Two-Loop Renormalization Group Equations for Soft Supersymmetry-Breaking Couplings

Michael T. Vaughn; Stephen P. Martin

REVIEW

Two-Loop Renormalization Group Equations for Soft Supersymmetry-Breaking Couplings

Stephen P. Martin, Michael T. Vaughn

High Energy Physics - Phenomenology · Submitted 23 Nov 1993 · 35 references

Pith's verdict: The standard two-loop RGE reference for soft SUSY-breaking couplings; the equations are complete and cross-checked, with one acknowledged prescription dependence in Eq. (2.20) that propagates into every MSSM scalar-mass beta function.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on October 29, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 1 major objection · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-ph/9311340 (version 4)

Pith IDGQBXVASY, Pith's permanent record of this paper and its review

CitationMichael T. Vaughn, Stephen P. Martin. "Two-Loop Renormalization Group Equations for Soft Supersymmetry-Breaking Couplings." arXiv:hep-ph/9311340 (1993). https://pith.science/paper/GQBXVASY

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Bottom line: this is the reference paper for two-loop running of soft SUSY-breaking couplings, and it holds up. The new content is (2.13)-(2.20) — the two-loop beta functions for h^{ijk}, b^{ij}, and (m^2)^j_i — plus the full MSSM specialization (4.18)-(4.43). The rest of Section II reviews known gauge, superpotential, and gaugino-mass results and says so plainly, which is the right call for a paper people will use as a lookup table [abstract:s0]. The equations are written out completely, with no compressed notation, so you can implement them directly. The N=2 finiteness check at the end of Section II is a real test, not decoration: the general formulas collapse to zero under the appropriate constraints, and that is hard to fake. The Note added is also honest — it records that Eq. (2.20) and some matrix orderings in Section IV were wrong in the original version and were fixed after comparison with Yamada and Jack-Jones. Where I would push is narrow. The derivation is deliberately not shown (we decline to exhibit the details here), and for most of the paper that is fine, because the dummy-scalar bookkeeping is mechanical and the results are corroborated. Eq. (2.20) is the exception. The Note added says the corrected coefficient corresponds to a prescription for the epsilon-scalar mass, chosen so the two-loop betas and the pole-mass relations do not depend on it, and then defers the argument to a note to be reported on elsewhere. That is a scheme choice asserted rather than derived, and the support at that point is two preprints rather than published work. Since (2.20) feeds every scalar-mass equation in Section IV, that is the one place where I would want the follow-up before trusting the last term at full precision. Proportion: a narrow caveat on an otherwise solid, heavily used reference. The h and b results are fine. Who it is for: anyone running soft parameters numerically or checking high-scale boundary conditions. Not a paper to read for ideas; a paper to cite and implement. Peer review: yes, send it out.

Referee report

Significance

If correct, this is the definitive reference for two-loop running of soft supersymmetry-breaking parameters, and it has plainly been used that way. The value of the paper is the explicit, complete set of equations rather than a conceptual advance, and the authors have done the field the service of printing every coefficient, keeping complex phases general, and separating M from M* throughout. Two features deserve explicit credit. First, the N=2 finiteness check described at the end of Section II is a genuine, nontrivial consistency test of the whole set of general formulas, not a cosmetic remark. Second, the Note added is a model of transparent correction: it identifies the erroneous coefficient in Eq. (2.20), states that the Section IV results were corrected accordingly, and credits the groups that found the discrepancy. The results are falsifiable in the practical sense that they can be checked against direct DRED calculations and against spectrum codes, and they have been.

Major objections (1)

[Eq. (2.20) and Note added] This is the one place where the manuscript's own text leaves a condition unmet, and the authors acknowledge it. The Note added states that the corrected coefficient of the last term in Eq. (2.20), 8 delta^i_j g^4 C(i)(Tr[S(r)m^2] - C(G)MM^dagger), 'corresponds to this prescription' for the unphysical epsilon-scalar mass, and that this prescription is related to the one used in [35] 'by a simple coupling constant redefinition for the scalar masses, of the type mentioned there.' Neither the subtraction condition nor the redefinition is exhibited, and the justification is deferred to a note 'to be reported on elsewhere.' Because Eq. (2.20) feeds every MSSM scalar-mass equation (4.30)-(4.43), a reader cannot verify from this manuscript alone that the printed equation is prescription-independent at the order claimed. I recommend a short appendix giving the epsilon-scalar mass counterterm treatment and the explicit redefinition relating the two prescriptions, or at minimum a precise statement of the subtraction condition that defines the scheme. I want to be clear that this is a transparency and reproducibility request, not a challenge to the result: the equation as printed is corroborated by the independent calculations of [34] and [35], and the risk that it is wrong is correspondingly low.

Minor objections (7)

[Note added] The Note added says that 'the correct ordering of matrices in some of the terms in the beta functions of Sec. IV' has been fixed, but it does not say which terms. Since users will compare the printed Section IV against the original preprint and against spectrum codes built on it, the affected equations or terms should be identified explicitly, for example in a short list or by marking the changed lines.
[Section I] The sentence 'This process is straightforward but rather tedious, and we decline to exhibit the details here' is candid, and the authors are entitled to omit routine algebra. What remains unresolved is reproducibility: the dummy-field assignments and the symmetry factors that connect the Machacek-Vaughn quartic-coupling results to the trilinear and bilinear soft terms are the substance of the calculation, and a compact table of those assignments would let a reader check the translation without redoing it.
[Section II, end] The N=2 finiteness check is described as 'an amusing exercise' but no explicit demonstration is given. One worked example, stating the representation content and soft-coupling constraints under which the beta functions are claimed to vanish, would convert an assertion into a verifiable test and would strengthen the paper's most valuable internal cross-check.
[Section V] The statement that the two-loop beta functions 'generally make a difference of several percent' for sparticle masses is not supported by any numbers in this paper. Either cite the numerical study on which the estimate is based or soften the claim to indicate that it is a rough expectation.
[Section III] The treatment of multiple U(1) factors in Eqs. (3.17)-(3.18) is careful, but Fayet-Iliopoulos D-terms, which can appear for U(1) factors and which affect the running of scalar masses, are not discussed. A sentence stating whether they are outside the scope of the paper, and why, would prevent readers from assuming they are covered.
[Eq. (2.6) and Section II] The text notes that the heights of indices on the last term of Eq. (2.6) have been reversed relative to [20], but the convention for h_{ijk} versus h^{ijk} is not stated in one place. A single sentence fixing the convention at the start of Section II would make the comparison with Eqs. (2.13)-(2.14) and with the earlier literature straightforward.
[References] References [34] and [35] are cited as preprints. If published versions now exist, they should be given, since these are the two independent confirmations on which the Note added relies.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The paper supplies the complete two-loop renormalization group equations for every soft supersymmetry-breaking coupling in a general softly broken supersymmetric model, and works them out explicitly for the Minimal Supersymmetric Standard M

The reading

If supersymmetry is realized at low energies, the masses of the unseen partner particles are not free parameters at the high scale — they are produced by running a handful of soft-breaking couplings down from unification. Getting sparticle masses right to the few-percent level therefore requires two-loop accuracy in those couplings, not just one loop. This paper computes, for the first time, the two-loop $\beta$ functions for all three classes of soft-breaking couplings in a general $N=1$ supersymmetric gauge theory: the trilinear scalar couplings $h^{ijk}$, the bilinear $b^{ij}$, and the scalar mass-squared matrix $(m^2)^j_i$. It then specializes every equation to the Minimal Supersymmetric Standard Model, writing out the running of $h_u, h_d, h_e$, of $B$, and of all seven scalar mass-squared parameters including the off-diagonal family-space matrices. The authors report that the two-loop terms typically shift sparticle masses by several percent relative to one-loop running, which is the size that matters when spectra are used to discriminate between competing proposals for how supersymmetry is broken.

What carries the argument

The engine is a translation trick. Two-loop beta functions for a general renormalizable theory were already known in dimensional regularization, but dimensional regularization breaks supersymmetry by mismatching gauge boson and gaugino degrees of freedom, so supersymmetric results must be quoted in dimensional reduction instead. The authors take the known general results and promote dimensionful couplings to dimensionless ones by adjoining non-propagating dummy scalar fields: a fermion mass becomes a Yukawa coupling to a dummy scalar, a scalar mass-squared becomes a quartic coupling with two dummy scalars, and a trilinear scalar coupling becomes a quartic coupling with one dummy scalar. Careful combinatorics on symmetry factors then extracts the beta function for the dimensionful coupling. Finally the results are converted from modified minimal subtraction to the supersymmetric modified minimal subtraction scheme using a one-loop dictionary supplied in the authors' earlier work, including one-loop mass counterterms for the unphysical epsilon-scalar fields that dimensional reduction introduces. The paper states that this procedure is straightforward but tedious and declines to exhibit the intermediate steps.

Extended reading notes

Core claim

The claim is completeness: the paper presents the full set of two-loop renormalization group equations for the soft supersymmetry-breaking sector of a general softly broken $N=1$ supersymmetric model, closing the last gap left by earlier work on gauge couplings, gaugino masses and superpotential parameters. The new results are Eqs. (2.13)-(2.14) for the trilinear couplings $h^{ijk}$, Eqs. (2.16)-(2.17) for the bilinear $b^{ij}$, and Eqs. (2.19)-(2.20) for the scalar masses $(m^2)^j_i$, each written as a one-loop piece plus a two-loop piece, with no assumption about the complex phases of the parameters. Section III gives the replacement rules needed when the gauge group is a product of simple factors and for the special case of more than one $U(1)$. Section IV then translates everything into the MSSM, listing the two-loop $\beta$ functions for $h_u, h_d, h_e$ (Eqs. 4.18-4.23), for $B$ (4.25-4.26), and for $m^2_{H_u}, m^2_{H_d}, m^2_Q, m^2_L, m^2_u, m^2_d, m^2_e$ (4.30-4.43), the last five being $3$ by $3$ matrices in family space. A closing note records that the coefficient of the final $g^4 C(i)$ term in Eq. (2.20) and the ordering of matrices in some Section IV terms were corrected after comparison with independent calculations, and that the version given here is the corrected one.

Load-bearing premise

Everything rests on the claim that the two-loop dimensional-reduction beta functions can be assembled by applying a dummy-field trick to the known minimal-subtraction results and then converting with a one-loop dictionary, including the one-loop masses given to the unphysical epsilon scalars — a step the authors describe as straightforward but tedious and do not show, and the step where the coefficient in Eq. (2.20) was originally wrong and had to be corrected.

Editorial extensions

If this is right

Sparticle mass spectra computed from high-scale inputs can be quoted at two-loop accuracy in the soft sector, removing a several-percent systematic from the running that was previously uncontrolled.
Comparisons between competing proposals for the soft-breaking terms at the input scale — universal scalar masses, universal trilinears, unified gaugino masses — become sharp at the percent level rather than the ten-percent level.
The general-model equations apply to extensions of the MSSM with extra chiral multiplets or extra gauge factors, so the same machinery can be reused when the low-energy spectrum is not minimal.
The equations preserve the known finiteness of $N=2$ supersymmetric Yang-Mills theory when the soft couplings are constrained appropriately, which the authors offer as a consistency check on the whole set.
Because phases are kept arbitrary throughout, the two-loop running of CP-violating soft parameters — and hence of predicted electric dipole moments — can be followed consistently.

Reading between the lines

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

The value of a paper like this is mostly as a reference implementation: the equations are long enough that independent re-derivations are the only real check, and the note recording a corrected coefficient in Eq. (2.20) suggests that any code built on the original preprint version should be re-validated against the revised equations.
Once two-loop soft running is standard, the dominant uncertainty in sparticle mass predictions shifts elsewhere — to the conversion between running and pole masses and to threshold corrections at the supersymmetry and unification scales — so the marginal return on going to three loops is likely small compared with getting those matching conditions right.
The dummy-field technique used here is generic and could be applied to any softly broken theory whose dimensionless two-loop beta functions are known, which makes the method itself reusable beyond supersymmetry.
A practical test of the whole set would be to run a few benchmark points through two independent spectrum codes that implement these equations and check that the predicted mass differences agree at the sub-percent level.

Original abstract

We compute the two-loop renormalization group equations for all soft supersymmetry-breaking couplings in a general softly broken N=1 supersymmetric model. We also specialize these results to the Minimal Supersymmetric Standard Model.

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, 2,255 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.04 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

We compute the two-loop renormalization group equations for all soft supersymmetry-breaking couplings in a general softly broken N=1 supersymmetric model. We also specialize these results to the Minimal Supersymmetric Standard Model.

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 compute the two-loop renormalization group equations for all soft supersymmetry-breaking couplings in a general softly broken N=1 supersymmetric model.	contribution 0.91	0.96	0.63	asserted 1.00
s1	We also specialize these results to the Minimal Supersymmetric Standard Model.	contribution 0.96	0.93	0.12	asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.97.

What was examined, and what was not

sources read

arXiv listing abstract v4

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

References

Works cited (35)

[20]

S. P. Martin and M. T. Vaughn, Phys. Lett. 318B, 331, (1993)

View on Pith 1993
[34]

Y. Yamada, “Two-loop renormalization group equations f or soft SUSY breaking scalar interactions: supergraph method”, KEK-TH-383, KEK Preprint 93-182, UT- 665, January 1994

View on Pith 1994
[35]

I. Jack and D. R. T. Jones, “Soft supersymmetry breaking a nd ﬁniteness” Liverpool preprint LTH-334, May 1994. 28

View on Pith 1994
[1]

For reviews, see H. P. Nilles, Phys. Rep. 110, 1, (1984) or H. E. Haber and G. L. Kane, Phys. Rep. 117, 75, (1985)

View on Pith 1984
[2]

L. Girardello and M. T. Grisaru, Nucl. Phys. B194, 65, (1982)

View on Pith 1982
[3]

P. Langacker, in Proceedings of the PASCOS90 Symposium, E ds. P. Nath and S. Reu- croft, (World Scientiﬁc, Singapore 1990) J. Ellis, S. Kelle y, and D. Nanopoulos, Phys. Lett. 260B, 131, (1991); U. Amaldi, W. de Boer, and H. Furstenau, Phys. Lett. 260B, 447, (1991); P. Langacker and M. Luo, Phys. Rev. D44, 817, (1991)

View on Pith 1991
[4]

A. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49, 970, (1982); H. P. Nilles, Phys. Lett. 115B, 193, (1982); L. E. Ib´ a˜ nez, Phys. Lett. 118B, 73, (1982); R. Barbieri, S. Ferrara, and C. Savoy, Phys. Lett. 119B, 343, (1982); L. Hall, J. Lykken and S. Weinberg, Phys. Rev. D27, 2359, (1983); P. Nath, R. Arnowitt and A. H. Chamseddine, Nucl. Phy...

View on Pith 1982
[5]

A. Lahanas and D. Nanopoulos, Phys. Rep. 145, 1, (1987)

View on Pith 1987
[6]

G. G. Ross and R. G. Roberts, Nucl. Phys. B377, 571, (1992)

View on Pith 1992
[7]

L. E. Ib´ a˜ nez and G. G. Ross, “Electroweak Breaking in Sup ersymmetric Models”, CERN-TH.6412/92, in Perspectives in Higgs Physics , edited by G. Kane (World Scientiﬁc, Singapore, 1993)

View on Pith 1993
[8]

R. Arnowitt and P. Nath, SSCL-Preprint-229 (1993); Phys. Rev. Lett. 69, 725, (1992); Phys. Rev. D46, 3981, (1992); P. Nath and R. Arnowitt, Phys. Lett. B289, 368, (1992); ibid, B287 (1992) 89; Phys. Rev. Lett. 70, 3696, (1993)

View on Pith 1993
[9]

S. Kelley, J. Lopez, D. Nanopoulos, H. Pois, and K. Yuan, Phys. Lett. B273, 423, (1991); Nucl. Phys. B398, 3, (1993); J. Lopez, D. Nanopoulos, and A. Zichichi, Phys. 25 Lett. , B291, 255, (1992); J. Lopez, D. Nanopoulos, and H.Pois, Phys. Rev. D47, 2468, (1993)

View on Pith 1991
[10]

M. Drees and M. Nojiri, Phys. Rev. D47, 376, (1993)

View on Pith 1993
[11]

T. Elliott, S. F. King, and P. L. White, Phys. Lett. B305, 71, (1993); Phys. Lett. B314, 56, (1993); “Radiative Corrections to Higgs Boson Masses i n the Next to Minimal Supersymmetric Standard Model”, SHEP-92/93-21 , (hep-ph 9308309)

View on Pith 1993
[12]

R. G. Roberts and L. Roszkowski, Phys. Lett. B309, 329, (1993)

View on Pith 1993
[13]

S. P. Martin and P. Ramond, Phys. Rev. D48, 5365, (1993)

View on Pith 1993
[14]

P. Ramond, “Renormalization Group Study of the Minimal S upersymmetric Stan- dard Model: No Scale Models”, invited talk at workshop “Rece nt Advances in the Superworld”, Houston TX, April 1993; D. J. Casta˜ no, E. J. Pi ard, and P. Ramond, “Renormalization group study of the standard model and its e xtensions: II. The minimal supersymmetric standard model...

View on Pith 1993
[15]

W. de Boer, R. Ehret, and D. I. Kazakov, “Constraints on SU SY Masses in Sup- ersymmetric Grand Uniﬁed Theories”, IEKP-KA-93-13, (hep- ph 9308238), August 1993

View on Pith 1993
[16]

V. Barger, M. S. Berger, and P. Ohmann, Phys. Rev. D 47, 1093, (1993); “The Super- symmetric Particle Spectrum”, Wisconsin preprint MAD-PH- 801, (hep-ph 9311269), November 1993

View on Pith 1993
[17]

G. L. Kane, C. Kolda, L. Roszkowski, and J. D. Wells, “Stud y of Constrained Minimal Supersymmetry”, Michigan preprint UM-TH-93-24

View on Pith
[18]

D. R. T. Jones, Nucl. Phys. B87, 127, (1975); D. R. T. Jones and L. Mezincescu, 26 Phys. Lett. 136B, 242, (1984)

View on Pith 1975
[19]

P. West, Phys. Lett. 137B, 371, (1984); D. R. T. Jones and L. Mezincescu, Phys. Lett. 138B, 293, (1984)

View on Pith 1984
[21]

Y. Yamada, Phys. Rev. Lett. 72, 25, (1994)

View on Pith 1994
[22]

M. E. Machacek and M. T. Vaughn, Nucl. Phys. B222, 83, (1983)

View on Pith 1983
[23]

M. E. Machacek and M. T. Vaughn, Nucl. Phys. B236, 221, (1984)

View on Pith 1984
[24]

M. E. Machacek and M. T. Vaughn, Nucl. Phys. B249, 70, (1984)

View on Pith 1984
[25]

G. ’t Hooft and M. Veltman, Nucl. Phys. B44, 189, (1972)

View on Pith 1972
[26]

W. Siegel, Phys. Lett. 84B, 193, (1979); D. M. Capper, D. R. T. Jones and P. van Ni- euwenhuizen, Nucl. Phys. B167, 479, (1980)

View on Pith 1979
[27]

W. A. Bardeen, A. J. Buras, D. W. Duke and T. W. Muta, Phys. Rev. D18, 3998, (1978)

View on Pith 1978
[28]

I. Jack, D. R. T. Jones and K. L. Roberts, Z. Phys. C62, 161, (1994); “Equivalence of Dimensional Reduction and Dimensional Regularisation” , Liverpool preprint LTH- 329 1994

View on Pith 1994
[29]

I. Jack, Phys. Lett. 147B, 405, (1984) and I. Jack and H. Osborn, Nucl. Phys. B249, 472, (1985)

View on Pith 1984
[30]

J. Wess and B. Zumino, Phys. Lett. 49B, 52, (1974); J. Iliopoulos and B. Zu- mino, Nucl. Phys. B76, 310, (1974); S. Ferrara, J. Iliopoulos and B. Zumino, Nucl. Phys. B77, 413, (1974); B. Zumino, Nucl. Phys. B89, 535, (1975); S. Ferrara and 27 O. Piguet, Nucl. Phys. B93, 261, (1975); M. Grisaru, W. Siegel, and M. Rocek, Nucl. Phys. B159, 429, (1979)

View on Pith 1974
[31]

P. S. Howe, K. S. Stelle, and P. West, Phys. Lett. 124B, 55, (1983)

View on Pith 1983
[32]

A. Parkes and P. West, Phys. Lett. 127B, 353, (1983); J.-M. Fr` ere, L. Mezincescu, and Y.-P. Yao, Phys. Rev. D29, 1196, (1984); Phys. Rev. D30, 2238, (1984)

View on Pith 1983
[33]

See for instance, “JLC-I”, KEK report 92-16, the JLC grou p, December 1992, and references contained therein

View on Pith 1992

Forward citations

Cited by 38 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Renormalization of general Effective Field Theories: Formalism and renormalization of bosonic operators
hep-ph 2025-01 conditional novelty 7.0 of 10

The authors compute, for the first time, the one-loop renormalization group equations of the bosonic operators of a completely general EFT up to mass dimension 6.
The Higgs Mass Bound in Gauge Extensions of the Minimal Supersymmetric Standard Model
hep-ph 2003-09 conditional novelty 7.0 of 10

An extra asymptotically free SU(2) gauge group, broken at a few TeV with a non-decoupling D-term, lifts the MSSM Higgs mass to 200 to 350 GeV while keeping the weak scale natural.
The first-order phase transitions and gravitational waves from an effective theory inspired by four-stack intersecting D6-brane models
hep-ph 2026-11 accept novelty 6.0 of 10

A four-stack intersecting D6-brane Pati-Salam model can host a strongly supercooled first-order phase transition at the TeV scale, generating a stochastic gravitational-wave background peaked at 0.611 Hz with amplitud...
Entanglement Bounds on Cardy Growth in Higher Dimensions
hep-th 2026-10 conditional novelty 6.0 of 10

Thermal entropy in unitary conformal field theories is conjectured to obey f≤2.9983...F in three dimensions and f≤4π²a in four dimensions, with both bounds saturated by a free real scalar.
Full two-loop QCD corrections to the Higgs mass in the MSSM with heavy superpartners
hep-ph 2019-08 accept novelty 6.0 of 10

The two-loop QCD threshold corrections that mix strong and electroweak couplings in the MSSM Higgs-mass EFT calculation are computed for the first time, completing the two-loop QCD matching at the SUSY scale.

Show 33 more citing papers

Precise prediction for the Higgs-Boson Masses in the $\mu\nu$SSM
hep-ph 2017-12 conditional novelty 6.0 of 10

A complete one-loop renormalization of the mu-nu SSM neutral scalar sector shows its SM-like Higgs mass prediction agrees with the NMSSM to within a few keV, and the model can host a 96 GeV scalar matching LEP and CMS...
FlexibleSUSY 2.0: Extensions to investigate the phenomenology of SUSY and non-SUSY models
hep-ph 2017-10 accept novelty 6.0 of 10

The paper documents FlexibleSUSY 2.0, which extends an automated spectrum generator to non-supersymmetric models, complex parameters, W-mass prediction from muon decay, muon g-2, a semi-analytic boundary value solver,...
Precision tools and models to narrow in on the 750 GeV diphoton resonance
hep-ph 2016-02 accept novelty 6.0 of 10

Forty SARAH model files plus automated NLO and N3LO diphoton and digluon decay rates let any weakly coupled 750 GeV explanation be checked with the same precision machinery.
Lepton flavor violation in low-scale seesaw models: SUSY and non-SUSY contributions
hep-ph 2014-08 conditional novelty 6.0 of 10

Adding all supersymmetric and non-supersymmetric loop contributions at once, rather than the dominant one alone, can reduce predicted lepton-flavor-violating rates by up to an order of magnitude in the supersymmetric ...
Renormalization of vacuum expectation values in spontaneously broken gauge theories
hep-ph 2013-05 conditional novelty 6.0 of 10

The VEV counterterm in R_xi gauge equals the scalar field renormalization plus the anomalous dimension of a background field, yielding explicit one-loop and Yukawa-enhanced two-loop VEV beta functions for generic and ...
A tool box for implementing supersymmetric models
hep-ph 2011-09 accept novelty 6.0 of 10

A single model definition in SARAH is automatically translated into working SPheno, CalcHep, micrOMEGAs, WHIZARD and HiggsBounds inputs, and the generated MSSM spectrum agrees with stock SPheno at the 10^-4 level.
Running soft parameters in SUSY models with multiple U(1) gauge factors
hep-ph 2011-07 conditional novelty 6.0 of 10

The authors extend the standard two-loop RGEs for softly broken SUSY to gauge groups with several abelian factors, giving explicit matrix-valued substitution rules for gauge couplings and gaugino masses.
Light MSSM Higgs boson mass to three-loop accuracy
hep-ph 2010-05 conditional novelty 6.0 of 10

Three-loop supersymmetric QCD corrections shift the predicted light MSSM Higgs mass by a few hundred MeV, opposite in sign to the two-loop term, leaving a residual theory uncertainty estimated below about 1 GeV.
Neutrino-induced Electroweak Symmetry Breaking in Supersymmetric SO(10) Unification
hep-ph 2006-04 conditional novelty 6.0 of 10

Neutrino couplings and large right-handed sneutrino masses enable radiative electroweak symmetry breaking in SO(10) models, reconciling Yukawa unification with the measured bottom quark mass and b to s gamma rate.
Split Supersymmetry
hep-ph 2004-06 accept novelty 6.0 of 10

Among all weak-scale particle additions that unify the Standard Model gauge couplings, the only real-fermion option with a dark matter candidate is Split Supersymmetry, whose higgsino or W-ino dark matter is fixed to ...
Multi-TeV Scalars are Natural in Minimal Supergravity
hep-ph 1999-08 accept novelty 6.0 of 10

Renormalization group focusing of the up-type Higgs soft mass makes multi-TeV squarks and sleptons natural in minimal supergravity when the top Yukawa is held fixed at its measured value.
Living dangerously with decoupled first/second generation scalars: SUSY prospects at the LHC
hep-ph 2024-11 conditional novelty 5.0 of 10

In the NUHM3 SUSY model, a landscape draw to heavy first and second generation scalars forces the surviving natural parameter space to have light top squarks but heavy gluinos, with low-mass regions excluded by charge...
The Last Vestiges of Naturalness
hep-ph 2013-09 conditional novelty 5.0 of 10

Split families, baryonic R-parity violation, and Dirac gauginos each relax LHC gluino bounds only enough to bring fine-tuning to a few percent, not to remove it.
Vacuum Stability and Higgs Diphoton Decays in the MSSM
hep-ph 2012-11 conditional novelty 5.0 of 10

Including a radiatively corrected tau Yukawa and the residual tan-beta dependence relaxes the MSSM stau-mixing metastability bound enough to permit roughly a 50 percent enhancement of the Higgs diphoton partial width.
SARAH 3.2: Dirac Gauginos, UFO output, and more
hep-ph 2012-07 conditional novelty 5.0 of 10

SARAH 3.2 adds UFO model-file output for MadGraph 5, support for Dirac gaugino mass terms, and automated consistency checks, with MSSM cross sections validated against MadGraph's own model files to within numerical error.
The Higgs Sector and Fine-Tuning in the pMSSM
hep-ph 2012-06 conditional novelty 5.0 of 10

In two pMSSM scan sets, roughly 19 percent (neutralino LSP) and 9 percent (gravitino LSP) of models give a 125 GeV Higgs, but only 13 neutralino-LSP models combine that mass with fine-tuning below 100, and those featu...
Light Stau Phenomenology and the Higgs \gamma\gamma Rate
hep-ph 2012-05 conditional novelty 5.0 of 10

Light staus with large left-right mixing can enhance the 125 GeV Higgs diphoton rate by up to about 50 percent while leaving weak boson rates near Standard Model values, and stau-sneutrino associated production is the...
Anatomy of maximal stop mixing in the MSSM
hep-ph 2012-04 accept novelty 5.0 of 10

Maximal stop mixing in the MSSM, needed for a 125 GeV Higgs with sub-TeV stops, requires a large negative trilinear coupling at the GUT scale, which gaugino mediation can supply but gauge and radion mediation cannot.
Les Houches 2011: Physics at TeV Colliders New Physics Working Group Report
hep-ph 2012-03 conditional novelty 5.0 of 10

The report proposes a Dark Matter Les Houches Agreement file format, extends the Flavour and SUSY accords to dipole moments and see-saw models, and reports new tool modules for automated RGE derivation and model validation.
Higgs Mass and Muon Anomalous Magnetic Moment in Supersymmetric Models with Vector-Like Matters
hep-ph 2011-08 conditional novelty 5.0 of 10

In supersymmetric models extended by a TeV-scale vector-like 10 + 10bar pair, the muon g-2 anomaly and a Higgs mass up to about 135 GeV (GMSB) or 130 GeV (mSUGRA) can be explained together at large tan beta.
Calculating the renormalisation group equations of a SUSY model with Susyno
hep-ph 2011-06 conditional novelty 5.0 of 10

Susyno automates the derivation of two-loop renormalisation group equations for supersymmetric models built on any gauge group with at most one U(1) factor and any field content.
Automatic Calculation of supersymmetric Renormalization Group Equations and Self Energies
hep-ph 2010-02 conditional novelty 5.0 of 10

A Mathematica package that derives one- and two-loop renormalization group equations, one-loop self-energies and Feynman rules automatically from a supersymmetric model's gauge sector, particle content and superpotential.
Anatomy and Phenomenology of FCNC and CPV Effects in SUSY Theories
hep-ph 2009-09 accept novelty 5.0 of 10

Correlation patterns among B, K, D and lepton observables separate supersymmetric flavour models with right-handed currents from those with left-handed currents, and both from Little-Higgs and Randall-Sundrum alternatives.
Abelian Hidden Sectors at a GeV
hep-ph 2009-04 conditional novelty 5.0 of 10

Suppressed communication of supersymmetry breaking to a hidden U(1) sector naturally lands its mass scale near a GeV, and the resulting models can be viable dark matter sectors with collider signatures.
SOFTSUSY: a program for calculating supersymmetric spectra
hep-ph 2001-04 accept novelty 5.0 of 10

SOFTSUSY 3.3.3 is a documented, freely available C++ code that solves the MSSM renormalisation group equations iteratively and returns a full superparticle spectrum, with its approximations and failure flags stated ex...
The unification in an $\widehat {\mathfrak{s}\mathfrak{u}}(8)_{ k_U = 1}$ affine Lie algebra
hep-ph 2024-11 conditional novelty 4.0 of 10

An N=1 supersymmetric extension of an SU(8) flavor-unified model is shown to unify the three gauge couplings through a level-1 affine Lie algebra conformal embedding.
Running quark and lepton parameters at various scales
hep-ph 2013-06 accept novelty 4.0 of 10

The authors give updated values and one-sigma ranges for the running Yukawa couplings and CKM parameters at five energy scales up to grand unification, with supersymmetric threshold corrections parametrized by two variables.
SPheno 3.1: extensions including flavour, CP-phases and models beyond the MSSM
hep-ph 2011-04 conditional novelty 4.0 of 10

SPheno 3.1 adds flavour mixing, CP phases, R-parity violation, seesaw models, and low-energy observables to the SPheno SUSY spectrum calculator, documented through extended SLHA input and output blocks.
Exploring new models in all detail with SARAH
hep-ph 2015-03 conditional novelty 3.0 of 10

SARAH derives, from a short model definition, the full Lagrangian-level data of a supersymmetric model and exports it as a working spectrum generator and as Monte Carlo model files, demonstrated end to end on the B-L-SSM.
The Next-to-Minimal Supersymmetric Standard Model
hep-ph 2009-10 accept novelty 3.0 of 10

A comprehensive review of the NMSSM that consolidates its Higgs and neutralino sectors, radiative corrections, collider and low-energy phenomenology, dark matter, and variant scenarios into a single reference with exp...
Proton stability in grand unified theories, in strings, and in branes
hep-ph 2006-01 accept novelty 3.0 of 10

A comprehensive review concludes that proton decay remains a key test of unification and that planned megaton-scale detectors can probe many GUT, extra-dimensional, and string scenarios.
A Supersymmetry Primer
hep-ph 1997-09 accept novelty 1.0 of 10

A self-contained pedagogical introduction to supersymmetry and the MSSM, from Lagrangian construction through soft breaking to collider phenomenology.
Calculating $\beta$-function coefficients of Renormalization Group Equations
hep-ph 2019-07 unverdicted

A pedagogical review that reproduces step-by-step calculations of beta-function coefficients for RGEs from known formulae in the literature.

How later papers used it

183 later papers cite it

Evidence: substantial. 182 of the citing papers share no author with it, and 87 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 323 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

10 later papers by other authors are listed below: 9 used its method or tool and 1 compared its own results with it.

Uses its method or tool 9 of 81 papers with this label

Needed this paper's two-loop MSSM RGEs to run parameters down from M_X^0 in a fit at M_Z using NMSGUT superpotential parameters and Sugry-NUHM soft supersymmetry-breaking parameters.

New minimal supersymmetric SO(10) GUT phenomenology and its cosmological implications (2015) · 0.97 · paraphrased: the licence does not allow quoting
Used this paper's two-loop renormalization group evolution equations for the softly broken MSSM to evolve randomly chosen SUGRY-NUHM parameters together with the mu parameter and the rest of the superpotential parameters down from M_X^0 to M_Z.

NMSGUT-III: Grand Unification upended (2011) · 1.00 · paraphrased: the licence does not allow quoting
Used this paper's RGE expressions for a general supersymmetric model to obtain the 1-loop RGE for the Yukawa parameters and its own 2-loop RGE for the gauge coupling.

Higher-dimensional Higgs Representations in SGUT models (2010) · 0.93 · paraphrased: the licence does not allow quoting
Implemented this paper's full 2-loop RG equations for the soft SUSY-breaking parameters and the gauge and Yukawa couplings into Mathematica for its scan.

Determining the Structure of Supersymmetry-Breaking with Renormalization Group Invariants (2010) · 1.00 · paraphrased: the licence does not allow quoting
Adopted the Isajet code for the Higgs and superparticle mass spectrum based on 2-loop RGE running from this paper.

Landscape Higgs and sparticle mass predictions from a logarithmic soft term distribution (2020) · 1.00 · paraphrased: the licence does not allow quoting
Adapted this paper's RGEs, up to two-loop order, to its own notation.

Refining the predictions of supersymmetric CP-violating models: A top-down approach (2008) · 0.93 · paraphrased: the licence does not allow quoting
Used this paper's MSSM DR RG equations above the scales of the heavy sparticles.

Decoupling heavy sparticles in Effective SUSY scenarios: Unification, Higgs masses and tachyon bounds (2012) · 1.00 · paraphrased: the licence does not allow quoting
Implemented the complete flavour structures including CP-phases in the RGEs at the 2-loop level using the formulas given in this paper.

SPheno 3.1: extensions including flavour, CP-phases and models beyond the MSSM (2011) · 0.95 · paraphrased: the licence does not allow quoting
Took into account this paper's renormalization group effect on the average mass scale in m_e_L^2, m_e_R^2, m_q_L^2, and m_d_R^2 while neglecting Yukawa couplings.

Like-sign dimuon asymmetry of B0 meson and LFV in SU(5) SUSY GUT with S4 flavor symmetry (2011) · 0.94 · paraphrased: the licence does not allow quoting

Compares with it 1 of 4 papers with this label

Compared its own analytical results for the one- and two-loop RGEs for the MSSM with this paper.

Automatic Calculation of supersymmetric Renormalization Group Equations and Self Energies (2010) · 1.00 · paraphrased: the licence does not allow quoting

How the literature uses it

172 arXiv papers cite this in 312 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 · 171 sentences without a confident label

Builds on it 2 papers

In this section, we shall describe the generic method of constructing the fully general two-loop RGEs for softly-broken supersymmetric gauge theories out of the results of [ref] relevant to the case of at most a single abelian gauge-group factor.

Running soft parameters in SUSY models with multiple U(1) gauge factors · Results · 0.98
Finally, for the scalar mass squared terms, the generalization of the RGE result calculated in [ref] , adjusted to take $U(1)$ mixing into account, reads: [equation] where in the last term the index $β$ runs over all $U(1)$ factors.

A Simplified Scheme for GUT-inspired Theories with Multiple Abelian Factors · Section · 1.00

Uses its method or tool 78 papers

Susyno uses the generic two loop RGEs of [ref] .

Calculating the renormalisation group equations of a SUSY model with Susyno · Renormalisation group equations of a general model · 0.92
$M_Z$ scale fitting: The best fit found at $M_X$ scale along with five SUGRY-NUHM (Supergravity-Non universal Higgs mass) type soft Susy breaking parameters ($m_0$ (universal scalar mass), $m_1/2$ (universal gaugino mass), $A_0$ (universal trilinear coupling), $m_H^2,H̅^2$) is run down to $M_Z$ scale using two loop MSSM RGEs [ref] .

New minimal supersymmetric SO(10) GUT phenomenology and its cosmological implications · SM fermion fitting criteria in NMSGUT and its features · 1.00
We use the 2-loop RG equations for the SM [ref] and MSSM [ref] to extrapolate the central values of fermion yukawa data at the top mass mass scale (including neutrino mass splitting and mixing data), which were recently recalculated and summarized in [ref] , past the MSSM threshold (taken to be $1 TeV$ and defined by expressing all soft masses in units of the gluino mass fixed at $1 TeV$ ) up to the MSSM 1 loop gauge coupling unification scale $M_U=10^16.3 GeV$ while ignoring, for the time being, complications such as righthanded neutrino thresholds.

The New Minimal Supersymmetric GUT : Spectra, RG analysis and Fermion Fits · Numerical search and anatomy of specimens found · 0.99

Uses its data or values 3 papers

In this appendix, the two-loop RGEs for a generic softly broken SUSY model are reproduced from [ref] .

Renormalization in supersymmetric models · Running time · 0.88
With $Z_i(M,μ) = C(M) g(μ)^-2(N^2-1)/(b_0 N)$,[This has been obtained from $γ_i ≡(μd/dμ) logZ_i (M,μ) = g_c^28π^2 N^2-1N + 𝒪(g^4)$ taken from [ref] .] where $M$ is the cutoff and $C(M)$ a cutoff dependent constant, we obtain

On the Dynamical Origin of the $\eta'$ Potential and the Axion Mass · $N$-dependence of the holomorphic scale · 0.89
These are taken from [ref] ; here $t=log(Q/hμ̂)$, $Q$ being the renormalization scale: [equation]

A Hybrid Higgs · One loop MSSM $β$-functions · 0.89

Compares with it 4 papers

*Verification of the calculated RGEs We have compared the analytical results for the one- and two-loop RGEs calculated by for the MSSM with [ref] and for the NMSSM with [ref] .

Automatic Calculation of supersymmetric Renormalization Group Equations and Self Energies · Verification · 1.00
In particular, the RGEs generated by were compared with the results for the MSSM [ref] , the R-parity violating MSSM [ref] , the general NMSSM [ref] and some SU(5)-based models [ref] .

Calculating the renormalisation group equations of a SUSY model with Susyno · Tests/validation of · 1.00
In this appendix we sketch the calculation of the one–loop RGE in the ET [We have checked that our results are consistent with the MSSM [ref] and Split Supersymmetry [ref] limits.].

The Effective Theory of the Light Stop Scenario · Renormalization group equations · 0.97

Cites it as support 1 paper

However, this is not radiatively stable as $M_1$ contributes to the renormalization of $M_2$ at two-loop [ref] .], without motivating a hierarchy between the $κ_i$ and $μ$.

R-Parity Conservation from a Top Down Perspective · Bilinear R-parity Violation $&$ Neutrino Mass constraints · 0.97

Cites it as background 13 papers

These equations are extensively discussed in literature, see e.g. [ref] .

Automatic Calculation of supersymmetric Renormalization Group Equations and Self Energies · Renormalization group equations · 0.96
Renormalization group equations for general supersymmetric models are known up to 2-loop order [ref] .

Soft masses in SUSY SO(10) GUTs with low intermediate scales · Leading-log RGE invariants · 0.89
The 2-loop RGEs for the MSSM (effective theory of Susy-GUTs) and the soft Susy breaking parameters are well known [ref] , and are useful from low scale to GUT scale.

Study of baryon number and lepton flavour violation in the new minimal supersymmetric SO(10)GUT · Introduction · 0.87

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [20]

S. P. Martin and M. T. Vaughn, Phys. Lett. 318B, 331, (1993)

View on Pith 1993

[2] [34]

Y. Yamada, “Two-loop renormalization group equations f or soft SUSY breaking scalar interactions: supergraph method”, KEK-TH-383, KEK Preprint 93-182, UT- 665, January 1994

View on Pith 1994

[3] [35]

I. Jack and D. R. T. Jones, “Soft supersymmetry breaking a nd ﬁniteness” Liverpool preprint LTH-334, May 1994. 28

View on Pith 1994

[4] [1]

For reviews, see H. P. Nilles, Phys. Rep. 110, 1, (1984) or H. E. Haber and G. L. Kane, Phys. Rep. 117, 75, (1985)

View on Pith 1984

[5] [2]

L. Girardello and M. T. Grisaru, Nucl. Phys. B194, 65, (1982)

View on Pith 1982

[6] [3]

P. Langacker, in Proceedings of the PASCOS90 Symposium, E ds. P. Nath and S. Reu- croft, (World Scientiﬁc, Singapore 1990) J. Ellis, S. Kelle y, and D. Nanopoulos, Phys. Lett. 260B, 131, (1991); U. Amaldi, W. de Boer, and H. Furstenau, Phys. Lett. 260B, 447, (1991); P. Langacker and M. Luo, Phys. Rev. D44, 817, (1991)

View on Pith 1991

[7] [4]

A. Chamseddine, R. Arnowitt and P. Nath, Phys. Rev. Lett. 49, 970, (1982); H. P. Nilles, Phys. Lett. 115B, 193, (1982); L. E. Ib´ a˜ nez, Phys. Lett. 118B, 73, (1982); R. Barbieri, S. Ferrara, and C. Savoy, Phys. Lett. 119B, 343, (1982); L. Hall, J. Lykken and S. Weinberg, Phys. Rev. D27, 2359, (1983); P. Nath, R. Arnowitt and A. H. Chamseddine, Nucl. Phy...

View on Pith 1982

[8] [5]

A. Lahanas and D. Nanopoulos, Phys. Rep. 145, 1, (1987)

View on Pith 1987

[9] [6]

G. G. Ross and R. G. Roberts, Nucl. Phys. B377, 571, (1992)

View on Pith 1992

[10] [7]

L. E. Ib´ a˜ nez and G. G. Ross, “Electroweak Breaking in Sup ersymmetric Models”, CERN-TH.6412/92, in Perspectives in Higgs Physics , edited by G. Kane (World Scientiﬁc, Singapore, 1993)

View on Pith 1993

[11] [8]

R. Arnowitt and P. Nath, SSCL-Preprint-229 (1993); Phys. Rev. Lett. 69, 725, (1992); Phys. Rev. D46, 3981, (1992); P. Nath and R. Arnowitt, Phys. Lett. B289, 368, (1992); ibid, B287 (1992) 89; Phys. Rev. Lett. 70, 3696, (1993)

View on Pith 1993

[12] [9]

S. Kelley, J. Lopez, D. Nanopoulos, H. Pois, and K. Yuan, Phys. Lett. B273, 423, (1991); Nucl. Phys. B398, 3, (1993); J. Lopez, D. Nanopoulos, and A. Zichichi, Phys. 25 Lett. , B291, 255, (1992); J. Lopez, D. Nanopoulos, and H.Pois, Phys. Rev. D47, 2468, (1993)

View on Pith 1991

[13] [10]

M. Drees and M. Nojiri, Phys. Rev. D47, 376, (1993)

View on Pith 1993

[14] [11]

T. Elliott, S. F. King, and P. L. White, Phys. Lett. B305, 71, (1993); Phys. Lett. B314, 56, (1993); “Radiative Corrections to Higgs Boson Masses i n the Next to Minimal Supersymmetric Standard Model”, SHEP-92/93-21 , (hep-ph 9308309)

View on Pith 1993

[15] [12]

R. G. Roberts and L. Roszkowski, Phys. Lett. B309, 329, (1993)

View on Pith 1993

[16] [13]

S. P. Martin and P. Ramond, Phys. Rev. D48, 5365, (1993)

View on Pith 1993

[17] [14]

P. Ramond, “Renormalization Group Study of the Minimal S upersymmetric Stan- dard Model: No Scale Models”, invited talk at workshop “Rece nt Advances in the Superworld”, Houston TX, April 1993; D. J. Casta˜ no, E. J. Pi ard, and P. Ramond, “Renormalization group study of the standard model and its e xtensions: II. The minimal supersymmetric standard model...

View on Pith 1993

[18] [15]

W. de Boer, R. Ehret, and D. I. Kazakov, “Constraints on SU SY Masses in Sup- ersymmetric Grand Uniﬁed Theories”, IEKP-KA-93-13, (hep- ph 9308238), August 1993

View on Pith 1993

[19] [16]

V. Barger, M. S. Berger, and P. Ohmann, Phys. Rev. D 47, 1093, (1993); “The Super- symmetric Particle Spectrum”, Wisconsin preprint MAD-PH- 801, (hep-ph 9311269), November 1993

View on Pith 1993

[20] [17]

G. L. Kane, C. Kolda, L. Roszkowski, and J. D. Wells, “Stud y of Constrained Minimal Supersymmetry”, Michigan preprint UM-TH-93-24

View on Pith

[21] [18]

D. R. T. Jones, Nucl. Phys. B87, 127, (1975); D. R. T. Jones and L. Mezincescu, 26 Phys. Lett. 136B, 242, (1984)

View on Pith 1975

[22] [19]

P. West, Phys. Lett. 137B, 371, (1984); D. R. T. Jones and L. Mezincescu, Phys. Lett. 138B, 293, (1984)

View on Pith 1984

[23] [21]

Y. Yamada, Phys. Rev. Lett. 72, 25, (1994)

View on Pith 1994

[24] [22]

M. E. Machacek and M. T. Vaughn, Nucl. Phys. B222, 83, (1983)

View on Pith 1983

[25] [23]

M. E. Machacek and M. T. Vaughn, Nucl. Phys. B236, 221, (1984)

View on Pith 1984

[26] [24]

M. E. Machacek and M. T. Vaughn, Nucl. Phys. B249, 70, (1984)

View on Pith 1984

[27] [25]

G. ’t Hooft and M. Veltman, Nucl. Phys. B44, 189, (1972)

View on Pith 1972

[28] [26]

W. Siegel, Phys. Lett. 84B, 193, (1979); D. M. Capper, D. R. T. Jones and P. van Ni- euwenhuizen, Nucl. Phys. B167, 479, (1980)

View on Pith 1979

[29] [27]

W. A. Bardeen, A. J. Buras, D. W. Duke and T. W. Muta, Phys. Rev. D18, 3998, (1978)

View on Pith 1978

[30] [28]

I. Jack, D. R. T. Jones and K. L. Roberts, Z. Phys. C62, 161, (1994); “Equivalence of Dimensional Reduction and Dimensional Regularisation” , Liverpool preprint LTH- 329 1994

View on Pith 1994

[31] [29]

I. Jack, Phys. Lett. 147B, 405, (1984) and I. Jack and H. Osborn, Nucl. Phys. B249, 472, (1985)

View on Pith 1984

[32] [30]

J. Wess and B. Zumino, Phys. Lett. 49B, 52, (1974); J. Iliopoulos and B. Zu- mino, Nucl. Phys. B76, 310, (1974); S. Ferrara, J. Iliopoulos and B. Zumino, Nucl. Phys. B77, 413, (1974); B. Zumino, Nucl. Phys. B89, 535, (1975); S. Ferrara and 27 O. Piguet, Nucl. Phys. B93, 261, (1975); M. Grisaru, W. Siegel, and M. Rocek, Nucl. Phys. B159, 429, (1979)

View on Pith 1974

[33] [31]

P. S. Howe, K. S. Stelle, and P. West, Phys. Lett. 124B, 55, (1983)

View on Pith 1983

[34] [32]

A. Parkes and P. West, Phys. Lett. 127B, 353, (1983); J.-M. Fr` ere, L. Mezincescu, and Y.-P. Yao, Phys. Rev. D29, 1196, (1984); Phys. Rev. D30, 2238, (1984)

View on Pith 1983

[35] [33]

See for instance, “JLC-I”, KEK report 92-16, the JLC grou p, December 1992, and references contained therein

View on Pith 1992