Mu Problem and Hybrid Inflation in Supersymmetric SU(2)_L x SU(2)_R x U(1)_B-L

Aristotle Univ.); G. Dvali (CERN); G. Lazarides (CERN; Q. Shafi (Bartol Res. Inst.)

REVIEW

Mu Problem and Hybrid Inflation in Supersymmetric SU(2)_L x SU(2)_R x U(1)_B-L

G. Dvali (CERN), G. Lazarides (CERN, Aristotle Univ.), Q. Shafi (Bartol Res. Inst.)

High Energy Physics - Phenomenology · Submitted 10 Oct 1997 · 19 references

Pith's verdict: The singlet-does-everything construction is genuinely clever and checks out at leading order, but the vacuum selection and the deferred leptogenesis/inflation numbers are thinner than the abstract suggests.

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arXiv IDhep-ph/9710314 (version 1)

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CitationAristotle Univ.), G. Dvali (CERN), G. Lazarides (CERN, Q. Shafi (Bartol Res. Inst.). "Mu Problem and Hybrid Inflation in Supersymmetric SU(2)_L x SU(2)_R x U(1)_B-L." arXiv:hep-ph/9710314 (1997). https://pith.science/paper/63CWCOF7

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Worth your time if you work on the μ problem or on hybrid inflation, but read it as a model-building letter, not as a finished cosmology paper. The core idea is real: one gauge singlet S, appearing only linearly in W = S(κ l^c l̄^c + λ h² − κM²), breaks SU(2)_R × U(1)_{B−L}, generates μ = −(λ/κ)m_{3/2} and B_μ = −2(λ/κ)m_{3/2}² through the soft-induced tadpole, and gives a flat tree-level V = κ²M⁴ along the S direction. That triple duty is the paper's actual contribution, and it is not in the cited literature. The leading-order minimization is clean and I checked it: the tadpole 2κm_{3/2}M²S against 2κ²M²|S|² gives ⟨S⟩ = −m_{3/2}/κ, and the Higgs mass-matrix determinant argument for stability of the good minimum is correct. The R-symmetry charge assignment killing the dimension-five proton-decay operators is also tidy and correct as stated. Credit where due: this is honest, compact model building by people who know what they are doing, and the authors are upfront about the left-right asymmetric Higgs sector and the awkwardness of the SO(10) embedding (the massless l, l̄ doublets at l^c = l̄^c = M). Those are acknowledged, not hidden. The soft spots are the ones you would guess. Three headline claims — leptogenesis, the gravitino bound on T_R, and the near-unity spectral index — are announced and then waved off with 'the details are quite involved and will not be discussed here.' Some of that is the Phys. Lett. B format, but it means the abstract promises more than the body delivers. The eV tau neutrino rests on an asserted breaking scale 'about an order of magnitude lower than the MSSM unification scale,' with no RGE running shown. And the vacuum selection is the thinnest part: the good and bad minima are degenerate at leading order for a minimal Kähler, and lifting that needs a hand-chosen α with |α|² ≥ λ/κ > 1, justified by one sentence about cosmological evolution. The stress-test's sharper point lands: footnote 1 itself flags the (SS*)² threat to flatness, and the paper never checks whether one Kähler can give both the needed α and a small enough (SS*)² coefficient. That is a real gap, though it is a next-paper gap, not a hole in the μ-plus-inflation core. Send it out. A referee should push on the Kähler compatibility and demand at least a sketch of the reheating and spectral-index calculation.

Referee report

Significance

The economy of the construction is genuine and worth stating: one singlet, with the most general renormalizable superpotential allowed by the gauge group and the R-symmetry, yields μ and Bμ as derived quantities rather than inputs, and the same field drives hybrid inflation. The leading-order minimisation in Eqs. (3)–(5) is correct, and the determinant argument showing that the electroweak Higgs mass matrix has no negative eigenvalues, det = (λ^2/κ^2 − 1)^2 m_{3/2}^4 ≥ 0, is a real check rather than an assertion. The R-symmetry doing double duty as matter parity and as a suppressor of dimension-five proton decay is an attractive feature, and the framework makes falsifiable statements (n_s close to unity, an eV tau neutrino, no dimension-five proton decay). The paper is a Letter and ships no code or formal proofs; the assessment therefore rests on the derivations as written.

Major objections (4)

[§3, Eqs. (3)–(5) and the paragraph following Eq. (5)] The μ mechanism operates only in the 'good' vacuum h = 0, l^c l̄^c = M^2, and the paper itself shows that for a minimal Kähler potential the good and bad minima are degenerate to O(m_{3/2}^4). The proposed remedy is to replace (A − 2) by (A − 2α) and require |α|^2 ≥ λ/κ > 1, but the sole justification is the sentence 'As we will see, the cosmological evolution prefers the first choice,' with no trajectory analysis anywhere in the manuscript. This is load-bearing: if the bad minimum is populated, the doublets take the large vev and the μ mechanism does not operate. More sharply, footnote 1 identifies the (SS*)^2 term of a generic Kähler potential as the threat to the flatness of V = κ^2M^4, and the same non-minimal Kähler that generates α will generically generate that term. The manuscript never checks whether a single Kähler potential can simultaneously give |α|^2 ≥ λ/κ > 1 and a (SS*)^2 coefficient small enough to keep |η| << 1. The authors should either exhibit such a Kähler potential or state explicitly the tuning required, and should support the vacuum-selection sentence with an argument rather than a forward reference.
[§4, Eqs. (7)–(11)] The abstract and text claim a spectral index 'extremely close to unity', but no numerical value of n_s, no slow-roll parameters, and no COBE normalisation of κ^2M^4 are given. Eq. (10) supplies V_inf, yet the derivatives needed for ε and η are never computed, and the number of e-folds and the value of |S| at horizon exit are not stated. This matters because Eq. (10) is only the asymptotic form valid for |S| >> M; if the last 60 e-folds occur near the instability of Eq. (9), the asymptotic form does not apply and the near-unity n_s is not established. The authors should give n_s, the COBE normalisation of κM^2, and the range of |S| over which the last 60 e-folds are obtained.
[§4, paragraph following Eq. (11)] The abstract lists successful leptogenesis as a result, but the text states 'The details are quite involved and will not be discussed here.' At minimum the reheat temperature, the gravitino constraint, the CP asymmetry, the washout regime, and the resulting η_B should be estimated, or the claim should be softened in the abstract to match what is actually shown. Relatedly, the statement that the SU(2)_R × U(1)_{B−L} breaking scale 'turns out to be about an order of magnitude lower than the MSSM unification scale' is asserted without derivation, yet it is precisely what bounds the right-handed neutrino masses below 10^13 GeV and thereby allows an eV tau neutrino. The derivation of M and of the M_R bound should be shown.
[§2, paragraph following Eq. (6)] The abstract claims that baryon number is 'essentially conserved', and the text argues that the exact R-symmetry removes baryon-number-violating operators to all orders but that the R-symmetry must in any case be broken by the hidden sector, so higher-dimensional B-violating terms from a non-minimal Kähler potential are expected and are 'adequately suppressed'. No operator, coefficient, or lifetime estimate is given, so the suppression is not demonstrated. The authors should identify the leading R-violating operator generated by the hidden sector, estimate its coefficient, and compare the resulting proton lifetime with the Super-Kamiokande bound.

Minor objections (9)

[Eq. (2)] The phases φ and θ in the D-flat conditions l^c = e^{iφ} l̄^{c*} and h^(1)_i = e^{iθ} ε_ij h^(2)j* are introduced without discussion; the reader should be told that the relative phase is fixed by the F-term and whether the vacuum preserves or spontaneously breaks CP at this stage.
[Eq. (3)] The origin of the (A − 2) combination in the soft term proportional to S κ M^2 should be stated; it follows from the −3|W|^2 term of the supergravity scalar potential with a canonical Kähler metric, but as written the reader must reconstruct it.
[Eq. (10)] The argument of the logarithm should be written as |S|^2/Λ^2 rather than SS*/Λ^2, and the renormalisation scale Λ should be specified. The coefficient (κ^2 + λ^2)/8π^2 should also be checked against the actual mass spectrum, since the h multiplet enters with a different multiplicity from l^c and l̄^c.
[Paragraph following Eq. (4)] The relation μ^2 + m_{3/2}^2 = −Bμ is said to hold 'approximately' for λ ≠ κ, but the deviation is O((λ/κ − 1)^2); for the λ/κ > 1 preferred by vacuum selection this is not a small correction. The accuracy of the relation should be stated quantitatively.
[Eqs. (4)–(5)] The model predicts Bμ/μ^2 = −2κ/λ, which fixes tanβ in terms of κ/λ and is a testable consequence of the construction. This relation is worth stating explicitly, together with the range of κ/λ compatible with radiative electroweak symmetry breaking.
[Eq. (6)] For the neutrino mass operator l̄^c l̄^c l^c_a l^c_b/M_P, the family-index structure of h_ab and whether it can reproduce the observed neutrino mass hierarchy should be stated, as should whether this is the unique operator of its dimension allowed by the gauge and R-symmetries.
[§5, Eq. (12)] The accidental U(4) symmetry and the resulting massless l, l̄ doublets are acknowledged, and the SO(10) embedding is deferred. Even so, the color-triplet spectrum required to maintain gauge unification should be specified explicitly, since the statement that unification is unaffected is otherwise unsupported within this paper.
[Footnote 1] The requirement that the (SS*)^2 coefficient be 'somewhat smaller than unity' should be quantified in terms of the η bound, so the reader can see how much tuning of the Kähler potential is being assumed.
[References and typography] 'prefered' (p. 4) should read 'preferred'; 'Niewenhuizen' in ref. [8] should read 'Nieuwenhuizen'; the address line 'Aristotle Univers ity' has a stray space; and the spacing in 'K¨ ahler' is inconsistent throughout.

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The paper

The paper claims

A single singlet field in a left-right supersymmetric model is claimed to generate the MSSM mu parameter, its soft partner B mu, and a flat inflationary potential, all from one superpotential term.

The reading

Supersymmetric extensions of the Standard Model contain a mass parameter called $mu$ that sets the scale of the two Higgs doublets, and nothing in the minimal theory explains why it should sit near the weak scale rather than at the Planck scale. This paper argues that $mu$ need not be an input at all: if the Higgs doublets are pseudo-Goldstone modes of a broken $SU(2)_R times U(1)_{B-L}$ symmetry, supersymmetry breaking shifts the vacuum expectation value of a gauge-singlet field $S$ by an amount of order the gravitino mass, and the coupling $S h^{(1)}h^{(2)}$ then generates both $mu$ and its soft partner $B mu$ automatically. The same singlet, because it enters the superpotential only linearly, yields a flat inflaton potential with constant energy $kappa^2 M^4$, so the mechanism that fixes $mu$ also supplies hybrid inflation. The authors further claim that an unbroken R-symmetry keeps the proton stable and, through leptogenesis fed by the inflaton's oscillations, can produce the observed baryon asymmetry, while right-handed neutrinos at the $SU(2)_R$ breaking scale give eV-scale neutrinos as the hot dark matter component.

What carries the argument

The load-bearing object is the gauge-singlet chiral superfield $S$ that appears in the superpotential only linearly, with an R-charge equal to that of the superpotential itself. Linearity is what makes the argument work in both directions: it guarantees that the inflaton potential is flat at tree level and that the one-loop Coleman-Weinberg correction is the only slope, and it guarantees that $S$ has no self-interactions to stabilize it at zero, so the tadpole proportional to $M^2$ from the soft terms displaces it by an amount fixed by $m_{3/2}$ rather than by $M$. The pseudo-Goldstone character of the Higgs doublets, following from an accidental $U(4)$ symmetry of the potential when $lambda = kappa$, is what keeps the doublets light and makes the generated $mu^2$ and $B mu$ automatically of the same order. The same R-symmetry whose $Z_2$ subgroup is ordinary matter parity also forbids the dimension-five operators that would let the proton decay, and the $l^c$ vev provides seesaw masses for right-handed neutrinos.

Extended reading notes

Core claim

The paper's central claim is that one renormalizable superpotential, $W = S(kappa l^c bar{l}^c + lambda h^2 - kappa M^2)$, does three jobs at once in a supersymmetric $SU(3)_c times SU(2)_L times SU(2)_R times U(1)_{B-L}$ model. First, the singlet $S$ triggers breaking of $SU(2)_R times U(1)_{B-L}$ at a scale $M$ far above the weak scale through the vacuum expectation values of the doublet pair $l^c, bar{l}^c$. Second, after gravity-mediated supersymmetry breaking the tadpole induced by that large vev shifts $S$ away from zero by roughly $-m_{3/2}/kappa$, so that $mu = lambda langle S rangle = -lambda m_{3/2}/kappa$ and $B mu = -2 lambda m_{3/2}^2/kappa$ come out of order the gravitino mass without any fine tuning of $M$; the Higgs doublets are pseudo-Goldstones of an approximate $U(4)$ symmetry of the scalar potential and stay massless until supersymmetry breaking lifts the flat direction. Third, for $|S| > max(M, M sqrt{kappa/lambda})$ all gauge non-singlet fields acquire masses of order $|S|$ and decouple, leaving an effective superpotential $W_{infl} = -kappa S M^2$ linear in $S$, hence a constant tree-level vacuum energy $V_{tree} = kappa^2 M^4$ and a one-loop logarithmic slope $V_{infl} simeq kappa^2 M^4 [1 + ((kappa^2+lambda^2)/8 pi^2) ln(S S^*/Lambda^2)]$ that drives the field to the minimum. Inflation ends when the condition on $|S|$ fails and some field goes tachyonic; the authors prefer $kappa < lambda$ so that the system falls to the $SU(2)_R$-breaking vacuum rather than to the one with large electroweak doublet vevs.

Load-bearing premise

The argument needs the universe to settle in the vacuum with $h = 0$ and $l^c bar{l}^c = M^2$; the paper shows the competing minimum with large electroweak doublet vevs is degenerate with it to leading order and can only be disfavoured by choosing a non-minimal Kaehler parameter with $|alpha|^2 geq lambda/kappa > 1$, justified by the single sentence that cosmological evolution prefers that choice, with no trajectory analysis given.

Editorial extensions

If this is right

The MSSM $mu$ parameter stops being a free input: it is predicted as $mu = -lambda m_{3/2}/kappa$, of order the gravitino mass, with $B mu = -2 lambda m_{3/2}^2/kappa$, and the relation $mu^2 + m_{3/2}^2 = -B mu$ holds in the $U(4)$-symmetric limit $lambda = kappa$.
The same sector that breaks $SU(2)_R times U(1)_{B-L}$ at scale $M$ is the inflaton sector, so the scale of new physics and the scale of inflation are tied together, with $V = kappa^2 M^4$ during the flat phase.
The scalar spectral index comes out extremely close to unity, consistent with a mixed cold-plus-hot dark matter picture in which the hot component is a neutrino in the eV range.
Baryogenesis proceeds through leptogenesis: the inflaton oscillations produce right-handed neutrinos whose decays create a lepton asymmetry, partially converted to baryon number by sphalerons, with the reheat temperature compatible with the gravitino bound.
Proton decay from dimension-five operators is absent to all orders as long as the R-symmetry is exact in the superpotential, so the proton is essentially stable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because $mu$ is generated by the same shift that sets the inflaton mass scale, a measurement pinning the inflaton coupling $kappa$ and $M$ would indirectly fix the ratio $lambda/kappa$ and hence the size of $B mu$; this cross-link is a testable correlation the paper does not spell out.
The requirement $kappa < lambda$ is stated but not derived from the inflationary dynamics; a dedicated slow-roll analysis with the one-loop potential could turn it into a quantitative constraint on the $SU(2)_R$ breaking scale, which the paper only estimates as roughly an order of magnitude below the MSSM unification scale.
The eV-scale neutrino hot dark matter component is the most externally checkable element: cosmological bounds on the sum of neutrino masses tighten the allowed window independently of any collider or proton-decay test of the $mu$ mechanism.
The mechanism relies on gravity-mediated supersymmetry breaking to set the size of the shift; in a gauge-mediated version the displacement scale would change, so the claim that $mu$ comes out near the weak scale is tied to the mediation scheme rather than to the left-right structure alone.

Original abstract

We present a solution of the mu problem within a supersymmetric model based on the SU(2)_L x SU(2)_R x U(1)_{B-L} gauge group. We show that this solution implies a built-in hybrid inflationary scenario, which may result in a successful baryogenesis via leptogenesis. Baryon number is essentially conserved as a result of an R-symmetry. Neutrinos with mass in the eV range can provide the `hot' dark matter component.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.20 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.65 · compares with prior results 0.04 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a solution of the mu problem within a supersymmetric model based on the SU(2)_L x SU(2)_R x U(1)_{B-L} gauge group. We show that this solution implies a built-in hybrid inflationary scenario, which may result in a successful baryogenesis via leptogenesis. Baryon number is essentially conserved as a result of an R-symmetry. Neutrinos with mass in the eV range can provide the `hot' dark matter component.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	We present a solution of the mu problem within a supersymmetric model based on the SU(2)_L x SU(2)_R x U(1)_{B-L} gauge group.	contribution 1.00	0.94	0.46	asserted 1.00
s1	We show that this solution implies a built-in hybrid inflationary scenario, which may result in a successful baryogenesis via leptogenesis.	interpretation 0.88	0.89	0.47	hedged possibility 0.63
s2	Baryon number is essentially conserved as a result of an R-symmetry.	result 0.94	0.65	0.08	asserted 1.00
s3	Neutrinos with mass in the eV range can provide the `hot' dark matter component.	result 0.66	0.42	0.06	hedged possibility 0.73

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.86.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
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levels run

P abstract

not run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (19)

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Forward citations

Cited by 9 Pith papers

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A shifted-hybrid non-minimal Higgs inflation model deforms the Starobinsky attractor via a leading non-renormalizable superpotential operator to raise ns into the ACT-preferred range while keeping r ~ 10^{-3}-10^{-2}.

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Waterfall-phase e-foldings in R-symmetric SUSY hybrid inflation can produce a PTA-compatible scalar-induced gravitational wave background and, in SU(5), dilute monopoles to observable levels.
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Including a soft supersymmetry-breaking linear term in the inflaton potential lets minimal-Kaehler supersymmetric hybrid inflation produce the WMAP5 spectral index 0.963 instead of the previously predicted 0.985 or higher.

How the literature uses it

38 arXiv papers cite this in 61 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

The mu problem is solved, in this model, and the baryon asymmetry of the universe is produced through leptogenesis.

Leptogenesis in Supersymmetric Hybrid Inflation · 0.80

Papers citing it · 55 sentences without a confident label

Cites it as background 5 papers

The form of these soft terms comes from gravity-mediated SUSY breaking [ref] , which we employ throughout.

Supersymmetric Hybrid Inflation Redux · Introduction · 0.80
SUSY hybrid inflation [ref] provides fascinating framework to realize inflation within the grand unified theories (GUTs) of particle physics.

Radiative Symmetry breaking, Cosmic Strings and Observable Gravity Waves in $U(1)_R$ symmetric $SU(5) \times U(1)_{\chi}$ · Introduction · 0.96
Reference [ref] , in particular, considered supersymmetric (SUSY) models based on the gauge group $SU(2)_L ×SU(2)_R ×U(1)_B-L$.

Testing Spontaneous Parity Violation at the LHC · Introduction · 0.85

Discussion

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