Phenomenology of Two Higgs Doublet Models with Flavor Changing Neutral Currents

Amarjit Soni (BNL); David Atwood (CEBAF); Laura Reina (BNL)

REVIEW

Phenomenology of Two Higgs Doublet Models with Flavor Changing Neutral Currents

David Atwood (CEBAF), Laura Reina (BNL), Amarjit Soni (BNL)

High Energy Physics - Phenomenology · Submitted 6 Sep 1996 · 61 references

Pith's verdict: A solid, unusually self-aware constraint catalogue for model III; the one soft joint is the negative loop result that licenses λ_ct, λ_tt ~ O(1), which is asserted without an error estimate.

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CitationAmarjit Soni (BNL), David Atwood (CEBAF), Laura Reina (BNL). "Phenomenology of Two Higgs Doublet Models with Flavor Changing Neutral Currents." arXiv:hep-ph/9609279 (1996). https://pith.science/paper/DEFX5OWH

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Editor's note

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This is the paper people cite when they need to know what model III can and cannot do, and that reputation is earned. The model is Cheng–Sher, and the authors say so up front. What they add is a systematic constraint catalogue: tree-level ΔF=2 bounds on λ_sd, λ_bd, λ_uc (Fig. 9), a table that labels tree versus loop dominance case by case (Table II), and a full one-loop recalculation of the Ztc, γtc and gtc form factors in Appendix B. That last piece is real work — they state which pieces of Luke–Savage they confirm, which they dispute, and they report the numerical consequence for t→cγ (Fig. 3 against Fig. 2 of Ref. [7]). They also state their checks: separate pole cancellation for the h0+A0+mixed set and for the charged set, Ward identities, and reproduction of the SM mixing results as a normalization anchor. That is the kind of bookkeeping I want to see in a paper of this type, and Sec. VI items i–iv are more explicit about approximations than most work in this area bothers to be. The soft spot is the one the stress-test flags, and it does land. The chain is: tree-level bounds kill the first generation; once it decouples, the one-loop contributions are 'never definitely larger than the SM,' so λ_ct and λ_tt survive at O(1); hence the headline rates. That middle step is a negative result from a dominant-diagram approximation, no QCD, VIA with one B-parameter for all eight operators of Eq. (32). The authors concede the window is bounded by 'as long as ... not much larger than 1,' but they give no error estimate, and that is the number everything downstream hangs on. It is a fixable gap, not a wrong answer — an uncertainty band on the loop contribution would settle it. Two smaller things: the Rb and ρ0 inputs are summarized from a companion paper listed as 'to appear,' so those numbers cannot be checked from this text; and case 3 is a tuned 50–70 GeV corner selected to chase the then-current Rb, which the authors admit. The Rb anomaly has since gone away, so that branch reads as history. Who should read it: anyone working on tree-level FCNC in extended scalar sectors, and anyone who needs the form factors. Send it out. The constraint table and Appendix B are worth referee time even where the conclusions are conditional.

Referee report

Significance

The paper's value is as a catalogue: it collects the relevant low-energy constraints on tree-level flavor-changing scalar couplings in one place, states its approximations explicitly (Sec. VI, items i-iv), and anchors its loop calculation by reproducing the Standard Model mixing results as a normalization check and by verifying separate pole cancellation for the h0+A0+mixed set and the charged-scalar set, together with the Ward identities (Appendix B). The predictions are falsifiable and stated in a form experimentalists can use: R_tc/λ^4 in Fig. 2, the branching ratios of Table I, R_bs in Fig. 11, and the mixing entries of Tables II and III. The two-scenario structure of Sec. VII, which separates the case in which the R_b anomaly is enforced from the case in which it is set aside, is an honest way to handle an experimental situation the authors themselves flag as unsettled. The paper is also candid that the surviving parameter space does not permit a large enhancement of the tc cross section, which is a more conservative statement than the abstract's emphasis on experimental effort might suggest.

Major objections (2)

[Sec. VI, items (ii)-(iv) and the paragraph following Table II] The statement that 'we do not find any region of the parameter space in which the one loop contributions in case 2 are definitely larger than the SM prediction' is the step that licenses λ_ct, λ_tt ~ O(1) and therefore the rates of Secs. III, IV and VIII, but it is a negative result quoted without an error estimate. The calculation retains only the dominant heavy-quark box (item ii), neglects QCD (item iv), and evaluates all eight operators of Eq. (32) with a single common B-parameter per meson (item iii, 'Clearly this is only an approximation'). Please add: (a) for one benchmark mass set, the ratio of the retained single-scalar box to the neglected two-scalar and W-H+ boxes of Fig. 7, so the reader can see how much room the approximation leaves; (b) the shift in the Case 2 entries of Table II when B_S and B_P are varied independently of B_VLL by, say, a factor of two; (c) an explicit statement of how the bound on λ_ct and λ_sb would move if the neglected topologies added coherently. Because the loop amplitude scales as λ², a factor-of-few underestimate would shift the bound on λ_ct only by a factor of order unity, so the conclusion is probably robust; but the paper should say so quantitatively rather than leave the reader to infer it.
[Sec. VIII, Table III, and the benchmark 'λ_bb ≃ 40 and λ_sb ≃ 10'] Table III quotes ΔM_Bs > 9 × 10^-12 GeV for Case 3, which is consistent with the experimental lower bound of Eq. (61) but spans many orders of magnitude and does not tell the reader what the scenario actually predicts. Evaluating Eq. (37) at the benchmark used later in the same section (λ_sb ≈ 10, m_A ≈ 60 GeV, with M_P^{Bs} = -f_B² M_B⁴/(m_b+m_s)² ≈ -1.2 GeV² and ξ_sb ≈ 0.033) gives ΔM_Bs of order 10^-6 GeV, i.e. x_s of order 10^6, and even λ_sb ≈ 1 with m_A ≈ 60 GeV gives x_s of order 10^4. Please state the predicted ΔM_Bs (or x_s) explicitly for the benchmark point and for the range of λ_sb admitted by the b → sc̄c bound, and comment on whether such a large x_s is acceptable. As written, the sentence that 'B_s0 - B̄_s0 mixing does not prevent ξ_sb from belonging to that small region' is true only in the weak sense that a lower bound cannot exclude a large mass difference; the model in fact predicts a very large one, which is a striking and testable signature that deserves to be stated as such.

Minor objections (9)

[Eqs. (19) and (21)] As printed, the right-hand sides of Eqs. (19) and (21) are dimensionless, whereas Eq. (20) correctly carries an overall 1/m_t multiplying a mass-squared bracket. An overall factor of order m_t appears to be missing from the t → cγ and t → cg rates; please restore it so the three rates are mutually consistent.
[Eqs. (13)-(15)] The overall β^4 (1 - β²/3) prefactor combined with the 1/(1-β²) weighting of the |C|² and |D|² dipole form factors gives a threshold behavior that differs from the standard β(3-β²)/2 shape for a two-body final state with one massive and one massless fermion. A sentence explaining the origin of this weighting, and confirming the relative normalization of the (A,B) and (C,D) terms, would let readers reproduce Fig. 2.
[Fig. 2 caption versus Sec. VI] The labels 'case 1', 'case 2' and 'case 3' denote mass patterns in Eq. (18) but coupling scenarios in Sec. VI and Sec. VII. The collision is confusing; please rename one of the two sets (for example, 'pattern 1-3' for the masses).
[Table II, Case 1 rows] The Case 1 entries for K0 and B_d0 mixing exceed the experimental values quoted in the same table, so Case 1 with λ = 1 is excluded by Fig. 9. The table should say so explicitly, and should state that its Case 1 entries correspond to λ at the bound rather than to λ = 1.
[Sec. VI, item (iii)] The dominant tree-level bound comes from the pseudoscalar matrix element M_P, which is fixed by PCAC and is therefore on firmer ground than the generic VIA used for the other operators. It would strengthen the paper to say this explicitly, since it means the bounds of Fig. 9 are less sensitive to the B-parameter approximation than the loop entries of Table II.
[Sec. III, paragraph following Fig. 2] The text refers to '10^4-10^5 µ+µ- events in a year of running for the next generation of e+e- colliders'; the final state should read e+e- events.
[Appendix B, comparison with Ref. [7]] The disagreement with Ref. [7] is asserted and illustrated only for a common scalar mass in Fig. 3. A short breakdown of the individual h0, A0, mixed and charged-scalar contributions to the form factors at one benchmark point would let readers locate the source of the discrepancy, particularly for the A0 and mixed terms where the paper reports the largest differences.
[Table III, row 2] The entry is labelled Br(B0 → X_s µ+µ-), while the text and the other rows use B → X_s µ+µ-. The superscript should be dropped for consistency.
[Sec. VII, item 1] The window 50 GeV ≤ m_h ~ m_A < 70 GeV is close to the then-current experimental lower bound on a light neutral scalar. Please state which bound is being applied, so the reader can judge how much of the window is genuinely available.

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The paper

The paper claims

A second Higgs doublet could let the top quark decay into a charm, and the existing data may already allow it.

The reading

The Standard Model forbids flavor-changing neutral currents at tree level, and the usual fix for two-Higgs-doublet models is an invented discrete symmetry that forbids them forever. This paper asks what happens if you drop that symmetry and let the extra doublet couple quarks of different flavors directly, with the new couplings scaled by $1/v$ times the square root of the product of the two quark masses, so that heavy-quark flavor change is large while light-quark flavor change stays tiny. The authors then walk the model through every low-energy constraint they can find: $K^0$-$/bar K^0$, $B^0_d$-$/bar B^0_d$ and $D^0$-$/bar D^0$ mixing, $B/rightarrow X_s/gamma$, the $/rho$ parameter, and $R_b$. Their conclusion is that the model survives, but only in two narrow configurations, and in those surviving configurations it makes sharp predictions: top-charm production at electron and muon colliders, rare top decays enhanced by up to four orders of magnitude over the Standard Model, and an observable $Z/rightarrow b/bar s$ rate. The value of the exercise is that a model usually dismissed as ruled out turns out to have a live corner, and that corner is testable with machines already being planned.

What carries the argument

The load-bearing device is the Cheng-Sher ansatz of Eq. (10), $/xi^{U,D}_{ij} = /lambda_{ij} /sqrt{m_i m_j}/v$, which converts an arbitrary complex matrix of new Yukawa couplings into a one-parameter-per-quark-pair family in which the top quark's flavor change is automatically the largest and the first family's is automatically the smallest. This hierarchy is what lets the authors keep large $t/rightarrow c$ effects while satisfying the very tight bounds from $K^0$-$/bar K^0$ mixing, which has been measured for decades and pins down any light-quark flavor violation. Around that ansatz the paper assembles a constraint pipeline: tree-level $h^0$ and $A^0$ exchange for $/Delta F = 2$ amplitudes, one-loop box and penguin diagrams with charged scalars, the full one-loop $Ztc$ and $/gamma tc$ form factors given in Appendix B, and the $/rho$ parameter computed from the scalar self-energies in Eq. (54). The form-factor calculation is also offered as a correction to earlier work: the authors state that their results for the $A^0$ and mixed $h^0$-$A^0$ contributions differ from Ref. [7], that this shifts $t/rightarrow c/gamma$ by more than an order of magnitude, and that in Ref. [7] the divergent terms in the $A^Z$ form factor appear not to cancel.

Extended reading notes

Core claim

The paper establishes that a two-Higgs-doublet model with tree-level flavor-changing scalar couplings, called model III, is not excluded by data, provided the couplings follow the Cheng-Sher mass-proportionality ansatz $/xi^{U,D}_{ij} = /lambda_{ij}/sqrt{m_i m_j}/v$ and provided the flavor-changing couplings of the first quark family are strongly suppressed. Two surviving scenarios are identified. In the first, which applies if one insists on reproducing the measured $R_b$, the neutral scalar and pseudoscalar must both sit in a narrow window of roughly 50 to 70 GeV, the charged scalar between 150 and 200 GeV, and the down-type couplings $/lambda_{bb}$ and $/lambda_{sb}$ must be much larger than one while the up-type ones stay small. In the second, which applies if the $R_b$ discrepancy is set aside as an unresolved experimental issue, only the first-family couplings need be small, the rest can be of order one, and $B/rightarrow X_s/gamma$ falls within two $\sigma$ of experiment once the charged scalar exceeds about 600 GeV. In both scenarios the model predicts $e^+e^- /rightarrow t/bar c + /bar t c$ at a rate up to $R_{tc} /sim 10^{-5}$ for $/lambda /simeq 1$, rare top decays with branching ratios reaching $10^{-7}$ for $t/rightarrow c/gamma$, $10^{-6}$ for $t/rightarrow cZ$ and $10^{-4}$ for $t/rightarrow cg$, and, in the $R_b$-driven scenario, a $Z/rightarrow b/bar s$ rate up to an order of magnitude above the Standard Model.

Load-bearing premise

Everything rests on the assumption that the new flavor-changing couplings really do scale as the square root of the product of the two quark masses, with one free multiplier per quark pair, and that the hadronic matrix elements of the resulting operators can be estimated by vacuum insertion with a single common bag parameter per meson, an approximation the authors themselves flag as crude.

Editorial extensions

If this is right

A lepton collider running at 500 GeV with a few times $10^{33}$ cm$^{-2}$s$^{-1}$ should be able to see or exclude $e^+e^- /rightarrow t/bar c + /bar t c$ for $/lambda$ near one and scalar masses of a few hundred GeV.
A muon collider tuned to the neutral Higgs resonance would produce $t/bar c$ through the tree-level flavor-changing coupling, giving a clean signal with no Standard Model background and, through top helicity in the decay distributions, separate access to the left- and right-handed couplings.
Rare top decays $t/rightarrow c/gamma$, $cZ$ and $cg$ become reachable at the next generation of hadron and lepton colliders, with $t/rightarrow cg$ reaching $10^{-4}$ in the most favorable mass region.
In the $R_b$-driven scenario, $B/rightarrow X_s/mu^+/mu^-$ and $B^0_s /rightarrow /mu^+/mu^-$ can be enhanced by up to two orders of magnitude over the Standard Model, and $Z/rightarrow b/bar s$ becomes a searchable channel at the $Z$ pole.
An improved measurement of $D^0$-$/bar D^0$ mixing by a single order of magnitude would tighten the constraint on the common $/lambda$ roughly to the level already set by kaon and $B_d$ mixing.

Reading between the lines

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

The paper's structure suggests a general lesson beyond this specific model: the reason flavor-changing neutral currents are usually forbidden by fiat is that the light-quark bounds are so severe, but if the new couplings track quark mass they inherit the mass hierarchy automatically, so the same data that kills generic models leaves a mass-ordered corner alive. That argument transfers to any new s
The two surviving scenarios are distinguished almost entirely by whether one believes the $R_b$ excess. If that excess is a measurement artifact, the model's parameter space widens considerably and the charged scalar can be heavy; if it is real, the neutral scalars are pinned into a window of tens of GeV, which is a region the LEP-era Higgs searches would have probed directly.
The authors note that heavy-flavor loop effects in $D^0$-$/bar D^0$ mixing can exceed the Standard Model by two orders of magnitude in the $R_b$-driven scenario while still sitting well below the current bound. That makes charm mixing an unusually clean discriminator for this class of models, since the Standard Model prediction there is small and the experimental reach is improving.
The disagreement with Ref. [7] over the $A^0$ and mixed contributions, and the observation about uncancelled divergences, is the kind of internal inconsistency that an independent recalculation could settle quickly; if the earlier result is right instead, the quoted rare-decay rates shift by an order of magnitude and the discovery reach moves accordingly.

Original abstract

A comprehensive phenomenological analysis of a two Higgs doublet model, with flavor changing scalar currents at the tree level, called model III, is presented. Constraints from existing experimental information especially on (Delta F)=2 processes are systematically incorporated. Constraints emerging from rare B-decays, Z-->b\bar b, and the \rho parameter are also examined. Experimental implications for e^+e^- (\mu^+\mu^-)-->t\bar c+\bar t c, t-->c\gamma(Z,g), D^0-\bar D^0, and B^0_s-\bar B^0_s oscillations, and for e^+e^-(Z)-->b\bar s+s\bar b are investigated and experimental effort towards these is stressed. We also emphasize the importance of clarifying the experimental issues pertaining to Z-->b\bar b.

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, 4,356 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.10 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.12 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

A comprehensive phenomenological analysis of a two Higgs doublet model, with flavor changing scalar currents at the tree level, called model III, is presented. Constraints from existing experimental information especially on (Delta F)=2 processes are systematically incorporated. Constraints emerging from rare B-decays, Z-->b\bar b, and the \rho parameter are also examined. Experimental implications for e^+e^- (\mu^+\mu^-)-->t\bar c+\bar t c, t-->c\gamma(Z,g), D^0-\bar D^0, and B^0_s-\bar B^0_s oscillations, and for e^+e^-(Z)-->b\bar s+s\bar b are investigated and experimental effort towards these is stressed. We also emphasize the importance of clarifying the experimental issues pertaining to Z-->b\bar b.

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	A comprehensive phenomenological analysis of a two Higgs doublet model, with flavor changing scalar currents at the tree level, called model III, is presented.	contribution 1.00	0.88	0.10	asserted 0.99
s1	Constraints from existing experimental information especially on (Delta F)=2 processes are systematically incorporated.	method 0.94	0.73	0.04	asserted 0.99
s2	Constraints emerging from rare B-decays, Z-->b\bar b, and the \rho parameter are also examined.	method 0.53	0.68	0.04	asserted 0.92
s3	Experimental implications for e^+e^- (\mu^+\mu^-)-->t\bar c+\bar t c, t-->c\gamma(Z,g), D^0-\bar D^0, and B^0_s-\bar B^0_s oscillations, and for e^+e^-(Z)-->b\bar s+s\bar b are investigated and experimental effort towards these is stressed.	contribution 0.58	0.65	0.13	asserted 0.90
s4	We also emphasize the importance of clarifying the experimental issues pertaining to Z-->b\bar b.	future work 0.98	0.26	0.08	asserted 0.85

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from analysis of existing data 0.65.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. Example of one-loop contributions to the

**Figure 3.** FIG. 3. Branching fraction for

**Figure 4.** FIG. 4. Tree level contributions to

**Figure 6.** FIG. 6. Tree level contributions to each different

**Figure 7.** FIG. 7. Box diagrams which contribute at one loop to the

**Figure 8.** FIG. 8. Penguin diagrams which contribute at one loop to the

Showing 7 of the paper's 11 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-ph/9609279 by the authors.

References

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Due to the qualitative character of our analysis, at thi s point it suﬃces to seek consistency with the experiment at t he 2 σ- level. Indeed, we took as reference the SM calculation [45], which is already aﬀected by a large uncertainty, and compute d only the leading corrections due to the new scalar bosons of m odel III, i.e. without considering the com...

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The theoretical prediction of Br(B → Xsγ) from Ref. [45] includes some NLO QCD corrections and the res ult could change in the future by a complete NLO analysis. We could have used in our analysis the fully consistent LO result for Br(B → Xsγ), which is a little higher (see Ref. [44,45]), but this would not modify the qualitative results that we are giving

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D. Buskulic et al ., ALEPH Collaboration, CERN-PPE/96-30

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Even if the pole terms are not explicitly given in Ref. [7 ], they can be easily deduced from the corresponding logarit hmic term (log µ2) in each diagram. 24 APPENDIX A: FEYNMAN RULES FOR MODEL III In this appendix we summarize the Feynman rules for model III , which are used in the calculations presented in the paper. We choose to work in the ’t Hooft-F...

View on Pith

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How the literature uses it

67 arXiv papers cite this in 91 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 · 58 sentences without a confident label

Uses its method or tool 2 papers

For the mass splitting connected to $F^0 - F^0$ oscillations [ref] , we find ΔM_F = B_F^2 f_F^2 M_F ∑_α[ 1/m_H^α^2 ( |I^α_rs|^2 (1/6 + 1/6 M_F^2/(m_r + m_s)^2 ) + |J^α_rs|^2 (1/6 + 11/6 M_F^2/(m_r + m_s)^2 ) ) ].

Constraining Flavour Symmetries At The EW Scale II: The Fermion Processes · Meson oscillations · 0.88
To tame the tree-level effects of the nonstandard scalars in $ΔM_P$, we first write the relevant expression for the NP contribution to the meson mass difference ($ΔM^NP_P$) as [ref]

Crossed two Higgs-doublet models: reduction of Yukawa parameters in the low-scale limit of left-right symmetry and other avatars · Phenomenology of the x2HDMs · 0.86

Uses its data or values 3 papers

The expression for the mass difference due to the new scalar is [ref] [equation] We use a similar equation for the $K-K̅$ mixing mass difference and use the experimental value $ΔM_K^exp=(52.93 ±0.09)×10^8 s^-1$ [ref] .

Resolving the $(g-2)_{\mu}$ and $B$ anomalies with leptoquarks and a dark Higgs boson · Hadronic constraints · 0.80
For completeness the branching fraction ratios in the different scenarios considered are shown in Fig. [ref], where we have use the parameters $ξ_ττ=2.2 ×10^-2$, $ξ_τμ=2.1 ×10^-1$, and $ξ_μμ=0.12$ according to the allowed regions cited in the literature [ref] .

Constraints on the charged Higgs boson of the two Higgs doublet model type III · Introduction · 0.82
The SM values are from [ref] , for predictions in type-III 2HDM, in the MSSM and in $R$-parity violating ($$) SUSY models see [ref] , [ref] , and [ref] , respectively.

Top quark physics at the LHC · Flavour-changing neutral current (FCNC) decays: · 0.83

Compares with it 1 paper

These interactions are compared with the experimental upper bounds in several $ΔF=2$ neutral meson processes [ref] and found to be generically well below these bounds.

Two Higgs Pair Heterotic Vacua and Flavor-Changing Neutral Currents · Introduction · 0.80

Cites it as background 26 papers

If the parameter $tanβ$ is eliminated from the Lagrangian, we have the usual 2HDM type III [ref] , where the Lagrangian of the charged sector is given by [equation] where $K$ is the Cabbibo-Kobayashi-Maskawa (CKM) matrix and $ξ^U,D$ the flavor changing matrices.

Constraints on the charged Higgs boson of the two Higgs doublet model type III · Introduction · 0.85
where $L_eff$ is the low energy $ΔF=2$ effective Lagrangian arising from a variety of processes [ref] .

Two Higgs Pair Heterotic Vacua and Flavor-Changing Neutral Currents · Phenomenology · 0.88
In 1996, a very comprehensive analysis of the model by Atwood, Reina and Soni [ref] looked at a large variety of processes including effects on $Δm_B$, $t →c h$, rare $μ, τ$ and $B$ decays, $b →s γ$, rate top decays and flavor-changing Z decays.

Flavor Changing Neutral Currents in the Higgs Sector · The rise and fall of the Cheng-Sher ansatz · 0.82

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.

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Even if the pole terms are not explicitly given in Ref. [7 ], they can be easily deduced from the corresponding logarit hmic term (log µ2) in each diagram. 24 APPENDIX A: FEYNMAN RULES FOR MODEL III In this appendix we summarize the Feynman rules for model III , which are used in the calculations presented in the paper. We choose to work in the ’t Hooft-F...

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