Matching renormalisable couplings: simple schemes and a plot
Abstract
We discuss different choices that can be made when matching a general highenergy theory – with the restriction that it should not contain heavy gauge bosons – onto a general renormalisable effective field theory at one loop, with particular attention to the quartic scalar couplings and Yukawa couplings. This includes a generalisation of the counterterm scheme that was found to be useful in the case of highscale/split supersymmetry, but we show the important differences when there are new heavy scalar fields in singlet or triplet representations of SU(2). We also analytically compare our methods and choices with the approach of matching pole masses, proving the equivalence with one of our choices. We outline how to make the extraction of quartic couplings using pole masses more efficient, an approach that we hope will generalise beyond one loop. We give examples of the impact of different scheme choices in a toy model; we also discuss the MSSM and give the threshold corrections to the Higgs quartic coupling in Dirac gaugino models.
1 Introduction
In the absence of clear collider signals of new particles, there has been much recent interest in constraining deviations from the Standard Model (SM) in terms of effective operators. This approach to the “Standard Model Effective Field Theory” has primarily been interested in higherdimensional operators that encode new effective interactions, for example recent work on calculating these in general theories can be found in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. However, there is also important information that can be extracted by matching the renormalisable couplings of the SM. In particular, this is an increasingly important approach to calculating the Higgs mass from a topdown theory, providing a more accurate calculation than a fixedorder one once new particles that couple to the Higgs are above a few TeV. It is the only approach to constraining the Higgs mass in split supersymmetry [12, 13, 14] where new physics could be around 100–10\(^5\) TeV [15, 16]; highscale supersymmetry [15, 17, 18, 19, 20] where it could be around 10\(^7\)–10\(^9\) TeV; the FSSM [21, 22] where it could be as high as the GUT/Planck scale, etc. Moreover, there is also a parallel effort considering the lowenergy theory to be a simple nonsupersymmetric extension of the SM such as a TwoHiggsDoublet Model (THDM) [23, 24, 25, 26, 27], and then it is very interesting to match these theories to new physics at a (much) higher scale.
With this motivation, we require: (i) the extraction of the renormalisable couplings (gauge couplings, Yukawa couplings and scalar quartic couplings) in the lowenergy theory from observables; (ii) renormalisation group equations (RGEs) for the lowenergy theory; and (iii) threshold corrections at the matching scale which we shall denote throughout \(M \). The RGEs for general renormalisable field theories have been known for some time up to two loop order [28, 29, 30, 31, 32, 33, 34, 35] and can be obtained for any model by SARAH [36, 37, 38, 39] or PyR@TE [40, 41], and higher loop orders are available for the SM. On the other hand, for (i) and (iii) the information is less complete: when the lowenergy theory is the SM, the Higgs mass is used to extract the running quartic coupling, and the extraction of all couplings can be performed at two loop order (with some three or fourloop corrections known), e.g. in [18, 19, 42, 43, 44], but for general models in SARAH it can be done only at one loop order, with twoloop corrections to the Higgs mass in the limit of vanishing electroweak gauge couplings [45, 46, 47]. Furthermore, threshold corrections to the Higgs quartic coupling have been computed explicitly for some models or scenarios such as split/highscale supersymmetry, up to full oneloop plus leading twoloop order [15, 20, 48, 49, 50], and even recently up to leading threeloop order in [51]. These corrections are implemented in public codes for the Higgs mass calculation such as SusyHD [49], MhEFT [24], FlexibleSUSY [52] and FeynHiggs [53, 54, 55]. The codes FlexibleEFTHiggs [56] and SARAH [57] also allow oneloop matching of a general theory to the SM as the lowenergy theory via matching of pole masses. Finally, the code MatchingTools [58] matches two general theories to each other, but only at the tree level.
While it is vital to reduce the error in the extraction of the top Yukawa couplings and strong gauge coupling, the need for precision in the extraction of lowenergy parameters and especially matching is particularly important for quartic couplings, which are wellknown to be highly sensitive to quantum corrections, as stressed e.g. in [59]. The purpose of the running to high scales in the bottomup approach is to constrain the scale of new physics or investigate the (scale of) instabilities of the potential, and these depend logarithmically on the scale, thus the scale depends exponentially on small differences in the lowenergy parameters.
 1.
Mixing between heavy and light states is inevitable in models with additional Higgs doublets, and then there are quantum corrections to the mixing angle(s). This has been investigated in the case of one extra doublet [20, 27, 60] and it was found that a judicious choice of counterterms allows the calculation to be simplified (so that the mixing angle \(\beta \) is not modified). We show how this can be generalised beyond one additional doublet.
 2.
In the presence of heavy SU(2) singlets or triplets, a trilinear coupling with two light Higgs fields is possible, and then the quartic coupling receives a correction at tree level when integrating out the heavy states. The presence of trilinear couplings with two light Higgs scalars moreover leads to infrared divergences in the amplitudes which cancel in the threshold corrections: we explicitly show how these cancel and how they can be simply dealt with.
 3.
In the presence of gauge singlets, tadpoles are generated before electroweak symmetry breaking. We describe four different approaches to dealing with them.
 4.
We show that the threshold corrections to the Higgs quartic, under the assumption that there are no heavy gauge bosons, are independent of gauge couplings^{1} at one loop, which is not immediately obvious.
 5.
It is clear that cubic scalar couplings in the low energy theory should be at most of the order of the mass scale of the lowenergy theory, which we denote \(\zeta \). However, if we insist on including such couplings that do not decouple as we take \(\zeta \rightarrow 0\) then we find that we must include higherdimensional operators to cancel the infrared divergences. We describe this explicitly in Sect. 3.4.
 6.
As a result of points (1) and (4) we give, in Sect. 4, what we believe is the simplest possible prescription for matching general scalar quartic couplings.
 7.
As mentioned above, an alternative approach to matching quartic or Yukawa couplings when the lowenergy theory is the SM is to match pole masses in the two theories. However, given that there are different possible choices for parameter definitions when we perform a “conventional” matching calculation, it is not immediately obvious how to compare the definitions in the two approaches (i.e. to know what we actually obtain from the polematching calculation!). This has been seen in the case of high scale/split SUSY in [20, 27, 60], where the pole mass calculation gives a result equivalent to the “counterterm” approach to the angle \(\beta \), which we define in Sect. 3. In Sect. 5 we derive the matching conditions for a general highenergy theory using the pole matching approach, and show the correspondence with the EFT calculation.
 8.
As a result of the derivation in Sect. 5, we propose in Sect. 5.1 a simple and explicitly infrared safe prescription for matching Higgs quartic couplings where we only need to evaluate twopoint scalar amplitudes.
2 Deriving the matching conditions
We shall match our two theories together at some scale \(M \), assuming that all “heavy” fields have masses of this order, and take the mass scale of our lowenergy theory to be \(\zeta \ll M \). We shall have in mind that this hierarchy can be of more than oneloop order, but in any case since we are only matching at one loop we can treat masses that are suppressed by one loop compared to the scale \(M \) – i.e. all \(m_{{{p}}{{q}}}^2\) – as effectively zero. For example, taking the SM as lowenergy theory, \(\zeta \sim v \sim m_h\). Then it is convenient to take the limit \(\zeta \rightarrow 0\) in the loop functions for the final expressions, as terms of order \(\zeta \) would lead to corrections to the result suppressed by powers of \(v/M \), i.e. equivalent to higherdimensional operators.
We shall now briefly review three methods of deriving the matching conditions between these two theories.
2.1 Diagrammatic
The conventional approach to matching theories is to compare Feynman diagram calculations. The approach in the next subsection (using path integrals) corresponds to calculating 1PI diagrams, but at the expense of obtaining a noncanonically normalised lowenergy theory. If we want to insist that our lowenergy theory has canonical kinetic terms, and want to match directly using diagrams, then the obvious and essentially only approach is to match Smatrix elements in the two theories. The simplest way to do this is to take \(\zeta \rightarrow 0\) first, making sure that the pole masses (not just the treelevel masses) of all the light particles are also set to zero, and then matching the results in the two theories as the total external momentum is taken to zero.
2.2 Effective action: path integral approach
2.3 Effective action: equations of motion method
 1.
In some favourable cases a discrete symmetry, which is broken at the same time as electroweak symmetry (or not at all), forbids such a tadpole (such as in e.g. the \(\mathbb {Z}_2\)symmetric singletextension of the SM or the \(\mathbb {Z}_3\)symmetric NMSSM in the unbroken phase).
 2.
We may have the freedom to adjust the treelevel tadpole term \(t_{{\mathcal {P}}}\) already in the basis of Eq. (2.1) so that the total tadpole including quantum corrections is zero, without needing to make any shifts of the form (2.3). This is the case if we specify the highenergy theory by just a matching scale and the dimensionless parameters, for example if we scan over supersymmetric models without specifying a mediation mechanism.
 3.We can assume that the tadpole equation is satisfied at tree level (so that \(t_{{\mathcal {P}}}=0\)). Then we solve (2.11) treating \(\delta t_{{\mathcal {P}}}\) as a oneloop perturbation of the treelevel tadpole condition. I.e. since we have \(\langle \Phi _{{\mathcal {P}}}\rangle = 0\) at tree level with all nonsinglet field expectation values set to zero, the solution to (2.11) iswhich effectively means shifting$$\begin{aligned} \Phi _{{\mathcal {P}}}=  \frac{1}{m_{{{\mathcal {P}}}}^2} \delta t_{{\mathcal {P}}}+ \mathcal {O}(\Phi _{{i}}^2) + \mathcal {O}(2\ \mathrm {loops}) \end{aligned}$$In this way we can compute around the treelevel vacuum; in the case that the treelevel expectation value is small or vanishing – in the basis (2.1) before any shifts – this option would appear to be the most appropriate choice.$$\begin{aligned}&\delta m_{{{i}}{{j}}}^2 \rightarrow \delta m_{{{i}}{{j}}}^2  \frac{\delta t_{{\mathcal {P}}}}{m_{{\mathcal {P}}}^2} \,a_{{{\mathcal {P}}}{{i}}{{j}}}, \nonumber \\&\delta a_{{{i}}{{j}}{{k}}} \rightarrow \delta a_{{{i}}{{j}}{{k}}}  \frac{\delta t_{{\mathcal {P}}}}{m_{{\mathcal {P}}}^2}\, \tilde{\lambda }_{{{\mathcal {P}}}{{i}}{{j}}{{k}}} . \end{aligned}$$(2.12)
 4.We can assume that the tadpole equation is satisfied at loop level (so that \(t_{{\mathcal {P}}}+ \delta t_{{{\mathcal {P}}}} =0\)) after making shifts of the form (2.3). In so doing, we can trade a different dimensionful parameter for each singlet tadpole equation, order by order in perturbation theory. This is the standard approach in pole mass calculations, where the typical choice is to eliminate masssquared parameters, but this is the most complicated from the EFT point of view because we want to fix the masses in order to perform the matching. The treelevel tadpole equations for the singlets in the basis before the shifts (2.3) readand, for the typical choice of adjusting the diagonal terms \(m_{{{\mathcal {P}}}}^2\), the oneloop mass shift becomes$$\begin{aligned} m_{{{\mathcal {P}}}}^2 v_{{\mathcal {P}}}&=  t_{{\mathcal {P}}} \frac{1}{2} a_{{{\mathcal {P}}}{{\mathcal {Q}}}{{\mathcal {R}}}} v_{{\mathcal {Q}}}v_{{\mathcal {R}}}\nonumber \\&\quad  \frac{1}{6} \tilde{\lambda }_{{{\mathcal {P}}}{{\mathcal {Q}}}{{\mathcal {R}}}{{\mathcal {S}}}} v_{{\mathcal {Q}}}v_{{\mathcal {R}}}v_{{\mathcal {S}}}\end{aligned}$$(2.13)where the tadpole \(\delta t_{{\mathcal {P}}}\) is computed at the minimum of the potential. Note that if we have a case where \(v_{{\mathcal {P}}}=0\) for all \({{\mathcal {P}}}\) then this approach reduces to option 2: we shall throughout assume when we refer to option 4 that the expectation values of all the singlets concerned are nonvanishing in the original basis.$$\begin{aligned} \delta m_{{{\mathcal {P}}}{{\mathcal {Q}}}}^2 \rightarrow \delta m_{{{\mathcal {P}}}{{\mathcal {Q}}}}^2  \frac{\delta t_{{\mathcal {P}}}}{v_{{\mathcal {P}}}}\,\delta _{{{\mathcal {P}}}{{\mathcal {Q}}}}, \end{aligned}$$(2.14)
3 Mixing and matching
In this section we shall discuss the effects of infrared safety and gauge dependence of the matching, and also derive the matrices \(\delta U\) that encode the effects of mixing of the light and heavy degrees of freedom, employing the “perturbative masses” approach; in Sect. 4 we shall show an alternative.
3.1 Infrared safety
If we compute the shifts with small or vanishing masses for the “light” fields, then the corrections \(\delta \lambda \) will contain large/divergent logarithms of the form \(\log \frac{m_{{p}}^2}{M^2}\), \(m_{{p}}\) being the light masses and M the mass scale of heavy states, at which the matching is performed. Clearly these should cancel against the corresponding corrections in the highenergy theory, so that the resulting shift is infrared finite. In the case that the theory contains no couplings of the form \(a_{{{\mathcal {P}}}{{p}}{{q}}}\) or \(a_{{{p}}{{q}}{{r}}}\) the infrared divergent corrections in \(\delta \tilde{\lambda }\) are identical to those in \(\delta \lambda \) and so the subtraction is straightforward. On the other hand, once we allow for these other types of coupling the cancellation of infrared divergences becomes more subtle.
 Subtracting an infrared divergent piece and taking the limit as \(\zeta \rightarrow 0\), e.g.$$\begin{aligned} \overline{P}_{SS} (0,0) \equiv \lim _{x\rightarrow 0} \bigg [ P_{SS} (x,x)  \log \frac{x}{M^2} \bigg ]= 0. \end{aligned}$$

Taking the loop integral to only be over the “hard” momenta, as described in Eq. (2.8).

Regularising the infrared divergences using dimensional regularisation and discarding the divergent terms \(\propto \frac{1}{\epsilon _{IR}}\).
Finally, we shall see in the next section that we must compute \(\delta m^2_{{{\mathcal {P}}}{{p}}}\) and \(\delta Z_{{{p}}{{q}}}\), which in principle could contain infrared divergences. However, a divergence that is not trivially equal to the same contribution in the lowenergy theory could only appear from a scalar loop, and the absence of the offending terms at one loop is guaranteed by forbidding couplings of the form \(a_{{{p}}{{q}}{{r}}}\). Hence we need make no distinction between \(\delta m^2_{{{\mathcal {P}}}{{p}}} \) and \(\overline{\delta }m^2_{{{\mathcal {P}}}{{p}}}\), etc.
3.2 Mixing
3.3 Gauge dependence
Since we take all gauge groups to be unbroken in the limit \(\zeta \rightarrow 0\), we may expect that gauge couplings ought to induce no net contribution to \(\lambda _{{{p}}{{q}}{{r}}{{s}}}\). Indeed, if there are no trilinear couplings in the theory, then this is immediately obvious: the gauge contributions to \(\delta \tilde{\lambda }_{{{p}}{{q}}{{r}}{{s}}}\) and \(\delta \lambda _{{{p}}{{q}}{{r}}{{s}}}\) are identical in this case, because the unbroken gauge interactions cannot mix heavy and light fields and certainly the corrections of quartic order in the gauge couplings – i.e. the first row of diagrams in Fig. 1 – must always be equal.^{3} For corrections of quadratic order in the gauge couplings, the second row of diagrams in Fig. 1 all contain only massless/light fields in the loops, and so we expect them not to contribute. However, once we include trilinear couplings, there are diagrams such as those given in Fig. 2 which are individually nonzero after infrared regulation, and so it is possible that there could be some residual dependence on the gauge couplings. However, this cancels out, as we show below.
Hence we can indeed neglect gauge contributions at one loop, as there is no gauge contribution to \(\delta m_{{{p}}{{\mathcal {P}}}}^2\) and \(\delta _{g^2} Z^H = \delta _{g^2} Z^L\). However, it is important to note that we require all of the separate pieces together in order to cancel the gauge dependence, which will be relevant in Sect. 4.
3.4 Trilinear couplings and higherdimensional operators
If we reason with orders of magnitude, it is natural to assume that couplings \(a_{{{\mathcal {P}}}{{q}}{{r}}}\) and \(a_{{{\mathcal {P}}}{{\mathcal {Q}}}{{r}}}\) are of the order of a heavy mass, say M, times numerical factors of \(\mathcal {O}(1)\). From this we can easily see that all of the above terms are of order \(a_{{{p}}{{q}}{{r}}}/M\) (and even \((a_{{{p}}{{q}}{{r}}}/M)^2\) for the last one). As we could expect \(a_{{{p}}{{q}}{{r}}}\) to be of the order of a light mass (e.g. \(m_{{{p}}}\sim \zeta \)), it would seem natural that the above terms be suppressed at least as \(\mathcal {O}(\zeta /M)\) – and therefore also go to zero in the limit \(\zeta \rightarrow 0\). The finite part of the matching is then exactly the same as that obtained previously.
However, one can still want to understand what happens if the trilinear couplings between light states are not of the order of \(\zeta \). Having very large trilinear couplings in the lowenergy theory could potentially cause a breakdown of perturbativity and/or unitarity, as well as expectation values in the lowenergy theory of the order of the heavy masses. Nevertheless, it is actually still possible in such a case to cancel all of the IR divergences, by taking into account higherdimensional operators.
More specifically, one can deduce from the form of the divergent terms in Eq. (3.13) that the required new operators are a dimension5 operator \(c_5^{{{p}}{{q}}{{r}}{{s}}{{t}}}\phi _{{p}}\phi _{{q}}\phi _{{r}}\phi _{{s}}\phi _{{t}}\) and a dimension6 operator \(k_6^{{{p}}{{q}}{{r}}{{s}}}\phi _{{p}}\phi _{{q}}\partial _\mu \phi _{{r}}\partial ^\mu \phi _{{s}}\) (a correction to the kinetic term of the scalars). The former will cancel out with the first five lines of Eq. (3.13), while the latter compensates the last remaining term.
3.4.1 Higherdimensional operators in a toy model
3.4.2 Discussion of the dimension5 operator for a general theory
3.4.3 Discussion of the dimension6 operator for a general theory
4 Nonminimal counterterm approach
Finally, a more extreme counterterm choice would be to use pole masses for all states, both light and heavy, without taking the limit \(\zeta \rightarrow 0\). This would technically remove the problem of infrared divergences, but replace it with a practical one (the computations would become much more cumbersome, with numerically large logarithms, unless the limit of \(\zeta \rightarrow 0 \) were taken analytically, when they would reduce to the expressions above).
5 Comparison with the pole matching approach
Furthermore, we find that it is straightforward to make a connection between the pole mass matching and the EFT approach for the treatment of the singlet expectation values: the third line in Eq. (5.24) vanishes for options 2 or 4 for the singlet tadpoles, and gives exactly the shifts (2.12) for option 3, where \(t_K =0\). This was not necessarily obvious, since the definitions are subtly different (in the pole matching procedure, the conditions are specified at \(v_1 \ne 0\)). Note that the treatment of the singlet tadpoles in the polemass matching approach is commonly chosen to be option 4.
5.1 Efficient computation of the matching
However, we must also take care with the gauge dependence in the presence of heavy triplet scalars (such as in Dirac gaugino models). In that case, if we set the gauge contributions to zero in the matching, then we must also set them to zero in the heavy tadpole relationship between \(m_I^2\) and \(v_I\) (5.14) – otherwise we will reintroduce gauge dependence into the result.
6 Examples
6.1 Pole matching in the MSSM
The calculation in Sect. 5 is perhaps couched in unfamiliar terms, so it is useful to present the standard example of split or highscale supersymmetry, where the MSSM scalars are heavy and, when integrated out, yield a scalar sector that is just that of the SM, so ideal for application of the pole matching procedure.
The above illustrates the equivalence between the polematching procedure and the EFT calculation for the MSSM matching to the SM, and is much simpler than an explicit termbyterm derivation in e.g. [56].
6.2 Dirac gauginos
In the context of matching a heavy theory onto the SM, Dirac gaugino models are particularly interesting because they contain both singlet and triplet scalars, which are the most general possibilities for the presence of a coupling \(a_{I11}\) at \(\mathcal {O}(\zeta ^0) \) with a SM doublet: SU(2) gauge invariance forbids other representations (although in the most general case we would also be allowed triplets carrying hypercharge \(\pm \,1\)). Moreover, in many scenarios a hierarchy between the singlet/triplet states and the Higgs is natural, which comes from a large Dirac gaugino mass, so an EFT approach to the Higgs mass calculation is particularly appropriate. Indeed first attempts were made in this direction in [21, 22, 26]; in [21, 22] a Diracgaugino model was matched onto the SM – without (most) threshold corrections – while in [26] the Minimal Dirac Gaugino Supersymmetric Standard Model (MDGSSM) and Minimal Rsymmetric Supersymmetric Standard Model (MRSSM) were matched onto the THDM, giving oneloop threshold corrections in the limit that the Dirac gaugino masses were small. Here we shall consider the oneloop threshold corrections of the MDGSSM matching onto the SM plus higgsinos in the limit that the Dirac gaugino masses are large.
7 Comparing two approaches to mixingangle renormalisation
A last useful illustration of our results is to compare for a simple toy model the “perturbative” and “nonminimal counterterm” approaches to the renormalisation of the mixing between light and heavy states.
7.1 “Perturbative masses” approach
7.2 “Nonminimal counterterm” approach
7.3 Numerical example
Figure 7 shows the values that we find for \(\lambda ^{hhhh}\) respectively in the “perturbative” (lightred curves) and the “nonminimal counterterm” (blue curves) schemes, at tree level (dashed lines) and oneloop level (solid lines), as a function of the coupling \(\lambda ^{1122}\) of the nondiagonal basis. At tree level, one can observe a large difference between the quartic couplings obtained in the two schemes. This can be understood because the mixing between h and H is small at tree level, but the looplevel mixing \(\overline{\delta }m^2_{hH}\) is large, therefore the relative effect of the loopinduced mixing is large and the mixing angle is modified significantly between the two schemes.
At oneloop, we see that the loop corrections are much larger in the “perturbative” scheme than the “nonminimal counterterm” scheme; again, this comes from the fact that the looplevel mixing term – proportional to \(\overline{\delta }m_{hH}^2\) – is large for the parameter points we considered. However, while the loop corrections differ in magnitude, the oneloop results for \(\lambda ^{hhhh}\) in the two approaches are close. The differences that appear for increasing \(\lambda ^{1122}\) can be interpreted as indications of the importance of twoloop corrections. A simple way to estimate the typical size of the twoloop corrections to the matching is to compute the matching relation (7.6) using for the couplings appearing in the oneloop terms the values obtained in the “perturbative” scheme – i.e. we use Eq. (7.6) with \(\tilde{\lambda }^{hhhh}_\text {c.t.}\), \(\tilde{\lambda }^{hhHH}\), \(\tilde{\lambda }^{hhSS}\), and \(\tilde{\lambda }^{hhhH}\) – as the difference with using all couplings computed in the “counterterm” scheme is a twoloop effect. Doing so, we obtain the dotdashed curve in Fig. 7, which is still close to the result of the “perturbative masses” scheme and only differs significantly for large \(\lambda ^{1122}\) – this indeed confirms missing twoloop corrections as the origin of the difference between the solid curves for \(\lambda ^{hhhh}\).
Before ending this section, a final comment is at hand about the choice of inputs and of scheme when integrating out heavy fields. If we had proceeded naively – or incorrectly – and had not specified the scheme in which the diagonalbasis couplings are given, or in which they are computed from other inputs (such as in Eq. (7.8)), we could have obtained widely different results for \(\lambda ^{hhhh}\). Indeed for a given value of \(\tilde{\lambda }^{hhhh}\), depending on the scheme that it is considered to be given (or computed) in, the loop corrections that are added to it change drastically – as we saw in the above.
8 Threshold corrections to Yukawa couplings
9 Outlook
We have described how to match renormalisable couplings between general theories and explained the different choices that can be made. Our aim is to simplify the calculation of the matching as much as possible, since already at one loop the expressions are rather long; we provide what we expect to be the simplest possible prescription for matching onto the SM using only twopoint scalar amplitudes in Sect. 5.1, and the simplest general prescription in Eq. (4.10).
The logical extension is to pursue our approach(es) at two loops. Beyond one loop, we expect the use of mass counterterms to become more important to simplify the removal of infrared divergences: in particular, if the hierarchy between \(\zeta \) and \(M \) is comparable to or greater than one loop order (so that the scales are highly tuned) then we expect the “naive perturbative” approach should break down, because we will not be able to treat the “light” states in the loops as massless. Investigating this and its relationship to the Goldstone Boson Catastrophe [47, 71, 72, 73] will be the subject of future work.
Footnotes
 1.
Note that, in the discussion of a general field theory, “gauge couplings” refers strictly to the interactions of the gauge bosons. In supersymmetric theories some of the scalar and Yukawa interactions may be related to the gauge couplings, but for the sake of our discussion they are treated just like all other scalar and Yukawa interactions.
 2.
Note that we will also use indices \(\{{{p}}, {{q}},{{r}},\ldots \}\) for states of the highenergy theory that can be identified as light and therefore correspond to states in the lowenergy theory.
 3.
On the other hand, there is a difference if we compute the corrections in different schemes; if we match a theory in the \({ \overline{\mathrm{DR}}'}\) scheme onto a theory in the \({\overline{\mathrm{MS}}}\) one then there is a shift to the quartic couplings of quartic order in the gauge couplings, see e.g. [10, 64] for general formulae.
 4.
We do not include the divergent parts of the counterterms in \(\delta _{ct} Z,\, \delta _{ct} m^2\) as they have already been implicitly subtracted.
 5.
Since the light masses need tuning to remain small, we see that we should either adjust the treelevel masses order by order in perturbation theory, or take \(\delta _{ct} m^2_{{{p}}{{q}}} =  \Pi _{{{p}}{{q}}}^{HET} (0) + \Pi _{{{p}}{{q}}}^{LET} (0).\)
 6.
Elsewhere we use \(\{I,J,K,L\}\) for fermions: in this section we do not have explicit fermion indices so there should hopefully be no confusion.
 7.
 8.
Note that in Sect. 4, we discussed the choice of counterterm for the heavy masses, however, as there are no trilinear couplings in this model we do not need to worry about this here for the matching condition for the quartic coupling.
 9.
Scalar trilinears \(a_{ijk}\) are of order \(\zeta \) and as we have taken the limit \(\zeta \rightarrow 0\), there are no scalar trilinear couplings in our setting and hence no corrections to them either – note that these corrections could in principle be obtained easily from the results in Appendix B.2.1 together with the modified loop functions defined in Appendix A.2.
Notes
Acknowledgements
We thank Sebastian Paßehr for useful discussions. MDG thanks Florian Staub and Martin Gabelmann for related discussions. We acknowledge support from French state funds managed by the Agence Nationale de la Recherche (ANR), in the context of the LABEX ILP (ANR11IDEX000402, ANR10LABX63), and MDG and PS acknowledge support from the ANR Grant “HiggsAutomator” (ANR15CE310002). PS also acknowledges support by the European Research Council (ERC) under the Advanced Grant “Higgs@LHC” (ERC2012ADG 20120216321133). During parts of this work, JB was supported by a scholarship from the Fondation CFM and by a fellowship from the Japan Society for the Promotion of Science.
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