SMEFT as a slice of HEFT’s parameter space

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Introduction: two EFTs for the electroweak sector
Beyond the Standard Model (BSM) searches in experiments at the LHC and other machines have come to be used to constrain the well-known SMEFT expansion, There, the various operators are organized by canonical (field and derivative) dimension as exposed by the powers of Λ made explicit, and their c i coefficients bound by experiment [1][2][3][4].The particle content of the electroweak symmetry breaking sector is packaged in a Higgs doublet field, just as in the SM, There, the Cartesian coordinates ϕ a can be rearranged into the polar decomposition in terms of the ω i Goldstone bosons (which set the orientation of H through the unitary matrix U(ω)) and the radial coordinate h SMEFT (with |H| = (v + h SMEFT )/ √ 2).If, alternatively, no assumption about the Higgs-boson h being part of a doublet H is made, the appropriate effective theory is rather HEFT, directly written in terms of h (see [5] for a recent review) (3) The h HEFT Higgs (or for simplicity, h) is exposed in multiplicative functions rescaling the traditional operators of the old electroweak chiral Lagrangian; for an alternative form see [6].
It was believed that one could indistinctly convert them into one another, as between Cartesian and polar coordinates, by wrapping or unwrapping the H field to expose h and the Goldstone bosons ω i , or to reconstruct it from them.But several works led by the San Diego group [7][8][9][10] have clarified that while HEFT can always be cast as a SMEFT, the converse is not always true, there are some requirements on the "flare" function of the Higgs field n , that we will shortly expose; saliently, F needs to have a zero h * and satisfy certain analyticity properties.
In our recent contributions [11,12], we have translated these criteria into practical conditions that amount to finding certain correlations among the HEFT parameters that, if violated by experimental data (also interpretable in terms of HEFT coefficients [13,14]), would falsify SMEFT.Here we quickly overview these developments.

Regularity conditions for the applicability of SMEFT
Alonso, Jenkins and Manohar [15] formulated HEFT in geometrical terms in the space of Higgs-fields and detailed the regularity conditions that the HEFT Lagrangian needs to satisfy to support a SMEFT, employing an elegant covariant formalism.We will proceed in a more pedestrian way, converting SMEFT into HEFT and viceversa with The change from SMEFT to HEFT is straightforward and always possible, with the canonical, nonlinear change of variables given in differential form as where the flare-function is provided by However, the reciprocal conversion h HEFT = F −1 (1 + h SMEFT /v) 2 from HEFT to SMEFT is not always possible, because of the need to reconstruct squared operators of the Higgs doublet field H necessary for SMEFT, such as The extra |H| 2 on the right hand side of the second equation appears then in a denominator , As SMEFT requires a Taylor expansion as in Eq. ( 1), this singularity precludes its existence and needs to be cancelled by the preceding bracket in the second line of Eq. ( 6).
The existence of that zero h * ≡ F −1 (0) of F , and the analyticity required for a power series expansion for F , the Higgs potential V, etc., become necessary requirements for a HEFT Lagrangian density to be expressible as a SMEFT.Our findings, in agreement with [10], can be abstracted as follows.
1.The F function in the HEFT Lagrangian density of Eq. ( 3) F (h * ) = 0 must have a double zero; S U(2) × S U( 2) is a good global symmetry there.
2. There, its second derivative must satisfy 3. All odd derivatives vanish, The restrictions over F at the symmetric zero h * (h = −v in the SM) that guarantee the existence of a SMEFT are theoretically illuminating but not too practical, since our measurements are taken around the vacuum h = 0.

Translating the geometrical conditions into constraints over the accessible HEFT parameter space
A key contribution of our earlier work has been to translate those conditions into restrictions over the a i at the physical vacuum h = 0. To do it, we match two series expansions, the one around h = 0 (setting a 0 := 1 and taking h normalized to v, so that v = 1) and the one around h * , expanding in terms of a * j = F ( j) (h * )/ j!, By matching the two expansions around the two different points it is easy to read off the coefficients a * j that encode the conditions over F in terms of the a i , accessible to experiment.The resulting correlations among the HEFT conditions are given in Table 1.The relation between the three lowest-order coefficients involving one, two and three h bosons, (respectively a 1 , a 2 , a 3 ) is shown in Figure 1.Finally, the numerical intervals for these HEFT parameters a i currently consistent with experimental data are given in Table 2.
We have next examined a couple of other HEFT multiplicative form-factor functions that incorporate powers of h to further operators of the electroweak Lagrangian; our second example here is the nonderivative V(H) Higgs-potential, accessible at "low" √ s as the absence of derivatives minimizes the number of momentum powers.The traditional SM form written in terms of h acquires additional non-renormalizable couplings in HEFT, organized in a power-series expansion with v 3 = 1, v 4 = 1/4 and v n≥5 = 0 in the SM.For the applicability of SMEFT, the coefficients of this series must satisfy the constraints listed in Table 3 and Figure 2. From the ATLAS [17] ,  [11]) introduces the correction Δa 1 ∝ c H that appears in the first row Table 1.Correlations among the a i HEFT coefficients necessary for SMEFT to exist, at the orders Λ −2 and Λ −4 , given in terms of Δa 1 := a 1 − 2 = 2a − 2 and Δa 2 := a 2 − 1 = b − 1 (so that all entries in the table vanish in the SM, with all equalities becoming 0 = 0).Notice that the r.h.s. of each identity in the second column shows the O(Λ −4 ) corrections to the relations of the first column.The third one assumes the perturbativity of the SMEFT expansion.

Correlations
Correlations Λ −4 Assuming accurate at order Λ −2 accurate at order Λ −4 SMEFT perturbativity Correlations for the HEFT coefficients from Table 1 that need to be satisfied for SMEFT to be a valid EFT of the electroweak symmetry breaking sector.The solid diagonal is the correlation of order Λ −2 , that becomes broadened as the indicated band (greenish online) at order Λ −4 .
Table 2. Bounds on the a i HEFT coefficients that need to be satisfied for SMEFT to exist that can be Taylor-expanded around the physical h = 0 vacuum according to (in the SM c 1 = 1 and c i≥2 = 0 simplify).The correlations among these coefficients induced by SMEFT at order 1/Λ 2 are also given, in the last row of Table 3 and in Figure 3.
Table 3.First two rows: correlations among the coefficients Δv 3 := v 3 − 1, Δv 4 := v 4 − 1/4, v 5 and v 6 of the HEFT Higgs potential expansion in Eq. ( 9) that need to hold, at O(1/Λ 2 ), if SMEFT is a valid description of the electroweak sector.The third row includes the leading correlations for the Yukawa G(h) function of Eq. ( 11), constraining c 2 and c 3 by c 1 and a 1 (from the correction to the value of the symmetric point h * ).We make use of current 95% confidence interval for the top Yukawa coupling c 1 ∈ [0.84, 1.22] [18].

Conclusion: Testing the embedding of SMEFT into HEFT at colliders
The SM is compatible, to date, with the measurements of the electroweak sector.It is a particular case of SMEFT and as a matter of course, all correlations discussed in this article among the coefficients of HEFT (many trivially) are satisfied.
Nevertheless, if the high-luminosity LHC of a future accelerator such as a new e − e + finds cracks in the SM, testing [21] whether both SMEFT and HEFT or only HEFT can be applied will become an issue.
The ωω → nh processes with n a varying number of Higgs particles would be key to disentangling the EWSBS.They give direct access to the a i coefficients of the flare function F , and hence to their correlations, as listed in Table 1. Figure 4 shows the typical Feynman diagrams describing the contact part of these processes that yield direct access to the a i coefficients.
. Contact diagrams which provide a i ; in a given physical process they are summed to t-channel exchanges between the two initial Goldstone bosons (wiggly lines) ω i , as given in [11].
For example, the BSM part of the two-to two-body amplitude is given by a single SMEFT coefficient, At this same order, the two-to three-body one receives a contribution from the a 3 coefficient and also t-channel exchanges; in HEFT, these are controlled by a 1 and a 2 , and are independent.However, in SMEFT all three coefficients are functions of the same c H and appear to cancel, T SMEFT ωω→hhh (s) = 0 + O(Λ −4 ).We have superimposed the SMEFT line (red online) which represents the O(1/Λ 2 ) correlation between these two parameters.
This shows that current data is much more sensitive to a 1 than a 2 , but the gentle sloping almost parallel to the SMEFT slice through that HEFT parameter space can also signal that the measurement is more sensitive to the separation from SMEFT than to the coefficient of SMEFT itself.This might be due to the fact that c H only brings an overall renormalization of h and no additional derivatives, so it does not change the shape of any kinematic distributions, it is a parameter of more difficult access.The SM (and therefore SMEFT too) is seen to comfortably sit within the 1 − σ band, in good agreement with the data.
Current experimental constraints are not really significant for n ≥ 2 and it will be challenging to obtain information on the coefficients for higher number of Higgs bosons n.Better reconstruction techniques at the high-luminosity LHC run but especially a future high-energy collider (either hadronic or muonic) will hopefully improve the situation.
Likewise, to exploit the correlation in Figure 3, an experimental determination of the t t → hh coupling c 2 needs to be undertaken.If it would be measured, SMEFT can be tested by comparing to c 2 ∈ [−0.27, 0.35] (where uncertainties have been added linearly).Similar measurements in the Yukawa sector, with an increasing number of Higgs bosons, would lead to further tests of the SMEFT framework.Similar considerations apply to the Higgs potential V(h).
To conclude, while other authors have also proposed methods to try to distinguish SMEFT from HEFT [23], we believe that our work offers clear criteria that depend only on the availability of sufficiently precise experimental data, and eventually higher order EFT computations [24] to match that precision.

7 4 a 3 == 1 6 a 5 Figure 1 .
Figure1.Correlations for the HEFT coefficients from Table1that need to be satisfied for SMEFT to be a valid EFT of the electroweak symmetry breaking sector.The solid diagonal is the correlation of order Λ −2 , that becomes broadened as the indicated band (greenish online) at order Λ −4 .

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Figure5displays the currently allowed experimental band (highlighted in yellow) for the first two coefficients, a 1 and a 2 (in the popular κ notation in which the experimental cross section is scaled to the SM one, these are κ V and κ 2V )1 .

Figure 5 .
Figure 5. Experimental CMS exclusion plot in the a 1 , a 2 plane (reproduced from[22] under the Creative Commons BY 4.0 license.)We have superimposed the SMEFT slice through this parameter space, that is seen to run parallel to one of the branches of the experimental exclusion band.