and Λ.

It follows therefore that it is possible to realize Higgs inflation through the Higgs-curvature coupling (corresponding to quartic UV-dependence the ψ_μ induces on the Higgs mass). It turns out that Higgs inflation is decided by how heavy ψ_μ is and how crowded the NP is. It is interesting that the ψ_μ hidden in the SM spectrum enables successful Higgs inflation if gravity is incorporated into the SM as in <cit.>. 



§ SPIN-3/2 FIELD AS DARK MATTER
 
Dark matter (DM), forming one-forth of the matter in the Universe, must be electrically-neutral and long-lived. The negative searches <cit.> so far have added one more feature: The DM must have exceedingly suppressed interactions with the SM matter. It is not hard to see that the spin-3/2 fermion ψ_μ possesses all these properties.  Indeed, the constraint (<ref>) ensures that scattering processes in which ψ_μ is on its mass shell must all be forbidden simply because its interactions in (<ref>)  involves the vertex factor c_3/2γ^μ. This means that decays of ψ_μ as in Fig.<ref> as well as its co-annihilations with the self and other SM fields are all forbidden. Its density therefore does not change with time, and the observed DM relic density <cit.> must be its primordial density, which is determined by the  short-distance physics the ψ_μ descends from. It is not possible to calculate the relic density without knowing the short-distance physics. Its mass and couplings, on the other hand, can be probed via the known SM-scatterings as studied in Sec. 3 above. In consequence, the ψ_μ, as an inherently off-shell fermion hidden in the SM spectrum, possesses all the features required of a DM candidate.  

Of course, the ψ_μ is not the only DM candidate in the setup. The crowded NP sector, needed to incorporate gravity in a way solving the hierarchy problem (see Sec. 4 above), involves various fields which do not interact with the SM matter. They are viable candidates for non-ineracting DM as well as dark energy (see the detailed analysis in <cit.>). The non-interacting NP fields can therefore contribute to the total DM distribution in the Universe. It will be, of course, not possible to search for them directly or indirectly. In fact, they do not have to come to equilibrium with the SM matter. 

Interestingly, both ψ_μ and the secluded fields in the NP act as extra fields hidden in the SM spectrum. Unlike ψ_μ which reveal itself virtually, the NP singlets remain completely intact. The main implication is that, in DM phenomenology, one must keep in mind that there can exist an unobservable, undetectable component of the DM <cit.>. 




§ CONCLUSION AND OUTLOOK

In this work we have studied a massive spin-3/2 particle ψ_μ obeying the constraint (<ref>) and interacting with the SM via (<ref>). It hides in the SM spectrum as an
inherently off-shell field. We first discussed its collider signatures by studying ν_L h →ν_L h and e^-e^+→ W^+W^- scatterings in detail in Sec. 3. Following this, we turned to its loop effects and determined how it contributes to big and little hierarchy problems in the SM. Resolving the former by appropriately incorporating gravity, we show that Higgs field can inflate the Universe. Finally, we show that ψ_μ is a viable 
DM candidate, which can be indirectly probed via the scattering processes we have analyzed. 

The material presented in this work can be extended in various ways. A partial list would include:


  * Determining under what conditions right-handed neutrinos can lift the constraints on ψ_μ from the neutrino masses,


  * Improving the analyses of ν_L h →ν_L h and e^-e^+→ W^+W^- scatterings by including loop contributions,


  * Simulating e^-e^+→ W^+W^- at the ILC by taking into account planned detector acceptances and collider energies,


  * Performing a feasibility study of the proposed neutrino-Higgs collider associated with ν_L h →ν_L h scattering,


  * Exploring UV-naturalness by including right-handed neutrinos, and determining under what conditions the little hierarchy problem is softened.


  * Including effects of the right-handed neutrinos into Higgs inflation, and determining appropriate parameter space. 


  * Giving an in-depth analysis of the dark matter and dark energy by taking into account the spin-3/2 field, right-handed neutrinos and the secluded NP fields.  


  * Studying constraints on the masses of NP fields from nucleosynthesis and other processes in the early Universe. 

We will continue to study the spin-3/2 hidden field starting with some of these points. 


Acknowledgements.
This work is supported in part by the TUBITAK grant 115F212. We thank to the conscientious referee for enlightening comments and suggestions. 

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