Dirac Gaugino Masses and Supersoft Supersymmetry Breaking

Ann E. Nelson; Neal Weiner; Patrick J. Fox

REVIEW

Dirac Gaugino Masses and Supersoft Supersymmetry Breaking

Patrick J. Fox, Ann E. Nelson, Neal Weiner

High Energy Physics - Phenomenology · Submitted 11 Jun 2002 · 44 references

Pith's verdict: A genuinely new SUSY-breaking framework with a clean one-loop core; the abstract oversells positivity and the finiteness claim is operator-counting rather than proof.

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CitationAnn E. Nelson, Neal Weiner, Patrick J. Fox. "Dirac Gaugino Masses and Supersoft Supersymmetry Breaking." arXiv:hep-ph/0206096 (2002). https://pith.science/paper/UQJXUDNZ

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Editor's note

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Worth your time if you work on the SUSY flavor problem. The core idea is real and simple: put the gauge sector in N=2, break SUSY with a hidden U(1)' D-term, and the operator W'^α W^α A/M gives the gaugino a Dirac mass while the adjoint scalar gets a matching mass. The scalar loop then cancels the log divergence the gaugino loop would produce. Eqs. 2.11–2.12 do that computation explicitly and the answer is finite, flavor blind, and positive. That is the paper's actual contribution and it holds up under checking. The operator argument in Sec. 2.3 — the only gauge-invariant counterterm for Q†Q involving the spurion is Eq. 2.3, suppressed by Λ² — is standard and correct as far as it goes. The bachelor-field discussion in Sec. 3.2 and the pseudoscalar adjoint phenomenology are new territory and worth the read on their own. Credit where due: Hall-Randall and Dine-MacIntire are cited for the Dirac mass precursors, and the paper is clear about what it is adding to them. Now the soft spots, in order of how much they matter. First, the abstract says the sfermion masses are 'positive' without qualification; the body (Sec. 2.2, Sec. 3.1) is honest that positivity needs δ² > m_i², i.e. the (2.9) operator absent or cancelled against the messenger contribution at the ~1% level. That caveat belongs in the abstract. Second, the all-orders finiteness is an operator-completeness assertion, not a multi-loop proof; it is probably right, but the paper writes it as settled. Third, Tables 1–3 are illustrative — the authors say two-loop and threshold effects are dropped — and those numbers will get quoted anyway. Fourth, the tree-level Higgs quartic is nearly zero (Eq. 2.8), so the model leans entirely on the stop loop; the claim that heavy stops are natural because the log is cut off at the gaugino mass is plausible but not shown quantitatively. The D-term-only premise is a model-building choice, not a result, and Sec. 3.1 says so. None of this breaks the mechanism. Send it out; ask for the finiteness statement to be tightened and the positivity caveat moved into the abstract.

Referee report

Major objections (4)

[Sec. 4 and Introduction, item 1] The Introduction lists 'natural electroweak symmetry breaking' as a headline feature, and Sec. 4 asserts that 'heavier stop masses than in the MSSM are consistent with lack of excessive fine tuning' because the logarithm in Eq. (4.1) is cut off by the gaugino mass rather than by a much higher scale. This is a quantitative claim that is not backed by any tuning measure. With the spectra of Tables 1-3 (stops 1.7-3.5 TeV, gluinos 8-15 TeV) and m_Hu² of order -(1 TeV)², a standard Barbieri-Giudice measure would give a tuning of order 10²-10³. The paper should either compute such a measure explicitly or soften the naturalness claim in the Introduction and Sec. 4 to a statement about the absence of logarithmically enhanced sensitivity.
[Abstract and Secs. 2.2, 3.1] The abstract states without qualification that the squark and slepton masses are 'positive', but Eq. (2.12) is positive only when δ² > m_i², i.e. when the operator (2.9) is absent or cancelled. The authors acknowledge in Sec. 2.2 and again in Sec. 3.1 that (2.9) is generically generated by the messenger diagram of Eq. (3.1) at a loop factor above (2.4), requiring a ~1% fine tuning to cancel. Since positivity is one of the four properties advertised in the abstract, the abstract and the summary sentence of Sec. 2.3 should state the condition under which it holds, rather than presenting it as automatic.
[Sec. 2.3, Eqs. (2.10)-(2.13)] The all-orders finiteness argument rests on the assertion that 'the only possible such counterterm is proportional to (2.3)'. This is an operator-completeness claim that is stated rather than demonstrated. A reader would want either an explicit classification of the gauge-invariant supersymmetric counterterms involving the spurion W'^α/M and the visible fields (including those with additional adjoint insertions and higher-dimension Kähler terms), or a two-loop check that the logarithmic divergence cancels as it does at one loop in Eq. (2.11). As written, the argument is plausible and standard but is the load-bearing step for the paper's central claim, so it deserves a more explicit treatment.
[Sec. 2.3, Eq. (2.13)] Eq. (2.13) generalizes the one-loop result to the case where the adjoint has a supersymmetric Majorana mass M. The bracket is not manifestly positive, and the paper does not state for which range of M and δ² the result remains positive. Since the positivity of sfermion masses is central, the authors should state the sign of Eq. (2.13) as a function of M/m_i and δ/m_i, or at least note the region in which it can turn negative.

Minor objections (7)

[Table 1 caption] The caption entry 'mhu i500 GeV' appears garbled; please give the entry a legible label and state explicitly whether the tabulated Higgs entries are the signed mass squared, the mass parameter, or its magnitude, since the text states that m_h² is negative while the tables list positive numbers.
[Sec. 2.3] The text states that the pseudoscalar SU(2) and SU(3) ESPs 'will also receive finite, positive, large masses at one loop', but the mechanism is not shown, and the same paragraph notes that the operator (2.4) leaves the pseudoscalar massless at tree level. A sentence identifying the loop that generates this mass would help the reader.
[Sec. 4, Eq. (4.5)] The quantities γ and γ' are defined in terms of M_i, δ_i and m_i, but the notation is not tied back to the definitions of Sec. 2.2; a short reminder of what M_i and δ_i denote in this section would improve readability.
[Sec. 3.2.1] The statement that α_3 is asymptotically flat at one loop and the assumption α_i(M_GUT) = α_3(M_D) are strong; since two-loop running will make α_3 non-flat, the text should flag this as an approximation alongside the existing caveats about two-loop and threshold effects.
[Sec. 5.2] The estimate of a 'few seconds' lifetime for the SU(5) bachelors from dimension-five operators should be confronted with big-bang nucleosynthesis constraints on long-lived colored or charged relics, or the absence of such a discussion should be noted.
[Figure 2] The diagram labels in the source are difficult to parse; please ensure the published figure has legible and unambiguous labels for the gaugino, adjoint scalar, and MSSM matter lines.
[References] Reference [34] is cited for a side remark on the µ parameter; since it is a companion preprint, the text should make clear that the main argument of the present paper does not depend on it.

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The paper

The paper claims

A hidden D-term could give gauginos Dirac masses and leave squark masses finite, positive, and flavor-blind.

The reading

Supersymmetry can stabilize the hierarchy between the weak and Planck scales, but the minimal supersymmetric standard model buys that stability with roughly a hundred free soft-breaking parameters and, generically, flavor-changing and CP-violating effects that experiment does not see. This paper asks whether the gauge sector can be promoted to $N=2$ supersymmetry — gauginos plus adjoint chiral superfields — while the matter sector stays $N=1$, and whether supersymmetry breaking can then be transmitted by a single $D$-term of a hidden $U(1)'$ rather than by $F$-terms. The authors show that the operator $d^2theta$-integral of $W'^alpha W^j_alpha A_j/M$ gives each gaugino a Dirac mass $m_D = D'/M$ and, crucially, that the accompanying scalar-adjoint loop cancels the logarithmic divergence a gaugino mass would normally induce in squark and slepton masses. The payoff is a spectrum fixed by very few parameters, with positive, finite, flavor-universal scalar masses, a negative Higgs mass squared driven by top/stop loops, and no new source of flavor or CP violation. If it holds, this is a competitor to gauge mediation and anomaly mediation rather than a variant of the MSSM.

What carries the argument

The load-bearing object is the supersoft operator coupling the hidden field strength to the visible one and to the adjoint chiral superfield. Its supersymmetry breaking can be packaged in a spurion $W'^alpha/M = theta^alpha m_D$, so the question of whether scalar masses are renormalized becomes a question of which gauge-invariant supersymmetric counterterms can be built from that spurion. The authors argue the only candidate is the dimension-eight Kähler operator built from two hidden field strengths contracted with $Q^dagger Q$, suppressed by two powers of the cutoff, and therefore vanishing as the cutoff is taken to infinity. The explicit one-loop integral over gaugino, scalar, and pseudoscalar contributions confirms the cancellation term by term. Two secondary pieces of machinery do real work: the $D$-flatness that follows from integrating out the massive real adjoint (which suppresses the tree-level Higgs quartic unless explicit adjoint Majorana masses are added, in which case the quartic is rescaled by factors such as $(M_1^2 + delta_1^2 - 4m_1^2)/(M_1^2 + delta_1^2)$), and the wavefunction renormalization of the adjoint fields, which converts a universal $m_D$ at the mediation scale into the gauge-coupling-weighted gaugino masses at the weak scale.

Extended reading notes

Core claim

The claim is that a supersymmetric extension with $N=2$ gauge multiplets and $N=1$ matter admits a genuinely new kind of soft breaking, which the authors call supersoft. The breaking is carried by a single operator coupling the hidden $U(1)'$ field strength to the visible gauge field strength and the new adjoint superfield $A_j$, with a coefficient $1/M$. It generates a Dirac mass $m_D = D'/M$ pairing each gaugino with the adjoint fermion, and simultaneously gives the real scalar component of $A_j$ a mass, leaving the pseudoscalar massless in the minimal case. The central technical result is that the usual logarithmic divergence a gaugino mass would produce in squark and slepton masses is cancelled by the new scalar loop, so the one-loop scalar masses are finite and calculable, given by $m^2 = C_i(r)alpha_i m_i^2 log(delta^2/m_i^2)/pi$ and vanishing as $delta$ approaches $m_i$ from above. Because the breaking is transmitted only through gauge interactions, the resulting scalar masses are positive, CP-conserving, and flavor universal. The Higgs mass squared comes out negative through the two-loop top/stop contribution, so electroweak symmetry breaks radiatively. The spectrum is distinctive: gauginos in the multi-TeV range, squarks around 2–3 TeV, $R$-parity-even pseudoscalar adjoints, and, in unified versions, long-lived 'bachelor' fields carrying the quantum numbers of heavy GUT gauge bosons.

Load-bearing premise

The whole construction assumes that the hidden $U(1)'$ $D$-term is the only source of supersymmetry breaking reaching the visible sector, since any comparable $F$-term or Planck-suppressed Kähler contribution to sfermion masses would reintroduce the flavor violation and the cutoff sensitivity the mechanism is designed to eliminate.

Editorial extensions

If this is right

The soft scalar spectrum is fixed by gauge quantum numbers alone, so squark and slepton masses are flavor universal by construction and the usual supersymmetric flavor-changing and CP problems are absent without any degeneracy assumption.
Gauginos are heavy Dirac fermions in the multi-TeV range while squarks sit near 2–3 TeV, so the LHC signatures differ sharply from the MSSM: no light bino lightest supersymmetric particle, and cascade decays ending in Higgsinos, a gravitino, or a neutral bachelor.
The theory predicts new $R$-parity-even states — pseudoscalar adjoints and, in unified versions, bachelor fields with GUT gauge-boson quantum numbers — some of which can be long-lived or, in $SU(3)^3$ unification, serve as dark matter.
Because the corrections to the Higgs mass squared are not logarithmically enhanced, stop masses of several TeV do not create a fine-tuning problem, unlike in the MSSM or in gauge mediation.
The $\mu$ and $B\mu$ parameters can be generated from the same supersymmetry breaking, either through a Giudice–Masiero term when $D \sim (10^{11}\,\mathrm{GeV})^2$ or through a singlet coupled to messengers when the $D$-term is smaller.

Reading between the lines

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

If the mechanism is right, the ratio of gaugino to squark mass is a fixed prediction rather than a free choice, so a single measurement of, say, the gluino mass together with a squark mass would over-determine the mediation scale $M$ and the $D$-term, giving a sharp consistency test.
The same operator can be written for any real representation of a gauge group, which suggests the construction generalizes beyond the standard-model gauge group to any hidden sector whose $D$-term is the dominant breaking source; the finiteness argument depends only on gauge invariance and the spurion power counting, not on the details of the visible sector.
The near-massless pseudoscalar of the $U(1)$ adjoint is the weakest link phenomenologically: without an added Majorana mass or trilinear coupling it is a light, $R$-parity-even, standard-model-singlet scalar, and cosmology would likely require it to be either heavy or diluted.
The claim that heavy first- and second-generation sfermions evade flavor bounds while remaining natural sits in tension with the usual expectation that naturalness prefers light stops; the paper resolves this by the absence of logarithmic running, which is exactly the point an explicit two-loop calculation in a UV completion would need to confirm.

Original abstract

We introduce a new supersymmetric extension of the standard model in which the gauge sector contains complete N=2 supersymmetry multiplets. Supersymmetry breaking from the D-term vev of a hidden sector U(1) gauge field leads to Dirac soft supersymmetry breaking gaugino masses, and a new type of soft scalar trilinear couplings. The resulting squark and slepton masses are finite, calculable, positive and flavor universal. The Higgs soft mass squared is negative. The phenomenology of these theories differs significantly from the MSSM. We discuss a variety of possible origins for the soft operators and new fields, including models in both four and higher dimensions.

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.

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What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.11 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.63 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; show every role.

We introduce a new supersymmetric extension of the standard model in which the gauge sector contains complete N=2 supersymmetry multiplets. Supersymmetry breaking from the D-term vev of a hidden sector U(1) gauge field leads to Dirac soft supersymmetry breaking gaugino masses, and a new type of soft scalar trilinear couplings. The resulting squark and slepton masses are finite, calculable, positive and flavor universal. The Higgs soft mass squared is negative. The phenomenology of these theories differs significantly from the MSSM. We discuss a variety of possible origins for the soft operators and new fields, including models in both four and higher dimensions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We introduce a new supersymmetric extension of the standard model in which the gauge sector contains complete N=2 supersymmetry multiplets.	contribution 1.00	0.88	0.11	asserted 1.00
s1	Supersymmetry breaking from the D-term vev of a hidden sector U(1) gauge field leads to Dirac soft supersymmetry breaking gaugino masses, and a new type of soft scalar trilinear couplings.	contribution 0.46	0.75	0.32	asserted 1.00
s2	The resulting squark and slepton masses are finite, calculable, positive and flavor universal.	result 1.00	0.82	0.28	asserted 1.00
s3	The Higgs soft mass squared is negative.	result 1.00	0.79	0.10	asserted 1.00
s4	The phenomenology of these theories differs significantly from the MSSM.	interpretation 0.88	0.31	0.15	asserted 1.00
s5	We discuss a variety of possible origins for the soft operators and new fields, including models in both four and higher dimensions.	contribution 0.86	0.69	0.05	hedged possibility 0.54

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.82.

What was examined, and what was not

sources read

arXiv listing abstract v3

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.** Categorization of fields into N = 2 and N = 1 sectors. The Higgs fields could be part of either sector. make the total beta functions for the gauge groups excessively positive, leading again to low energy Landau poles, so we only extend the gauge sector to N = 2. As we shall see shortly, with an N = 2 gauge sector, a novel form of supersymmetry breaking can be added, which is not even logarithmically UV sensitive. This division into the N = 2 sector and the N = 1 sector is shown in figure 2. The MSSM matter fields embedded into N = 1 multiplets, while we can quite straightforwardly…

**Figure 2.** Loop contributions to scalar masses. The new contribution from the purely scalar loop cancels the logarithmic divergence resulting from a gaugino mass alone. for the masses involving this spurion. The only possible such counterterm is proportional to (2.3), and gives Z d 4 θ θ 2 θ 2 m4 D Λ2 Q †Q. (2.10) Since we have four powers of mD, we have to introduce another scale to make this dimensionfully consistent. Since the only other scale is the cutoff Λ, this operator is suppressed by Λ2 , and, in the limit that Λ → ∞, must vanish. Consequently, we conclude all radiative co

**Figure 3.** Integrating out messenger fields can generate the operator of (2.4). 3. Gauge Extended Models The presence of the new fields, and new operators gives a multitude of model building options. Many issues remain to be explored: grand unification, the origin of (2.4) and (2.9), as well as an explanation of the Higgs parameters µ and Bµ. Our discussion here should not by any means be considered exhaustive. 3.1 Origins of D-terms and gaugino/ESP mixing Up to this point we have discussed neither the origin of the term in (2.4), nor the source of the D-term which produces supersymmetry…

Figures reproduced from arXiv:hep-ph/0206096 by the authors.

References

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How later papers used it

109 later papers cite it

Evidence: substantial. 101 of the citing papers share no author with it, and 7 of those carry a confident label from Pith's citation classifier.

Pith read 25 of the 204 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

1 later paper by other authors is listed below: 1 compared its own results with it.

Compares with it 1 of 5 papers with this label

Found a result in agreement with the expressions in this paper.

Dirac Gauginos in General Gauge Mediation (2008) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

109 arXiv papers cite this in 203 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 · 145 sentences without a confident label

Contests it 1 paper

Previous microscopic derivations of supersoft Dirac gaugino masses were plagued with a problem: in the perturbation expansion one obtains a tachyonic mass for one component of the scalars in the adjoint representation [ref] .

Dirac Gauginos in General Gauge Mediation · Introduction · contest 0.72

Builds on it 1 paper

Motivated by stabilizing the hierarchy between the light scalars and the heavy fermions, we discuss one of the possible supersymmetric extensions of the varying kinetic mixing in the Dirac gaugino model [ref] with the superpotential [equation] In eq:dgugino, $W_j$ is the gauge field strength of the SM gauge group $G_SM, j$ where the label $j=1,2,3$ denotes the SM gauge groups $U(1)_Y$, $SU(2)_L$ and $SU(3)_c$, respectively, $W'$ is the gauge field strength of $U(1)_d$, and $A_j$ is the chiral multiplet.

Cosmologically Varying Kinetic Mixing · Varying Kinetic Mixing From Dirac Gaugino · 0.81

Compares with it 5 papers

If $H_1/2(0)$ is real, then we find g_r^2 δA_r = 4 m_D^2 logm_P^2/m_D^2 giving δm_f̃^2 = ∑_r=1^3 C_2 (f; r) g_r^2 (m_D^(r))^2/4π^2 log(m_P^(r)/m_D^(r))^2 , in agreement with the expressions in [ref] .

Dirac Gauginos in General Gauge Mediation · Adjoints and General Gauge Mediation · 0.99
Let us stress that the presence of supersymmetric couplings, as well as holomorphic and non-holomorphic masses for the adjoint scalars, improves the situation with respect to [ref] , where the quartic coupling vanishes for decoupled adjoint scalars (see also [ref] ).

Dirac gauginos, R symmetry and the 125 GeV Higgs · Tree level Higgs mass · 0.95
Since the gauge sector of the model is similar to that in [ref] , the model shares the good feature that unlike the MSSM, the usual logarithmic divergence from the stop contributions to the Higgs mass-squared parameter $m_H_u^2$ are cutoff by the Dirac gluino mass, leading to only a finite contribution.

"L=R" - $U(1)_R$ as the Origin of Leptonic 'RPV' · The Scalar Electroweak Sector · 0.86

Cites it as support 1 paper

One important restriction is that the Dirac wino cannot be light in the MRSSM, due to the structure of the wino supersoft operator [ref] .

\mu to e in R-symmetric Supersymmetry · A Simplified R-symmetric Model · 0.98

Cites it as background 47 papers

In supersoft models the radiatively generated squark and slepton masses are finite, flavor symmetric, positive, UV insensitive, and light compared to the gaugino masses [ref] .

New Supersoft Supersymmetry Breaking Operators and a Solution to the $\mu$ Problem · Introduction · 0.94
Generating Dirac gauginos through gauge mediation was proposed by Nelson et. al., [ref] [ref] , and larger explorations of Dirac gauginos in the formalism of general gauge mediation have been made [ref] .

Dirac Gauginos, Negative Supertraces and Gauge Mediation · Introduction · 0.86
The soft SUSY-breaking Lagrangian has been defined in Ref. [ref] , based on the discussion of supersoft supersymmetry breaking in Ref. [ref] .

Low-energy lepton physics in the MRSSM: $(g-2)_\mu$, $\mu \to e\gamma$ and $\mu\to e$ conversion · Model definition · 0.80

Discussion

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