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Cosmology of Supersymmetric Models with Low-energy Gauge Mediation

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Andre de Gouvea (UC Berkeley, LBNL), Takeo Moroi (LBNL), Hitoshi Murayama (UC Berkeley

High Energy Physics - Phenomenology · Submitted 8 Jan 1997 · 36 references

Pith's verdict: Appendix A's two-loop flat-direction potential is the durable contribution here; the gravitino bound recaps the authors' own earlier work, and the moduli-survival branch rests on a thermal-inflation sector that is posited rather than built.

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arXiv IDhep-ph/9701244 (version 1)
CitationAndre de Gouvea (UC Berkeley, Hitoshi Murayama (UC Berkeley, LBNL), Takeo Moroi (LBNL). "Cosmology of Supersymmetric Models with Low-energy Gauge Mediation." arXiv:hep-ph/9701244 (1997). https://pith.science/paper/hep-ph/9701244
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The paper

The paper claims

Low-energy gauge mediation can be made cosmologically viable, but only if a flat direction starts with an amplitude above about $10^{13}$ GeV so that Affleck-Dine baryogenesis generates the observed baryon asymmetry and survives the entropy

The reading

In models where supersymmetry breaking is transmitted to the Standard Model by ordinary gauge interactions at roughly $10^5$ GeV, the gravitino comes out light, around 100 keV. The paper first argues that such a gravitino is produced in the early universe in numbers that would overclose it, so some large entropy production at a temperature well below the usual reheating temperature is required — which makes ordinary high-temperature baryogenesis hard to use. The authors then ask whether the Affleck-Dine mechanism, in which a flat direction of the supersymmetric Standard Model carries a large amplitude and its rotation generates baryon number, can do the job. Because the supersymmetry-breaking masses switch off above the messenger scale, the flat-direction potential is not the usual parabola, so they compute it at two loops and find it grows only as $(ln|varphi|^2)^2$ until supergravity takes over near $7 imes 10^{11}$ GeV. Combining the two regimes, they conclude that Affleck-Dine baryogenesis yields a baryon-to-entropy ratio of order $10^{-10}$ and can survive even the enormous entropy release needed to dilute string moduli, provided a late thermal inflation with about five e-foldings is added.

What carries the argument

The load-bearing object is the two-loop effective potential of a MSSM flat direction in the presence of gauge-mediated supersymmetry breaking. Because the soft masses are proportional to $langle F_S angle^dagger langle F_S angle$ and hence must fall off at high momentum transfer, the potential is parabolic only for $|varphi| ll langle S angle$ and becomes a shallow logarithmic plateau above it; the supergravity contribution $m_{3/2}^2|varphi|^2$ then reasserts itself above $varphi_{eq}$. This two-regime potential, together with the assumed baryon-number-violating Kähler operator $l^* q^* u^c d^c/M_*^2$, controls when the field starts to roll, how much baryon number the rotating condensate deposits, and how much entropy its decay releases — the three numbers that decide whether the observed asymmetry survives.

Extended reading notes

Core claim

The paper's central claim is that Affleck-Dine baryogenesis works in low-energy gauge-mediated models, despite the fact that the flat-direction potential is qualitatively different from the hidden-sector case. The authors compute the two-loop effective potential generated by messenger and gauge-multiplet exchange in a flat-direction background and find it rises only logarithmically, $V(varphi) sim V_0 (ln |varphi|^2/langle S angle^2)^2$ with $V_0^{1/4} sim 3 imes 10^3$ GeV, above the messenger scale $langle S angle sim 3 imes 10^4$ GeV, while it is parabolic below. Adding the supergravity term $m_{3/2}^2|varphi|^2$, they locate a crossover amplitude $varphi_{eq} sim 7 imes 10^{11}$ GeV for $m_{3/2} = 100$ keV. Running the flat direction's motion through both regimes and including the entropy released when it decays, they obtain $n_B/s sim 70 imes (|varphi_0|/M_*)^2 (varphi_{dec}/10^5,{ m GeV}) sin 4 heta_0$ in the large-amplitude branch, independent of the gravitino mass, so the observed $10^{-10}$ is reachable for $|varphi_0| gtrsim 10^{13}$ GeV. They further note that string moduli, with masses of order the gravitino mass, live far longer than the age of the universe and would overclose it by more than 15 orders of magnitude, and that a late thermal inflation with $N gtrsim 5$ e-foldings and energy density below $(10^7,{ m GeV})^4$ can dilute them without erasing the baryon asymmetry.

Load-bearing premise

The initial amplitude of the flat direction is not derived from any inflationary model but simply parameterized as $varphi_0$, so every quantitative conclusion about the surviving baryon asymmetry is conditional on that amplitude being at or above roughly $10^{13}$ GeV, and near $10^{17}$ GeV for the moduli-dilution case.

Editorial extensions

If this is right

  • The gravitino mass in these models is bounded from below at roughly 70 keV by slepton searches, which is one to two orders of magnitude above the cosmological closure bound of $2h^2$ keV, so entropy production is mandatory rather than optional.
  • The decay of the Affleck-Dine flat direction itself can supply that entropy: for $|varphi_0| gtrsim 10^{14}$–$10^{15}$ GeV the dilution factor exceeds the roughly 50 required to bring relic gravitinos below closure.
  • String moduli, if they exist in this framework, are stable on cosmological timescales and overclose the universe by more than 15 orders of magnitude, so a late inflation with $N gtrsim 5$ and $rho_{inf} lesssim (10^7,{ m GeV})^4$ is needed.
  • A thermal inflation of the kind already proposed can supply exactly that, with $N simeq 5$ arising naturally from $ln(Lambda_{DSB}/m)$, and the Affleck-Dine asymmetry is large enough to survive it.
  • Because the flat-direction potential differs from the hidden-sector case, the resulting baryon asymmetry has a different dependence on the initial amplitude — quartic for small amplitudes, quadratic once entropy production dominates — which changes which inflationary models are compatible.

Reading between the lines

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

  • If the mechanism is right, the baryon asymmetry in gauge-mediated models is a fossil of the flat direction's initial amplitude, so measuring the primordial $n_B/s$ precisely would in principle constrain inflationary model building rather than the supersymmetric sector itself.
  • The same logarithmic plateau that rescues baryogenesis also makes the flat direction's oscillation very long-lived, which suggests the condensate could act as a late matter-dominated stage; that would leave a small imprint on the expansion history that is not explored here.
  • The paper's moduli argument applies to any light, gravitationally coupled scalar with mass near the gravitino mass, so the thermal-inflation requirement is arguably a generic feature of low-scale gauge mediation rather than a string-specific problem.
  • The stated $N gtrsim 5$ and $rho_{inf} lesssim (10^7,{ m GeV})^4$ window is narrow enough that a future measurement of tensor modes or of the reheating temperature could confirm or exclude this cosmological rescue.
Original abstract

We study the cosmology of supersymmetric models in which the supersymmetry breaking effects are mediated by gauge interactions at about the 10^5 GeV scale. We first point out that the gravitino is likely to overclose the Universe in this class of models. This requires an entropy production, which prefers a baryogenesis mechanism at a relatively low temperature. The Affleck-Dine mechanism for baryogenesis is one of the possibilities to generate enough baryon asymmetry, but the analysis is non-trivial since the shape of the potential for the flat direction differs substantially from the conventional hidden sector case. To see this, we first perform a 2-loop calculation to determine the shape of the potential. By combining the potential with the supergravity contribution, we then find that the Affleck-Dine baryogenesis works efficiently to generate sufficient baryon asymmetry. On the other hand, we also point out that string moduli fields, if present, are stable and their coherent oscillations overclose the Universe by more than 15 orders of magnitude. One needs a very late inflationary period with an e-folding of N \gtrsim 5 and an energy density of \lesssim (10^7 GeV)^4. A thermal inflation is enough for this purpose. Fortunately, the Affleck-Dine baryogenesis is so efficient that enough baryon asymmetry can survive the late inflation.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 7,582 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.59 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.97 · compares with prior results 0.29 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We study the cosmology of supersymmetric models in which the supersymmetry breaking effects are mediated by gauge interactions at about the 10^5 GeV scale. We first point out that the gravitino is likely to overclose the Universe in this class of models. This requires an entropy production, which prefers a baryogenesis mechanism at a relatively low temperature. The Affleck-Dine mechanism for baryogenesis is one of the possibilities to generate enough baryon asymmetry, but the analysis is non-trivial since the shape of the potential for the flat direction differs substantially from the conventional hidden sector case. To see this, we first perform a 2-loop calculation to determine the shape of the potential. By combining the potential with the supergravity contribution, we then find that the Affleck-Dine baryogenesis works efficiently to generate sufficient baryon asymmetry. On the other hand, we also point out that string moduli fields, if present, are stable and their coherent oscillations overclose the Universe by more than 15 orders of magnitude. One needs a very late inflationary period with an e-folding of N \gtrsim 5 and an energy density of \lesssim (10^7 GeV)^4. A thermal inflation is enough for this purpose. Fortunately, the Affleck-Dine baryogenesis is so efficient that enough baryon asymmetry can survive the late inflation.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study the cosmology of supersymmetric models in which the supersymmetry breaking effects are mediated by gauge interactions at about the 10^5 GeV scale. contribution 0.71 0.55 0.04 asserted 0.94
s1 We first point out that the gravitino is likely to overclose the Universe in this class of models. result 0.95 0.78 0.17 hedged possibility 1.00
s2 This requires an entropy production, which prefers a baryogenesis mechanism at a relatively low temperature. interpretation 0.89 0.53 0.06 asserted 0.79
s3 The Affleck-Dine mechanism for baryogenesis is one of the possibilities to generate enough baryon asymmetry, but the analysis is non-trivial since the shape of the potential for the flat direction differs substantially from the conventional hidden sector case. background 0.58 0.18 0.06 hedged possibility 0.91
s4 To see this, we first perform a 2-loop calculation to determine the shape of the potential. method 0.97 0.93 0.03 asserted 0.95
s5 By combining the potential with the supergravity contribution, we then find that the Affleck-Dine baryogenesis works efficiently to generate sufficient baryon asymmetry. result 1.00 0.95 0.27 asserted 1.00
s6 On the other hand, we also point out that string moduli fields, if present, are stable and their coherent oscillations overclose the Universe by more than 15 orders of magnitude. result 0.99 0.83 0.13 asserted 0.80
s7 One needs a very late inflationary period with an e-folding of N \gtrsim 5 and an energy density of \lesssim (10^7 GeV)^4. result 0.70 0.60 0.10 asserted 0.96
s8 A thermal inflation is enough for this purpose. result 0.56 0.35 0.07 asserted 0.92
s9 Fortunately, the Affleck-Dine baryogenesis is so efficient that enough baryon asymmetry can survive the late inflation. result 0.78 0.71 0.24 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.85.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

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

Figures
Figure 1. The upper bound on Tmax as a function of the gravitino mass from the requirement that the relic stable gravitinos do not overclose the Universe. We take the Hubble param￾eter to be H0 = 100 Mpc/km/sec. There is no constraint below m3/2 = 2 keV, which is represented by the vertical line. For smaller H0, the constraints become more stringent. The upper bound on Tmax shifts towards smaller Tmax as (H0) 2 . The vertical line moves towards smaller m3/2 also as (H0) 2 . Note that the current data prefer H0 ∼ 70 Mpc/km/sec. 5
Figure 2. The initial motion of the flat direction with the potential give
Figure 3. The resulting baryon-to-entropy ratio as a function of t
Figure 4. Feynman diagrams which contribute to the vacuum energy
Figure 5. A Feynman diagram with the gaugino of mass
Figure 6. Feynman diagrams with vacuum polarization due to (C1) mes
Figure 7. A plot of the effective potential Eq. (A.15) for
Figure 8. A plot of the effective potential Eq. (A.15) (or equivalently

Figures reproduced from arXiv:hep-ph/9701244 by the authors.

References
How the literature uses it

42 arXiv papers cite this in 58 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 citing it · 47 sentences without a confident label

Compares with it 1 paper
Cites it as support 2 papers
  • We will see instances where other constraints, such as overproduction of gravitinos [ref] , insure that the discrete symmetry is broken at a scale well above the reheating temperature; then domain walls have been inflated away provided the scale of inflation is not much below the GUT scale.

    Discrete R Symmetries and Domain Walls · Domain Walls and Discrete $R$ Symmetries · 0.98

  • i) The reheating temperature for our range of parameters is at most of $O(TeV)$, with gravitino at least of MeV mass, and hence the thermal production of gravitino is not significant [ref] .

    Sgoldstino-less inflation and low energy SUSY breaking · Gravitino abundance · 0.99

Cites it as background 8 papers
  • Here, $V_A$ is the A-term which generates the lepton asymmetry. $V_gauge$ denotes the potential coming from the gauge-mediated SUSY breaking [ref] , and $V_grav$ is due to the gravity mediation.

    Lepton Asymmetric Universe · Lepton asymmetry from L-ball · 0.88

  • In the gravity-mediated [ref] or gauge-mediated scenarios [ref] , the messengers correspond respectively either to supergravity fields or to some heavy particles charged under the gauge group of the standard model.

    Q-balls in flat potentials · Introduction · 0.97

  • The thermal production is always present as long as the Universe becomes radiation dominated after inflation [ref] .

    Non-thermal Gravitino Dark Matter in Gauge Mediation · Introduction · 0.88

Discussion

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