Double insertions of SMEFT operators in gluon fusion Higgs boson production

Deviations from the Standard Model (SM) can be parameterized in terms of the SM effective field theory (SMEFT), which is typically truncated at dimension-6. Including higher dimension operators -- as well as considering simultaneous insertions of multiple dimension-6 operators -- may be necessary in some processes, in order to correctly capture the properties of the underlying UV theory. As a step towards clarifying this in the Higgs boson production in gluon fusion process, we study double insertions of dimension-6 operators in the 1-loop virtual amplitude. We present needed Feynman rules up to $\mathcal{O}(1/\Lambda^4)$ and we numerically study the impact of various approximations to the $\mathcal{O}(1/\Lambda^4)$ expansion.


I. INTRODUCTION
Current measurements of LHC experiments are in excellent agreement with theoretical predictions, but with uncertainties at the O(5−20 %) level [1].As a result, the High Luminosity LHC program will be focussed on high precision measurements.It is expected that the experimental uncertainties will reduced to O(1 %) for many observables [2].This requires precise theoretical Standard Model (SM) predictions, but also precise computations in specific Beyond the Standard Model (BSM) scenarios to describe potentially emerging small non-SM signatures.A more general approach is also possible; BSM physics which contains no new light particles and which respects the SM gauge symmetries can be parameterized using the Standard Model effective field theory (SMEFT) [3].This consists of an expansion around the SM Lagrangian L SM in terms of an infinite tower of higher dimension operators, where Λ is chosen to be the scale of new physics, O d i are operators of dimension d, and C d i the corresponding dimensionless SMEFT Wilson coefficients (WC).Fits to the latter have been made using Higgs, di-boson, electroweak precision, and top data [4][5][6][7].Such analyses are usually done by terminating the series in Eq. ( 1) after dimension-6 operators.Yet, the need for precision calls for an investigation beyond O(1/Λ 2 ).At the next non-trivial order, this includes studying the impact of dimension-8 SMEFT operators, but also double insertions of dimension-6 operators [8][9][10][11][12][13][14][15][16].An amplitude, A i , for a lepton number conserving process can be parameterized in the SMEFT as a power series in 1/Λ 2 , where the α coefficients are process dependent.The terms proportional to C 6 j C 6 k /Λ 4 are the double insertions of interest here.The amplitude-squared corresponding to a cross section is then expanded generically as, If a coefficient is well constrained by data, it may be sufficient to retain only the O(1/Λ 2 ) contributions to observables.This is typically the case in fits to electroweak precision observables [17][18][19].However, for most of the SMEFT coefficients contributing to predictions for LHC observables, the O(1/Λ 4 ) terms play an important role.
Global fits [4][5][6][7] include the first term on the second line of Eq. ( 3) (required to make the cross sections positivedefinite), but the other terms of O(1/Λ 4 ) are more subtle.
For tree-level processes, the second term on the second line of Eq. (3) (which corresponds to a double insertion) is easily included [20,21] and can have important numerical effects [22].The dimension-8 contributions (first term on the third line of Eq. ( 3)) have been studied in only a few special cases and the numerical importance of these terms is not known in general [8][9][10]23].In the case where the new physics that generates the SMEFT coefficients corresponds to a strongly interacting theory, it has been argued that the dimension-8 contributions are small [24].
In the following, we present a preliminary investigation of the impact of double insertions on the inclusive gluon fusion Higgs boson production process.This production channel has recently been calculated in the SM to N 3 LO QCD [25][26][27].In the SMEFT, the NLO result with single insertions of dimension-6 operators is well known [28][29][30][31][32]. Gluon fusion Higgs production has also been calculated to all orders in v 2 /Λ 2 using the GeoSMEFT approach [33,34].Here, we present a study of the 1loop contributions to the gg → h amplitude including all terms of O 1/(16π 2 Λ 4 ) and we investigate the numerical effects of double insertions of a consistent subset of dimension-6 SMEFT operators.
The paper is organized as follows.Section II contains a brief description of the SMEFT to O 1/Λ 4 .The 1-loop calculation of gg → h to O 1/(16π 2 Λ 4 ) is presented in Section III, including the insertion of two dimension-6 operators in the 1-loop amplitude and the required counterterm for the gg → h process corresponding to the dimension-8 (φ † φ) 2 G A,µν G B µν operator.Numerical effects of the double insertions are investigated in Section IV, along with a discussion of the potential effects of neglected contributions.Finally, we conclude in Section V with a discussion of the path forward to a more complete study of the impact of O(1/Λ 4 ) effects.

II. SMEFT TO O Λ −4
We start by presenting the pieces of the dimension-6 SMEFT Lagrangian (in the Warsaw basis [35]) which are relevant for the calculation of the virtual 1-loop gg → h diagrams containing double insertions.All the remaining necessary terms of the Lagrangian can be found in Ref. [36].In the end of this section, we present the relationships up to O(1/Λ 4 ) between the original parameters of the Lagrangian and our input parameters [23].
We neglect finite contributions from dimension-8 terms.Although such contributions enter in the cross section at the same order as double insertions of dimension-6 operators, they can be treated separately, as they are not required to obtain a gauge-independent result.Yet, the dimension-8 operators are in general required to absorb ultraviolet (UV) divergences of O 1/Λ 4 .There is a single dimension-8 operator that can be used to this end [37,38], When renormalizing the theory, the counterterm δC G 2 φ 4 is generated from Eq. ( 4).Below, we present the result δC G 2 φ 4 using minimal subtraction.We work in minimal subtraction, which amounts to dropping all poles.A complete understanding of dimension-8 renormalization in the SMEFT, including fermionic operators, does not yet exist, although significant progress has been made in understanding the bosonic operators [39][40][41][42][43].

A. Lagrangian and field redefinitions
The relevant pieces of the dimension-6 SMEFT Lagrangian can be grouped into three terms, The first one is the Higgs Lagrangian, where φ represents the Higgs doublet, which we parametrize as Here, v T is the vacuum expectation value (vev) that minimizes the Higgs potential in the presence of the SMEFT operators, and h, φ 0 and φ + represent the Higgs, the neutral Goldstone, and the charged Goldstone boson fields, respectively.The second term in Eq. ( 5) is the QCD Lagrangian, with where g A µ is the gluon field.Finally, L fermions is the fermionic Lagrangian,

and we retain only the top quark contributions.
To ensure that all fields have canonical kinetic terms, we need to perform the following shifts, where with X h in Eq. (12a) defined as

B. Input Parameters
We choose as independent parameters where G F is the Fermi constant, α s is the strong coupling constant and M Z (M W ), M h and m t are the gauge boson, Higgs and top masses.
The expression for v T can be determined through the amplitude for muon decay, including double insertions of dimension-6 operators.Assuming flavor universality of the WCs which can be inverted to yield

+ 2C
(3) The parameters µ 2 and λ are fixed by the requirement that the coefficient of the Higgs tadpole contribution vanishes (i.e. that v T is the true vev) and that the mass of the Higgs field in the Lagrangian is given by M h .Using also Eq. ( 16), we find The top quark Yukawa coupling is determined by requiring that the mass of the top-quark field in Eq. ( 10) is given by m t , Finally, g 2 s can be related to 4πα s through the inverse transformation of Eq. (11c) and we find where we defined

III. CALCULATION
We now describe the 1-loop calculation of the gg → h amplitude to O 1/(16π 2 Λ 4 ) .The Feynman rules accurate to O(1/Λ 4 ) that are relevant for our calculation are given in Appendix B. Lorentz and gauge invariance imply that at any order, the amplitude for g A (p µ 1 )g B (p ν 2 ) → h must have the form, where, up to 1-loop,  2) and (6) also contribute with crossed initial states (not shown for compactness).with F 0 representing the tree-level SMEFT contribution, F V the virtual 1-loop amplitude and F CT the total counterterm.
The tree-level contribution is given by φl − 2C ll . ( F V is computed from the diagrams shown in Fig. 1, using the software FeynMaster [44][45][46][47].We use the true vev up to 1-loop order [48] and we work in the Param-eter Renormalized tadpole scheme [49].Analytic results for F V can be found in the auxiliary file submitted with this paper.Finally, F CT is determined by identifying the original parameters and fields in Eqs (4, 5) as bare parameters (with index "(0)") and by expanding them into renormalized quantities, where C X represents a generic WC.The expression for F CT is given in Appendix A. 1   This allows us to determine δC G 2 φ 4 by requiring Eq. ( 23) be free from divergences.We work in dimensional regularization, using D = 4 − 2ϵ for the spacetime dimension, and fix the counterterms of the WCs in the minimal subtraction scheme [50].We perform the calculation in two independent ways: i) we subtract known infrared (IR) poles using results of Ref. [51]; and ii) we use Package-X [52] and consider only UV poles.
It is sufficient to compute the counterterms in Eq. (A2) to order O(1/Λ 2 ), since Eq.(A2) is already O(1/Λ 2 ).δZ h and δZ g can be computed from the Higgs and gluon self energies at 1-loop, respectively; explicit expressions can be found in Appendix A. δG F is given by where the expressions for ∆r SM and ∆r EFT can be found in Appendix D of Ref. [53].The contributions from δC (3) φl and δC ll cancel when Eq. ( 26) is used in Eq. (A2).The contribution to δC φG of O(1/Λ 2 ) can be obtained from Refs [54]; we confirmed their result by requiring that Eq. 22 be finite to O(1/Λ 2 ) and present it in Eq. (A1).Combining these elements, we find the expression for δC G 2 φ 4 given in Eq. (A5).

IV. IMPACT OF DOUBLE INSERTIONS
To study the impact of double insertions on the 1-loop amplitude of the gluon fusion process, we compute the amplitude squared in two ways: i) we truncate the amplitude at O(1/Λ 2 ) and then compute the amplitude squared; ii) we compute the amplitude to O(1/Λ 4 ) and then truncate the amplitude squared at O(1/Λ 4 ).The first truncation is not sensitive to the double insertions of the dimension-6 operators, and we label it as "single".The second truncation is sensitive to the double insertions of SMEFT operators, and we label it as "double". 1As discussed in Section II, we ignore finite effects from dimension-8 operators (i.e.we set the renormalized WC C G 2 φ 4 to zero).We note that the latter is in fact a complete computation of the virtual amplitude up to O 1/Λ 4 at 1-loop, neglecting finite contributions from dimension-8 operators.Since the WC C φG contributes at tree-level, the double insertions proportional to C φG require the computation of 2-loop virtual graphs with single insertions of dimension-6 operators, along with 1-loop virtual graphs proportional to C φG to obtain an IR finite result.
As a first step in understanding the relevance of double insertions, we consider a scenario where C φG is generated at loop level and thus can be consistently set to zero after renormalization.This is a realistic scenario from a model building point of view.At tree-level, scalars, vector-like quarks, and vector particles in arbitrary representations that contribution to the dimension-6 SMEFT Lagragian do not generate C φG contributions [55].It is interesting to note that vector-like quarks generate C φG at 1-loop consistent with our assumption.When we set C φG = 0, there are no real corrections and we can study the numerical effects of the double insertions from the remaining operators using our finite results for the renormalized amplitude to construct a cross section normalized to the SM result. 2   For the numerical results reported below, we use M h = 125 GeV, M W = 80.377 GeV , M Z = 91.1876GeV, m t = 172 GeV, G F = 1.166 • 10 −5 GeV −2 and α s = 0.1179.
The renormalization scale µ is chosen to be equal to the Higgs mass M h .Finally, we write the virtual amplitude squared as, In the C φG = 0 limit that we are working in, Numerical results for a i and b ij in the 2 expansions at O(1/Λ4 ) are presented in Table I.
We first note that some contributions that contain C ll or C φl are present in the single but vanish in the double setup.From the Feynman diagrams shown in Fig. 1 it can be easily seen that these contributions are proportional to 1/(R 2 φ v 2 T ), which vanishes in the double expansion.Consequently, the functional dependence of the amplitude on these WCs in the two expansions is quite different; for example, we show this for the combination of C (3) φl and C tG in the upper plot in Fig. 2. In this figure we show the regions where |µ ggh − 1| is less than 5 %.For a given value of C (3) φl and C tG , the remaining coefficients C ll , C φ□ , C φD , and C tφ are varied over the region allowed by the 95% CL individual fits of Ref. [5]. 3 It is clear that the difference between the single and double insertion expansions has no phenomenological relevance, since the values of the parameters plotted are excluded by fits to Higgs data [4,5,7].We do not show it explicitly, but we have checked that the same conclusion holds for all other combinations that include C ll and/or C (3) φl . 2 We have explicitly checked the gauge independence of our results. 3Limits used in all figures for WCs not shown explicitly are −0.5 ≤ We also observe a non-trivial change in the coefficient of C tG and we show a fit in combination with C φ□ to the value of the SM amplitude squared in Fig. 2 (bottom).Also in this case, significant differences between single and double expansions only occur for values of the WCs far beyond current single parameter limits [5].
The biggest change is in the coefficient of C tG C tφ .For this combination of WCs, the allowed parameter space is available in Ref. [5] from 2-parameter fits to Higgs and Higgs plus top data at 95 % CL.In Fig. 3, we show these regions together with a fit to |µ ggh − 1| < 5%.The difference in the results for single and double expansions is small and demonstrates the power of including top data in the fits.While fits to Higgs data alone show a small sensitivity to the expansion, when top data is included with the Higgs data, there is again no difference between the two expansions in the region allowed by global fits. 4

V. CONCLUSIONS
We computed the 1-loop amplitude for the gluon fusion process gg → h including all contributions of dimension-6 operators up to O(1/(16π 2 Λ 4 )).This includes double insertions of dimension-6 operators and the relationships between parameters in the SMEFT Lagrangian and physical observables to this order.We derived the necessary Feynman rules that are valid up to O(1/Λ 4 ) and determined the required counterterm to obtain a UV finite result at this order.For our numerical studies, we considered the limit C φG = 0 which ensures that there are no infrared singularities.We note that this is a well motivated scenario, since in many BSM models C φG is only generated at 1-loop level.We then compared the gluon fusion cross section in different expansions up to O 1/Λ 4 and found that the impact of the double insertions is negligible for values of the WCs allowed by global fits and neglecting the unknown dimension-8 contributions.
An extension of this study including the effects of C φG and double insertions would require 2-loop virtual amplitudes with up to two insertions of dimension-6 SMEFT operators as well as real-virtual and double real emission contributions.We leave this exercise for future investigations.
Digital data associated with this research is contained in the auxiliary file attached to this paper.
and the O(1/Λ 2 ) contribution as δC 6 φG , The quantity F CT defined in Eq. 23 is given by The poles of δZ h and δZ g are respectively such that C tφ , (A3) C ll C tG − ϵ δC 8 φG . (A5)

Figure 1 .
Figure 1.Virtual 1-loop contributions to the gluon fusion to Higgs amplitude including contributions from both single and double insertions of dimension-6 SMEFT operators.Conventions used throughout the paper concerning 4-momenta, Lorentz indices and colour indices are shown in diagram (1).Note that diagrams (1), (2) and (6) also contribute with crossed initial states (not shown for compactness).

Figure 2 .
Figure2.Regions where |µ ggh −1| < 5% are shown for single insertions (squared blue) and double insertions (orange).The limits from global fits to individual operators at 95% CL are denoted by the black cross.[4,5,7].The WCs not shown are varied over values allowed by the 95% CL fits to individual coefficients of Ref.[4].

Figure 3 .
Figure 3. Allowed parameter space from a 2-parameter fit to Ctφ and CtG.Yellow (hashed) and green (fine hashed) ellipses show constraints from linear fits at 95% CL to Higgs data and Higgs plus top data respectively[5].Regions where |µ ggh − 1| < 5% are shown for single insertions (squared blue) and double insertions (orange).The WCs not shown are varied over values allowed by the 95% CL fits to individual coefficients of Ref.[4].

Table I .
(27)rical results for linear coefficients ai and coefficients bij of pairs of SMEFT WCs, c.f. Eq.(27).Results are shown with (third column) or without (second column) double insertions.In the fourth column we show the ratio of single coefficients over double coefficients.Ratios given as rational numbers are exact.Numerical values for physical parameters are reported in section IV.See text for further details.