Retrofitting O'Raifeartaigh Models with Dynamical Scales

Eva Silverstein; Jonathan L. Feng; Michael Dine

REVIEW

Retrofitting O'Raifeartaigh Models with Dynamical Scales

Michael Dine, Jonathan L. Feng, Eva Silverstein

High Energy Physics - Theory · Submitted 23 Aug 2006 · 27 references

Pith's verdict: A clean, reusable trick — small parameters replaced by W_alpha W^alpha/M_*^n enforced by a discrete R symmetry — with two worked examples; the stabilization check is order-of-magnitude but survives scrutiny at the quoted parameters.

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arXiv IDhep-th/0608159 (version 2)

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CitationEva Silverstein, Jonathan L. Feng, Michael Dine. "Retrofitting O'Raifeartaigh Models with Dynamical Scales." arXiv:hep-th/0608159 (2006). https://pith.science/paper/RQWZCVTK

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Editor's note

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The thing to take away here is the trick, not the models. Take an O'Raifeartaigh or Fayet model whose small parameters would respect an R symmetry if they transformed like the superpotential, couple in a pure SYM sector through W_alpha W^alpha/M_*^n, and those parameters become powers of Lambda^3/M_*. The discrete R symmetry does the forbidding, so it is natural in the technical sense. That packaging is genuinely useful and I have not seen it laid out this cleanly before. The two examples — the O'Raifeartaigh model with 5+5bar messengers in §2.1, and the Fayet model in §3 where the a, a~ sector generates eD ~ m_a^2 — are complete enough to lift into your own model building, messengers included. The §2.2 observation that metastability lets you drop chirality and vanishing-Witten-index requirements is a good model-building point, though it is in the air post-ISS rather than new here. On the stress-test worry: I checked it and it mostly does not land. The Coleman-Weinberg curvature from Eq. (2.6) is positive regardless of the renormalization scale — differentiating the log gives c^2/(16 pi^2 (mu^2+|Z2|^2)), positive definite; the sign of the log's argument only shifts the constant. And the control parameter works out: Z2*/M_* ~ Lambda^3/(lambda^4 M_*^3) = mu^2/(lambda^4 M_*^2), which at mu ~ 10^12, M_* ~ 10^15, lambda ~ 0.1 is about 5 x 10^-3. So the M_* expansion is controlled at the quoted numbers. The paper does not spell this out, but it is a two-line check. What is genuinely soft: everything is order-one scaling, there is no numerical minimization of the full potential, and the global structure is not analyzed — the authors say so themselves (SUSY vacua far away, cosmology deferred to [25,26]). The Witten index claim in §2.2 is asserted, not shown. The M-generation in Eq. (2.15) is explicitly not symmetry-enforced, and they flag that. The messenger-loop subleading claim is a scaling estimate. None of these is load-bearing for the method; they are the usual limits of a model-building note. Who it is for: anyone doing gauge mediation or dynamical SUSY breaking who needs a small scale without a small input. It is a note, not a treatise — fifteen pages, no numerics. Peer review: yes, send it out. It is a real construction with checkable steps.

Referee report

Significance

The method is simple, general, and clearly stated, and the paper is an early and influential entry in the metastable-SUSY-breaking program. Its central construction is not circular: Lambda is an independent input of the hidden gauge theory, and the discrete R symmetry forbids the small parameters rather than fitting them to reproduce a desired answer. The two worked examples are complete enough to be taken seriously as models of direct gauge mediation, and the paper is candid about the role of metastability and about the fact that SUSY vacua can appear far away in field space. The explicit messenger-sector mass formula (Eq. 2.7) and the Fayet-model scaling (Eqs. 3.8-3.9) are useful, checkable results. The paper would be strengthened by making the local-stability check quantitative rather than order-of-magnitude, but the central claim is defensible as it stands.

Major objections (3)

[§2.1, paragraph following Eq. (2.10)] The stabilization argument for Z2 is order-of-magnitude only: a tadpole of order Z2 Lambda^12/(M_*^9 lambda^2) is balanced against a Coleman-Weinberg mass of order lambda^2 |Z2|^2 Lambda^9/M_*^7, and the text concludes this is 'sufficient to stabilize Z2 close to its original minimum at the origin.' The minimum of V(Z2) = a Z2 + b |Z2|^2 sits at Z2* = -a/(2b), and the paper does not evaluate whether |Z2*|/M_* remains small at the quoted parameters (lambda ~ 0.1, m_eta ~ 10^11 GeV, M_* ~ 10^15 GeV). Since the entire M_*-expansion of Eqs. (2.2) and (2.8) presupposes |Z2|/M_* << 1, an explicit numerical value of Z2*/M_* at the quoted parameters should be given; otherwise the 'self-consistent' claim is asserted rather than demonstrated. On the sign question, the positive curvature is in fact manifest from Eq. (2.6) as written (the prefactor is positive and the logarithm increases with |Z2|^2), but the renormalization-scale dependence of the log argument should be stated, since the coefficient of |Z2|^2 depends on it.
[§2.1, Eq. (2.3) and Eq. (2.4)] The solution for phi_0 and the resulting F-terms are difficult to parse as printed: Eq. (2.3) contains symbols (mu_1, mu_4) that do not appear elsewhere in the model, and the structure of the expansion in mu/M_* and Z2/M_* is not made explicit. Since Eqs. (2.3)-(2.4) are the inputs to the messenger-mass estimate of Eq. (2.7) and to the stabilization discussion after Eq. (2.10), they should be rewritten cleanly, with the regime of validity stated and the leading corrections displayed.
[§3.2, Eq. (3.12) and following] The chain from eD = m_a^2 ~ Lambda^6/M_*^4 to the quoted F_X ~ Lambda^9/M_*^7 is not shown. The text notes that the symmetry also dresses M^2 X - lambda X^3/3 by W_alpha W^alpha/M_*^3 and adds an M X^2 term, but the effect of these replacements on the relation between X0 and M, and hence on F_X, is left implicit. Please display the intermediate step explicitly, including the M-dependence, so that the reader can verify the scaling and see under what relation between M and M_* it holds.

Minor objections (6)

[§2.2, Eq. (2.11) and following] The statement that with epsilon = 0 the model 'is the most general consistent with a discrete Z2N R symmetry' should specify up to which order in 1/M_* the classification is meant.
[§2.2, paragraph after Eq. (2.14)] The claim that for SU(N) the Witten index 'can be computed for non-zero epsilon, and it is equal to N' should be justified or given a reference, since it is used to support the non-chiral, non-vanishing-index character of the construction.
[§4, final paragraph] The assertion that 'the metastable vacua themselves are very long-lived' is not supported by a lifetime estimate in the text; either provide one or soften the statement to indicate that it is expected on general grounds.
[§2.1, Eq. (2.6) and §3.1, Eq. (3.6)] The Coleman-Weinberg expressions are written with a logarithm whose argument involves M_*; the renormalization scale at which these expressions are evaluated should be stated explicitly, since the coefficient of the leading |Z2|^2 (or X^2) term depends on it.
[§2.1, paragraph after Eq. (2.4)] The parenthetical remark that the suppression of F_Z2's coupling to messengers 'is forced on the model by the discrete symmetries it respects' would benefit from naming the specific symmetry charge assignment that enforces it.
[References] Reference [26] is listed as 'to appear'; since the text only says the cosmological question is under investigation there, this is acceptable, but the authors may wish to flag it as unpublished in the citation itself.

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The paper

The paper claims

The paper proposes a general recipe for replacing the hand-entered small mass parameters of O'Raifeartaigh and Fayet supersymmetry-breaking models with powers of a dynamical scale, so that every small number in the theory is generated by a纯

The reading

The paper's goal is to take the old-fashioned, perturbative models of supersymmetry breaking — O'Raifeartaigh models and Fayet models — and fix their most awkward feature: the small mass parameters are put in by hand, with no symmetry explaining why they are small. The authors propose a general procedure they call retrofitting: couple the model to a pure supersymmetric Yang-Mills sector whose gaugino condensate is $W_alpha W^alpha ~ Lambda^3$, and let that operator supply the would-be small parameter through a coupling suppressed by a high scale $M_*$. If the original model would have respected an $R$ symmetry when its small parameters transformed the way the superpotential does, the replacement is automatic and natural, because the symmetry forbids the bare parameter but allows the $W_alpha W^alpha$ coupling. The authors then check, in a complete worked example with messengers in the $5$ and $bar 5$ of $SU(5)$, that the local supersymmetry-breaking minimum survives this surgery. What matters if the claim holds is that simple, non-chiral, even Witten-index-nonzero models become usable as hidden sectors, and supersymmetry breaking can be mediated to the Standard Model either by gauge mediation or by gravity mediation without inventing new small scales.

What carries the argument

The load-bearing device is the gaugino condensate of a pure supersymmetric Yang-Mills sector: the chiral operator $W_alpha W^alpha$ acquires a holomorphic expectation value $~ Lambda^3$, and because it transforms nontrivially under the discrete $R$ symmetry, a coupling such as $Z, W_alpha W^alpha/M_*$ is symmetry-allowed exactly where a bare mass term would be forbidden. The symmetry thus does the bookkeeping: it forbids the small parameter and permits its dynamical replacement. The second piece of machinery is the Coleman-Weinberg potential, which lifts the classical flat direction of the O'Raifeartaigh model and supplies the curvature that holds the would-be modulus near the origin; the paper checks that the $phi$ loops dominate over messenger loops and that the tadpole induced by the $Z_2$ dependence of the gauge coupling is smaller than the Coleman-Weinberg mass, so the expansion about $Z = 0$ is self-consistent.

Extended reading notes

Core claim

The central claim is that a large class of perturbative supersymmetry-breaking models can be made natural by a single structural move: replace each small input parameter $m_i$ with a coupling of the form $int d^2theta, W_alpha W^alpha f(Z)/M_*^n$, so that at low energies $m_i$ becomes a power of the Yang-Mills dynamical scale $Lambda$ divided by powers of $M_*$. In the explicit example, the O'Raifeartaigh mass parameter $mu^2$ becomes $mu^2 prop Lambda^3/M_*$ through a coupling $lambda Z_2 W_alpha W^alpha/M_*$, and the resulting superpotential is $W = Z_1 phi^3/(3M_*) + lambda Lambda^3 e^{-12 Z_2/(b_0 M_*)} + Z_2(lambda phi^2/2[1 + lambda_1 Z_2/M_*] + phi eta tilde-eta/M_*) + lambda phi eta tilde-eta + lambda_2 (eta tilde-eta)^2/M_*$. The authors argue that the Coleman-Weinberg potential generated by integrating out $phi$ and the messengers still produces a positive curvature at the origin, so the local supersymmetry-breaking minimum is not destabilized, even though supersymmetric vacua appear far away in field space. They extend the same move to Fayet models, where the Fayet-Iliopoulos parameter $eD$ is traded for a dynamical $m_a^2 ~ Lambda^6/M_*^4$, and they note that the resulting models need not be chiral and can have non-vanishing Witten index.

Load-bearing premise

The claim that the local supersymmetry-breaking minimum survives rests on a leading-order expansion in $Z_2/M_*$ around the origin, where a tadpole of order $Z_2 Lambda^{12}/(M_*^9 lambda^2)$ is balanced against a Coleman-Weinberg mass of order $lambda^2 |Z_2|^2 Lambda^9/M_*^7$; this is a local perturbative check, not a global analysis of the potential.

Editorial extensions

If this is right

Complete models of direct gauge mediation follow, in which the messenger fields $eta, tilde-eta$ both stabilize the scalar potential and transmit supersymmetry breaking to the Standard Model, giving superpartner masses of order $tilde m ~ g^2 mu^4/(16 pi^2 M_*^3 lambda^2)$.
With $m_{eta,tilde-eta} ~ lambda mu ~ 10^{11}$ GeV and $M_* ~ 10^{15}$ GeV, the gauge-mediated contribution dominates gravity mediation for $F_{Z_i} lesssim 10^{20}$ GeV$^2$, so the construction is viable at intermediate messenger scales.
The same retrofitting step converts the Fayet-Iliopoulos parameter of a Fayet model into $m_a^2 ~ Lambda^6/M_*^4$, giving $F_X ~ Lambda^9/M_*^7$ and a naturally small $D$-term.
Because the resulting models need not be chiral and can carry non-vanishing Witten index, they enlarge the usable class of hidden sectors well beyond the chiral, index-zero constructions that dynamical supersymmetry breaking usually demands.
The models double as hidden sectors for gravity mediation, since no symmetry forbids coupling the singlet $Z$ to Standard Model gauginos, giving scalar and gaugino masses of the same order.

Reading between the lines

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

The recipe effectively turns the $R$ symmetry into a selection rule for mass hierarchies: any small parameter whose absence is enforced by an $R$ symmetry can be promoted to a power of $Lambda/M_*$, which suggests a systematic catalogue of which perturbative breaking models are retrofittable and which are not.
Because the supersymmetric vacua sit far away in field space while the local minimum sits near the origin, the cosmology of these models is a separate question from their existence; whether the early universe settles into the metastable vacuum is not settled by the potential calculation here.
The construction invites a check of whether the same gaugino-condensate trick can supply the messenger scale itself, which would push the messenger masses below $10^7$ GeV and remove the gravitino-overclosure problem that the paper notes for its own examples.
If the $Lambda^3$ replacement is generic, one expects a low-energy signature in which the ratios of small parameters in the hidden sector track powers of a single scale, so hidden-sector spectroscopy would be organized by one number rather than by several independent inputs.

Original abstract

We provide a method for obtaining simple models of supersymmetry breaking, with all small mass scales generated dynamically, and illustrate it with explicit examples. We start from models of perturbative supersymmetry breaking, such as O'Raifeartaigh and Fayet models, that would respect an $R$ symmetry if their small input parameters transformed as the superpotential does. By coupling the system to a pure supersymmetric Yang-Mills theory (or a more general supersymmetric gauge theory with dynamically small vacuum expectation values), these parameters are replaced by powers of its dynamical scale in a way that is naturally enforced by the symmetry. We show that supersymmetry breaking in these models may be straightforwardly mediated to the supersymmetric Standard Model, obtain complete models of direct gauge mediation, and comment on related model building strategies that arise in this simple framework.

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We provide a method for obtaining simple models of supersymmetry breaking, with all small mass scales generated dynamically, and illustrate it with explicit examples. We start from models of perturbative supersymmetry breaking, such as O'Raifeartaigh and Fayet models, that would respect an $R$ symmetry if their small input parameters transformed as the superpotential does. By coupling the system to a pure supersymmetric Yang-Mills theory (or a more general supersymmetric gauge theory with dynamically small vacuum expectation values), these parameters are replaced by powers of its dynamical scale in a way that is naturally enforced by the symmetry. We show that supersymmetry breaking in these models may be straightforwardly mediated to the supersymmetric Standard Model, obtain complete models of direct gauge mediation, and comment on related model building strategies that arise in this simple framework.

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s0	We provide a method for obtaining simple models of supersymmetry breaking, with all small mass scales generated dynamically, and illustrate it with explicit examples.	contribution 1.00	0.95	0.60	asserted 1.00
s1	We start from models of perturbative supersymmetry breaking, such as O'Raifeartaigh and Fayet models, that would respect an $R$ symmetry if their small input parameters transformed as the superpotential does.	method 0.57	0.27	0.04	asserted 0.93
s2	By coupling the system to a pure supersymmetric Yang-Mills theory (or a more general supersymmetric gauge theory with dynamically small vacuum expectation values), these parameters are replaced by powers of its dynamical scale in a way that is naturally enforced by the symmetry.	method 0.98	0.75	0.16	asserted 1.00
s3	We show that supersymmetry breaking in these models may be straightforwardly mediated to the supersymmetric Standard Model, obtain complete models of direct gauge mediation, and comment on related model building strategies that arise in this simple framework.	contribution 0.65	0.92	0.36	hedged possibility 0.53

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References

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Forward citations

Cited by 13 Pith papers

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Spontaneous R-Symmetry Breaking in O'Raifeartaigh Models
hep-th 2007-03 accept novelty 7.0 of 10

Spontaneous breaking of the R-symmetry in O'Raifeartaigh models requires at least one field with R-charge other than 0 or 2, and the simplest model with such a field does break it.
A Stealth Supersymmetry Sampler
hep-ph 2012-01 conditional novelty 6.0 of 10

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hep-th 2009-02 accept novelty 6.0 of 10

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How the literature uses it

52 arXiv papers cite this in 74 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 · 56 sentences without a confident label

Uses its method or tool 2 papers

I will use the method of retrofitting [ref] to generate the small parameter in ([ref]), $μ$, dynamically.[This model of inflation has been modified to include dynamically generated terms in [ref] , but in that paper they used a different technique.] This method begins by introducing a new supersymmetric sector.

Retrofitting Models of Inflation · Retrofitting · 0.98
Using conventional effective field theory philosophy, we could ascribe this by the strategy of retro-fitting [ref] .

A Pyramid Scheme for Particle Physics · Breaking $R$-symmetry and SUSY · 0.89

Compares with it 2 papers

The scenario presented here has many features in common with that presented in [ref] .

Dynamical SUSY Breaking in Intersecting Brane Models · Conclusions · 1.00
It presents a new viewpoint on the dynamical origin of the small $ϵ/μ$ different from the discussion in [ref] , where the strong gauge dynamics of the hidden sector is considered to explain the smallness.

Spontaneous Supersymmetry Breaking by Large-N Matrices · O'Raifeartaigh Model Coupled to SUSY Matrix Model · 0.99

Cites it as background 10 papers

Recently, these ideas regained a lot of attention [ref] , following the elegant work of Intriligator, Seiberg and Shih (ISS) [ref] .

Comments on the Meta-Stable Vacuum in N_f=N_c SQCD and Direct Mediation · Introduction · 0.86
This possibility has been sharpened recently by considerations of the landscape [ref] and of metastable supersymmetry breaking [ref] .

The Fate of Nearly Supersymmetric Vacua · The Stability of Nearly Supersymmetric Vacua · 0.88
Recent developments in dynamical supersymmetry breaking [ref] reopen these issues.

Dynamical Supersymmetry Breaking and Low Energy Gauge Mediation · Introduction · 0.88

Discussion

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