Complete On-Shell Renormalization Scheme for the Minimal Supersymmetric Higgs Sector

J.Rosiek; P.Chankowski; S.Pokorski

REVIEW

Complete On-Shell Renormalization Scheme for the Minimal Supersymmetric Higgs Sector

P.Chankowski, S.Pokorski, J.Rosiek

High Energy Physics - Phenomenology · Submitted 26 Mar 1993 · 19 references

Pith's verdict: A solid, honest technical reference for MSSM Higgs on-shell renormalization; the cross-section section is a Higgs-sector calculation and says so, so the missing electroweak terms are a precision caveat rather than a hidden flaw.

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CitationJ.Rosiek, P.Chankowski, S.Pokorski. "Complete On-Shell Renormalization Scheme for the Minimal Supersymmetric Higgs Sector." arXiv:hep-ph/9303309 (1993). https://pith.science/paper/E7KOXIE7

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This is the full-details paper behind the authors' earlier letters [5,6]: a complete one-loop on-shell renormalization of the MSSM Higgs and gauge sectors, with all 2- and 3-point Green's functions written out and the Higgs masses defined as exact poles of the 2x2 scalar propagator (Eq. 3.26). That last point is the real substance. They show numerically that the naive perturbative mass formula differs significantly from the exact poles, especially at small tan beta and at large tan beta near M_A ~ M_Z. That is a useful, checkable claim and it is the reason to read the paper rather than the letters. The scheme itself is carefully laid out: explicit renormalization conditions, counterterms, and divergence cancellation used as an internal check. The discussion of tan beta scheme dependence in the introduction is more candid than most of the literature and worth reading on its own. The appendix algebra is long and I did not verify it line by line; the authors say the FORTRAN is available on request, which is not the same as shipped code, so independent reproduction means real reimplementation work. That is a genuine cost, not a flaw in the physics. On the stress-test concern: Section 4 states plainly that corrections are included only in the Higgs sector, with the electron self-energy, the Z e+e- vertex, and box diagrams dropped and m_e set to zero. The authors acknowledge this. It is a real precision limitation for the quoted rates and the boundaries in Figs. 4-5, and a numerical estimate of the dropped terms would have strengthened the phenomenology. But it is an acknowledged approximation in a paper whose stated contribution is the renormalization scheme, not a complete electroweak prediction, so I would not weight it as load-bearing. The approximate mass formula (3.27-3.30) is quoted at 1-2 GeV accuracy with no derivation shown; minor, but worth a footnote. Who should read it: anyone doing precision MSSM Higgs phenomenology, or comparing EPA/RGE results against a diagrammatic reference. It is a reference paper, not a discussion piece. Recommendation: send it to a serious referee. The scheme is well-defined and internally checked, and the exact-pole treatment is a defensible improvement that deserves scrutiny rather than desk rejection.

Referee report

Significance

If the scheme is correct, this is a reference calculation: it supplies a diagrammatic, gauge-sector-complete alternative to the effective-potential and renormalization-group treatments, and it makes the scheme dependence of tan beta explicit rather than implicit. Several features deserve credit. The renormalization conditions are stated one by one and are reproducible. The choice to define the Higgs masses as exact poles of the 2x2 propagator, Eq. (3.26), is a substantive improvement over the perturbative formula (3.25), and the paper reports that the two differ significantly, which is a falsifiable statement others can check. Divergence cancellation between counterterms and bare two-point functions, and the finiteness of delta xi_{W,Z}^1 in (3.9)-(3.10), are stated as internal checks. The paper also reports that a large part of the formulae was independently checked by A. Dabelstein, and it offers the corresponding FORTRAN library on request, which is a real reproducibility asset for its era. The phenomenological statements in Section 5 are falsifiable predictions for e+e- colliders and are tied to explicit parameter sets in the figure captions.

Major objections (3)

[Section 1 and Section 3, Eq. (3.12); Figures 2b and 5] The paper's phenomenological range 1/2 <= tan beta <= m_t/m_b is licensed by the assertion in Section 1 that the renormalized tan beta defined here is 'effectively very close' to the Landau-gauge definition used in the effective-potential and renormalization-group approaches. That assertion is not backed by any numerical comparison. Eq. (3.12) fixes delta v_i to a purely divergent constant with vanishing finite part, which is a specific scheme choice; the paper itself argues at length in Section 1 that different choices for delta v_i can make tan beta 'arbitrarily different' already at one loop. The reader therefore needs at least one quantitative demonstration: for a representative point in the (M_A, tan beta) plane, compute the one-loop shift in tan beta between this scheme and the Landau-gauge delta v_i = 0 scheme, or equivalently show how much the contours of fixed tan beta in Fig. 2b move under that change. Without such a number, the upper limit used throughout Section 5 and Figs. 4-5 rests on an unquantified scheme-proximity assumption. This is a local addition and does not threaten the scheme itself.
[Section 4, Eqs. (4.15)-(4.26); Figures 4 and 5] Section 4 states that only Higgs-sector corrections are included and that the electron self-energy, the Z0 e+e- vertex and box diagrams are dropped, with m_e = 0. The authors acknowledge the approximation, so the issue is not its existence but its size. The paper elsewhere estimates the retained gauge self-energy corrections as 'of the order of a few percent', which sets the scale at which the neglected terms matter, yet no numerical estimate of the dropped terms is given anywhere. For the Bjorken process, where Z0 exchange dominates, a few-percent shift moves the sigma_0 contour boundaries in Figs. 4a-4c and 5a-5c, which are the paper's main experimental deliverable. Initial-state photon radiation, which is not mentioned at all and is not small at the 200-500 GeV energies considered, is a further unquantified item. I would ask for either a numerical estimate of the neglected electroweak and ISR contributions at two or three representative points, or an explicit statement of a theoretical uncertainty band attached to the contour boundaries, so that the reader can judge whether the quoted observability regions are robust against a few-percent shift.
[Appendix B, Eqs. (B.3)-(B.26)] The abstract claims that complete one-loop results for the 2- and 3-point Green's functions are 'explicitly given', but Appendix B does not meet that standard for the chargino and neutralino contributions. The matter-fermion auxiliary functions are written out (fff1a, fff1b in (B.1)-(B.2), fff2a, fff2b in (B.7)-(B.8), fff3 in (B.11), fff4 in (B.14)), and gf1i through gf10ij are defined in (B.3), (B.5), (B.9), (B.12), (B.15), (B.17), (B.19), (B.21), (B.23) and (B.25). However, the chargino and neutralino functions actually used in the final expressions, gC1i and gN1i in (B.4), gC2i and gN2i in (B.6), gC3i and gN3i in (B.10), gC4ij and gN4ij in (B.13), gC5ij and gN5ij in (B.16), gC6i in (B.18), gC7i in (B.20), gC8i in (B.22), gC9ij in (B.24) and gC10ij in (B.26), are never defined. A reader cannot reconstruct those terms from the manuscript. The offer of the FORTRAN library mitigates this substantially, but the wording 'explicitly given' should then be qualified, or the missing definitions supplied. This is the one place where the paper's own completeness claim is not met by the text.

Minor objections (7)

[Section 3, Eqs. (3.27)-(3.30)] The approximate mass formulae are stated to be accurate to 'typically 1-2 GeV' relative to the exact solutions of (3.26), but the region of parameter space over which this holds is not specified. Since the paper argues that the perturbative formula (3.25) fails badly precisely for small tan beta and for large tan beta with M_A near M_Z, the reader needs to know whether the approximation of (3.27)-(3.30) is reliable in those same corners. A short statement of the domain of validity, or a single comparison figure, would settle this.
[Section 2, Eq. (2.5)] As typeset, the gauge-fixing functionals contain repeated identical field symbols, for example the combination written as H_2^2 v_2 - H_2^2 v_2 + H_1^1 v_1 - H_1^1 v_1. Presumably one factor in each pair is the shifted field and the other the VEV, but the notation as printed does not distinguish them. Please make the shift explicit so that the linear terms generating the Goldstone-gauge mixing can be read off unambiguously.
[Section 3, Eq. (3.30) and Appendix A] The quantity epsilon in (3.30) is written in terms of m_t without saying whether this is the pole mass or a running mass, and the dimensional-reduction scale mu is never given a numerical value. Since the paper emphasizes scheme dependence throughout, both conventions should be stated explicitly, even if the final results are mu-independent.
[Section 5 and Figure 2a caption] The caption of Fig. 2a lists four soft-parameter sets for m_t = 120 GeV but only three for m_t = 180 GeV. If the fourth set was dropped deliberately, say so; otherwise the asymmetry looks like an omission. Relatedly, the sentence claiming max(sigma_Zh, sigma_Ah) > 0.05 pb over the whole (M_A, tan beta) range for m_t = 140 GeV and sqrt(s) = 200 GeV carries the hedge 'for a broad range of values of the soft supersymmetry breaking parameters' only in the following sentence about m_t = 180 GeV; the hedge should apply uniformly to both.
[Section 4, Eq. (4.15)] The propagator denominator uses |s - M_Z^2 - Pi_T^ZZ(s)|^2 while the photon-exchange interference is absorbed into the tilde vertices through Z_gammaZ in (4.19). Please confirm in the text that the gamma-Z mixing in the denominator is treated consistently with that rearrangement, since the two treatments are only equivalent when the mixing is included to the same order in both places.
[Appendix A, footnote 4] The footnote listing corrections to the couplings and sfermion mass matrices of ref. [12] is genuinely useful and should be easy to miss where it stands. Consider promoting the corrected sign convention for the v_1^2 - v_2^2 terms to the main text of Appendix A, since it affects the numerical results.
[Throughout] A number of typographical and bibliographic items should be corrected: 'deviced' for 'devised' (Section 1), 'infinitesimaly' for 'infinitesimally' (Section 2), 'Thompson limit' for 'Thomson limit' (Sections 3 and 5), 'Bjorken' for 'Bjorken' (Section 4 and figure captions), 'Pasarino' for 'Passarino' and 'Ridolfi' for 'Ridolfi' in the references, and 'Fort. Phys.' for 'Fortschr. Phys.' in refs. [8] and [15].

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The paper

The paper claims

A complete one-loop on-shell renormalization scheme for the MSSM Higgs and gauge sectors is presented, with explicit Green's functions and corrected Higgs masses and electron-positron production cross sections.

The reading

The Minimal Supersymmetric Standard Model has two Higgs doublets, so its Higgs sector contains five physical particles: two neutral scalars, one neutral pseudoscalar, and a charged pair. At tree level their masses and couplings are fixed by two parameters, usually the pseudoscalar mass and the ratio of vacuum expectation values, tan beta. Loop corrections from the top quark and its superpartners are known to be large, so a reliable prediction needs a consistent renormalization procedure. This paper supplies that procedure for the Higgs and gauge boson sectors. It defines counterterms for masses, vacuum expectation values, fields, gauge-fixing parameters, and mixing, then fixes their finite parts with on-shell conditions: physical gauge boson masses, no photon-Z mixing at zero momentum, vanishing tadpoles, and no A0-Z mixing at the pseudoscalar pole. The authors work in a gauge infinitesimally close to 't Hooft-Feynman and use dimensional reduction to preserve supersymmetry at one loop. They list the complete one-loop two-point and three-point Green's functions in appendices. For the Higgs masses, they do not use the usual perturbative formula. They solve for the exact poles of the 2x2 scalar propagator, which resums the leading higher-order effects. They then compute cross sections for e+e- -> Z h0/H0 and e+e- -> A0 h0/H0, including Higgs-sector one-loop corrections, external-line factors, and vertex corrections, while neglecting genuinely electroweak corrections and box diagrams. The numerical section scans MSSM parameters and shows where the lightest scalar is observable, where the heavier scalar takes over, and where the MSSM prediction becomes indistinguishable from a Standard Model Higgs boson. The paper also stresses that tan beta is scheme dependent beyond tree.

Extended reading notes

Core claim

The paper claims to provide a complete, internally consistent one-loop on-shell renormalization of the MSSM Higgs and gauge boson sectors, with all 2- and 3-point Green's functions explicitly given and with Higgs masses defined as exact poles of the 2x2 scalar propagator (Eq. 3.26). If correct, this supplies a reference calculation for MSSM Higgs masses and electron-positron production cross sections.

Load-bearing premise

The cross-section results assume that dropping genuinely electroweak corrections (electron self-energy, Z e+e- vertex, box diagrams) and setting the electron mass to zero leaves the Higgs-sector corrections as the numerically dominant effect. Section 4 states this approximation but gives no numerical estimate of the neglected terms, so the quoted production rates are not a complete electroweak prediction.

Original abstract

Systematic on-shell renormalization programme is carried out for the Higgs and gauge boson sectors of the Minimal Supersymmetric Standard Model. Complete 1-loop results for the 2- and 3-point Green's functions are explicitly given. The Higgs boson masses and the cross sections for the neutral scalar production in the $e^+e^-$ colliders are calculated.

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The abstract, sentence by sentence

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

Systematic on-shell renormalization programme is carried out for the Higgs and gauge boson sectors of the Minimal Supersymmetric Standard Model. Complete 1-loop results for the 2- and 3-point Green's functions are explicitly given. The Higgs boson masses and the cross sections for the neutral scalar production in the $e^+e^-$ colliders are calculated.

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	Systematic on-shell renormalization programme is carried out for the Higgs and gauge boson sectors of the Minimal Supersymmetric Standard Model.	contribution 0.79	0.92	0.17	asserted 1.00
s1	Complete 1-loop results for the 2- and 3-point Green's functions are explicitly given.	result 0.53	0.93	0.19	asserted 1.00
s2	The Higgs boson masses and the cross sections for the neutral scalar production in the $e^+e^-$ colliders are calculated.	result 0.89	0.92	0.33	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.74 · evidence from proof or derivation 0.93.

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Figures

**Figure 1.** Diagrams contributing at the 1-loop level to the processes e +e − → Z 0h 0 (H0 ) and e +e − → A0h 0 (H0 ), included in the calculations of Section 4. Figure 2a. Allowed mass regions for h 0 and H0 for the different values of mt and supersymmetric parameters. Supersymmetric parameters (Msq, Msl, Mgau, µ, A) read (in GeV): for mt = 120 GeV: solid (1000, 300, 1000, 500, 0), long-dashed (200, 100, 60, 50, 0), medium-dashed (1000, 300, 1000, 100, 1000), short-dashed (1000, 300,1000, 500, 1000); for mt = 180 GeV: solid (1000, 300, 500, 250, 500), long-dashed (200, 100, 100, 100, 0),…

**Figure 3.** Cross sections for the lighter scalar production as a function of Mh mass for fixed values of MA. Figure 4a. Regions in the (MA, Mh) plane in which at least one of the cross sections e +e − → Z 0h 0 or e +e − → A0h 0 is larger than assumed σ0, for the top quark mass mt = 140 GeV and for the different values of the σ0 and CMS energy. Figure 4b. Regions in the (MA, Mh) plane in which at least one of the cross sections e +e − → Z 0h 0 or e +e − → A0h 0 is larger than assumed σ0, for the top quark mass mt = 180 GeV and for the different values of the σ0 and CMS energy. F

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References

Works cited (19)

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View on Pith 1979

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How the literature uses it

40 arXiv papers cite this in 54 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 · 28 sentences without a confident label

Builds on it 2 papers

It is an extension of the DCPR scheme [ref] used in the context of the MSSM to the NMSSM [ref] , [equation] where $δZ_H_u$ and $δZ_H_d$ are the wave function renormalisation constants of the $H_u$ and $H_d$ doublets.

One-loop renormalisation of the NMSSM in SloopS : 1. the neutralino-chargino and sfermion sectors · $t_β$ from the Higgs sector · 1.00
The $DR$ condition for $t_β$ in this scheme is an extension of the DCPR scheme [ref] , used in the context of the MSSM, to the NMSSM [ref] , [equation] where $δZ_H_u$ and $δZ_H_d$ are the wave function renormalisation constants of the $H_u$ and $H_d$ doublets.

One-loop renormalisation of the NMSSM in SloopS 2: the Higgs sector · Mixed OS-$DR$ schemes · 1.00

Uses its method or tool 9 papers

We work in a linear $R_ξ$ gauge given by [equation] with [equation] The inverse propagator of the vector bosons is given at tree-level by [equation] with $V=W,Z$ and the transverse and longitudinal projectors are defined as: [equation] The multiplicative renormalization of the bare parameters is done in the following form: [equation] The $Z_ξ$s are fixed by the conditions that propagators mixing the Goldstone bosons with the vector bosons vanish [ref] and will not be discussed further here as they are not needed in the subsequent discussion.

Electroweak corrections to Neutralino and Chargino decays into a W-boson in the (N)MSSM · Wave-function and coupling renormalization · 0.90
It is fixed by forbidding transitions between the CP–odd Higgs boson $A$ and the $Z$ boson on the $A$ mass shell [ref] .

Radiative Corrections to the Neutralino Dark Matter Relic Density - an Effective Coupling Approach · Higgs Sector · 0.89
We thus have [equation] where, in the electroweak on-shell scheme [ref] supplemented with the DCPR [ref] definition of $tanβ$, [equation] Here, $δv/v$ is the common DCPR counterterm for the two Higgs doublets given in Eq. (3.11) of Ref. [ref] and $Δr$ [ref] contains those radiative corrections to the muon lifetime which the SM introduces on top of those derived in the QED-improved Fermi model.

Two-Loop Electroweak Corrections to the A^0 gamma gamma and A^0 g g Couplings of the CP-Odd Higgs Boson · $𝒪(x_t)$ and $𝒪(x_F)$ corrections to $Γ(A^0→γγ)$ · 0.87

Compares with it 1 paper

Refs. [ref] chose to work with on-shell fields and in $R_ξ$ gauge instead, and their shifts do not strictly minimize the one-loop effective potential.

The supersymmetric Higgs sector and B-Bbar mixing for large tan beta · Renormalization of $tanβ$ · 0.81

Cites it as background 13 papers

Significant efforts have also been devoted to the Higgs-mass calculation in the presence of CP-violating phases, as well as in non-minimal SUSY extensions of the SM.] is quite advanced: full one-loop corrections [ref] and two-loop corrections in the limit of vanishing external momentum [ref] are available, and the dominant momentum-dependent two-loop corrections [ref] as well as the dominant three-loop corrections [ref] have also been obtained.

Improved determination of the Higgs mass in the MSSM with heavy superpartners · Introduction · 0.97
The full one-loop result [ref] , the dominant two-loop corrections [ref] as well as partial three-loop results [ref] are known.

Precise prediction for the light MSSM Higgs boson mass combining effective field theory and fixed-order calculations · Introduction · 0.97
The complete one-loop result within the MSSM is known [ref] .

Prediction of the light CP-even Higgs-Boson Mass of the MSSM: Towards the ILC Precision · The Higgs and scalar top sectors of the MSSM · 0.88

Discussion

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