Effective Field Theory of Integrating out Sfermions in the MSSM: Complete One-Loop Analysis

We apply the covariant derivative expansion of the Coleman-Weinberg potential to the sfermion sector in the minimal supersymmetric standard model, matching it to the relevant dimension-6 operators in the standard model effective field theory at one-loop level. Emphasis is paid to nondegenerate large soft supersymmetry breaking mass squares, and the most general analytical Wilson coefficients are obtained for all pure bosonic dimension-6 operators. In addition to the non-logarithmic contributions, they generally have another logarithmic contributions. Various numerical results are shown, in particular the constraints in the large $X_t$ branch reproducing the $125$~GeV Higgs mass can be pushed to high values to almost completely probe the low stop mass region at the future FCC-ee experiment, even given the Higgs mass calculation uncertainty.


I. INTRODUCTION
The first task of next generation colliders such as the ILC, the CEPC and the FCC-ee is precision measurement, of Higgs physics as well as Z-pole physics and tri-gauge boson physics and so on. If there is new physics, it is expected to show up first as corrections to the standard model (SM) processes, subject to such precision measurements. The corrections can be naturally sorted by dimension-6 operators in the SM effective field theory (EFT), which has a total of 59 independent operators for one family of fermions as a complete basis [1]. Precision measurements can be translated into a set of constraints on (part of) the operators.
The operator level fitting process is (UV) model independent. On the other hand, in literature there are many popular new physics models with different motivations and merits. Previously the matching of new physics to the model independent operator constraints are mostly electroweak precision test (EWPT) and Higgs cross sections/decay branching ratios. For the most important subset of pure bosonic operators and at one-loop level which is usually leading with occasional exceptions, a method called the covariant derivative expansion (CDE) [2,3] has greatly facilitate the matching procedure, with advantage of being complete for all operators simultaneously, model independence in computation process, and giving analytical results.
It is an industry to scan for UV models and match them to the SM EFT [2][3][4][5][6], but before that some generalizations of the CDE technique need to be made. In addition to the generalization to fermionic degree of freedom (DOF) which has extra gamma matrices related contributions [3,6], for realistic model parameters the original assumption of degeneracy of large scales is too simplified to be dropped. Since the CDE algebra has already been formulated in matrix basis, it's not a hard generalization to use nondegenerate large scales, and use heavily the integration of Feynman parameterization at the textbook level after taking traces.
Here we do sample calculations for the sfermion sector of the minimal supersymmetric standard model (MSSM). We expand the Coleman-Weinberg (CW) potential with covariant derivatives, matching it to a maximal set of bosonic operators, getting their Wilson coefficients. In addition to the nondegeneracy of soft mass squares we keep the small bottom/tau Yukawa couplings, and the differences between the squark sector and the slepton sector are accounted by different SU (3) c representations and U (1) Y hypercharges. The analytical results at one-loop level should be complete and realistic to use for various purposes.
A lot of works have been done in the indirect precision constraints of the squark sector of the MSSM, including the one-loop level Feynman diagram calculation. To provide some new useful numerics here we consider the MSSM Higgs mass constraint in detail, in particular we solve the mixing term X t for the required SM like Higgs mass. There are in general two solutions for a required SM like Higgs mass, and we find the precision constraints are especially effective for the large X t branch.
This paper is organized as follows. In Section II, we briefly review the dimension-6 operator basis and the bosonic CDE formulism, then we discuss the structure needed for large mass nondegeneracy. Next in Section III we show the sfermion sector input to the CDE formulism, and then the main analytical results. Section IV is devoted to various numerical applications of the analytical results. We conclude in Section V. At last, Appendix A provides a list of useful auxiliary formulas for the integration of Feynman parameters.

Symbol Operator expression
Symbol Operator expression Symbol Operator expression The operator basis we use is listed in Table I, which includes only Higgs and electroweak gauge boson fields. The basis is natural in sense that it directly captures all the possibilities generated by the CDE formulism, before removing any redundant operators.
The bosonic CDE formulism given in [6] can be summarized as the dimension-6 terms generated by the one-loop integration where In Eq. (1) n B is the DOF for each entry, being 1 for a real bosonic DOF and 2 for a complex bosonic DOF. The dimension-2 auxiliary number u is introduced as a trick to regularize orders of commutators of ∂ ∂p s acting on (p 2 − M 2 )s [7]. A Wick rotation is performed and subscript "E" indicates Euclidean. The CDE assumes in the CW potential the double derivatives (indicated by ′ ) to the beyond SM fields acting on the Lagrangian potential terms can be decomposed as V ′′ = M 2 + δV ′′ , where M 2 is some large constant squared mass term which is irrelevant to Higgs vacuum expectation value (VEV), and δV ′′ on the other hand captures the spacetime dependent part (namely the SM Higgs) of potential terms, with not only the physical Higgs boson but the Nambu-Goldstone modes before electroweak symmetry breaking. At last˜generally indicates a Baker-Campbell-Hausdorff expansion with covariant derivatives and ∂ ∂p s, which can be understand as introducing spacetime dependence in the momentum space [8] (and the covariant generalization in gauge field ∂ → D = ∂ − igA is given in [9,10], in which theG terms arise).
Note that the δṼ ′′ andG are all matrices, with each entry coupling to a gauge DOF of new particle to be integrated out. So is the large spacetime independent mass term M 2 , which in all known models is a diagonal matrix (if M 2 is not diagonal the new physics has intrinsic mixing irrelevant to the SM, such a model seems difficult to get motivated). This directly shows the way of generalizing the CDE formulism with nondegenerate large mass parameters, namely treating 1/(p 2 E + M 2 + u) as a diagonal matrix (p 2 E + M 2 + u) −1 and doing the multiplication with full respect to their matrix nature and non-commutativity with the (δṼ ′′ +G)s. The ∂ ∂p action on any (p 2 E + M 2 + u) −1 factor (as well as the factor p in the first term ofG) will give a commutative p number on the numerator and introduce no ambiguity. Eventually the overall trace operation reduces the matrix to numbers.
The later jobs such as the textbook trick of Feynman parameterization for each term, the dimensional regularization/reduction integration over d d p E and the integration over du (with the the boundary u → ∞ dropped as MS subtraction) are quite tedious, but possible especially with a symbolic calculation tool such as the Mathematica.

III. THE SFERMION SECTOR
In the following we do the CDE matching for the squark sector explicitly. Without specifying the Y q , Y t , Y b values the slepton sector results can be correspondingly obtained, though one should keep in mind that the c GG vanishes and every other Wilson coefficient should be divided by a factor of 3 as the SU (3) c color multiplicity.
The CW potential matrix in the basis of (t L ,b L ,t R ,b R ) has large spacetime independent masses (suppressing the SU (3) c components) and the spacetime dependent terms can be written in a decomposition δV ′′ = δV ′′ which count the supersymmetric F-term, D-term and the trilinear X-term contributions respectively. Here the Yukawa couplings are defined as their SM values such as y t = √ 2m t /v = y MSSM t sin β with v = 246 GeV, and we have ignored any possible CP phases. Plugging them into Eq. (1) and collecting the dimension-6 operators as described in the appendix of [6], we tabulate the resulting Wilson coefficient for each operator in the following long table.

Non-logarithmic Contributions
Logarithmic Contributions Here we have a few comments: • In general every Wilson coefficient has a non-logarithmic contribution and logarithmic contribution, the latter of which is proportional to the logarithm of the soft breaking mass ratio. This is the general way the scales nondegeneracy comes. In that case the denominators of both contributions contain factors of certain powers of the nondegeneracy, with the logarithmic one higher by one in power, consistent with the behavior of the Feynman parameter integration listed in the appendix. It is straightforward to check that, while each contribution is singular in the degeneracy limit, the combination of the two are finite and reproducing an expansion around the degenerated results.
• For a hierarchy of the large scales, only the non-logarithmic contribution survives, the logarithmic contribution will always be more suppressed by one extra large scale square. It is straightforward to read them out, for example the leading contribution to c T is Note that in the second hierarchy even the leading contributions are suppressed by Mq.
• The symmetry between the up type and the down type is manifested in the top-bottom switching t ↔ b, which by definition is in some cases that the "down" type quantities also appear in a "up" type term. The SU (2) L couplings includes a t 3 induced flipping between the up and down sector, since they are the same for both squarks and sleptons we do not keep it explicitly like the hypercharge Y t (Y b ). Their effect is to flipping the sign of g 2 → −g 2 . And note the sign flip of the c W B , c B and c HB in the top-bottom switching.
• In addition to the above operators, there are "universal" contributions to the pure gauge boson operators. Since the new particle couplings to SM gauge group are always within a gauge representation so that share the same large scale, for example in our sfermion case the SU (2) L gauge bosons only couple to left hand sfermions so that only feel the Mq(Ml), and the other SU (3) c and U (1) Y gauge bosons couplings are block diagonal in the M 2 + δV ′′ matrix, they do not show the above nondegeneracy property. Here we adapt the results in [3] for completeness Only the operator O 3W is directly constrained by the tri-gauge boson precision experiments, and is generally expected to have a relatively low sensitivity. We do not include them in the following fit.
• In the following numerical works we will focus on the four most stringent constraints, but completely matching to EFT indeed has an advantage of providing much more information. For example, following [3] the hZγ, hW W , hZZ can also be calculated and constrained with the Wilson coefficients.
• The degenerated mass case result is obtained in [2,3]. The nondegenerate mass result for c GG , c W W , c BB and c W B is obtained in [4], however the latter three are in a different basis. Here it is straightforward to check that for the Higgs diphoton coupling the combination of terms proportional to y 2 t X 2 t from the latter three operators conspires to cancel the (M 2 q − M 2 t ) −1 and the logarithmic term, going back to a form consistent with the Higgs low energy theorem. The alternative Feynman diagram calculation dates back to [11].

IV. NUMERICAL CONSTRAINTS
The afore calculated Wilson coefficients can be used straightforwardly to transfer the model independent constraints to a set of constraints of the sfermion parameters. In the following we will ignore the subdominant slepton sector contribution, and in the squark sector we will ignore the much smaller bottom Yukawa couplings 1 , and for simplicity we further assume the right handed soft squark mass Mb = Mt which presumably will not make significant differences if relaxed. While the results in Table II depend on the two soft breaking parameters Mq and Mt, they can be translated to the physical masses of the two stop squarks, through mixing determined by diagonalization of stop mixing matrix Note that the mixing matrix is just the first and third rows and columns of the matrix in Eq. (4,5,6,7), while the Higgs components are replaced by their VEVs. We will ignore any loop correction to the above squark mass matrix. There are four most stringent operator level constraints in the SM EFT. Two of them are the EWPT Peskin-Takeuchi T and S parameters [12], and the other two correspond to two channels of Higgs production/decay coupled to gluon pairs and photon pairs, which are generated at one-loop level in the SM even in leading order. The four constraints in our operator basis of Table I where M hgg and M hγγ is the SM amplitude of the (looped) hgg and hγγ couplings. Note in the Higgs precision experiments usually measurements are not for ∆Γ hgg Γ SM hgg ( ∆Γ(hγγ) ΓSM(hγγ) ) or ∆BR BR but for ∆σ·BR σ·BR , and there are subdominant contributions induced by other operators beyond the interference of new physics and the SM amplitude, but for simplicity we will ignore all of them. The current best measurements and projected future sensitivities are listed in Table III  A lot of numerical fittings have already been performed in the literature [4,20,21]. In Fig. 1 we focus on the 2σ constraints of the T and S operators as well as the hgg coupling for comparison with [4] for different choices of X t , with nondegenerate Mq and Mt substituted by the physical stop masses Mt 1 and Mt 2 . The X t introduces some splitting between the two stop masses, so for nonzero X t a region with Mt 1 ≃ Mt 2 is inaccessible and shaded gray in the plots (also regions with very small Mt 1 and large Mt 2 ). We assumet 1 has larger mixing component of left handed stop andt 2 of right handed stop, so that the bottom right corner of each plot shows the constraints for Mq < Mt, while the Mq > Mt case is shown in the top left corner. Since the X t = 0 slice will not satisfy the SM like Higgs mass constraint of 122 GeV < M h < 128 GeV in the MSSM anyway (except for exponentially large soft mass) we will not show such result, and the three plots shown are for X t = 1 TeV, 3 TeV and 6Mt 1 Mt 2 respectively.
In all the three panels the most constraining operator arises between the T and hgg. On the Mq < Mt side a sizeable T is generated as the most stringent one, while on the other side T is suppressed and the Mq-Mt symmetric  hgg takes over, but the T parameter is still somehow competitive if Mt 2 is close to Mt 1 , especially for large X t and more aggressive experiments. In the first panel the constraints vanish at Mt 1 ≃ X t or Mt 2 ≃ X t of 1 TeV, that's the "blind spot" discussed in [20,21]. Except for the blind spot, the constraint for the heavier stops will naively extend to large values, that's another interesting feature of the indirect constraint method. We have checked that the constraints depend very weakly with tan β, which can be understood through the fact that it always comes in the cos 2β factor of the small D-term.
In supersymmetric theories the SM like Higgs mass is calculable, from the D-term tree level contribution and the large radiative corrections mainly from the top/stop sector. For stop masses of a few TeVs, the X t term needs to contribute significantly to tune the Higgs mass to the measured 125 GeV, which predicts the X t to be a considerable value with roughly the same scaling with stop masses, potentially contributing significant X 2 t and X 4 t terms of the Wilson coefficients. So the SM like Higgs mass imposes another constraint of the parameter region, and it is interesting to stick to that parameter slice. Note that there is still some uncertainty in calculating the SM like Higgs mass, especially in the low SUSY scale region, different code may give up to 3 GeV discrepancy. We use the FeynHiggs 2.11.2 [22] to calculate the MSSM Higgs mass (for an estimation of hierarchical stops two-loop contribution, see Eq. (5.3) of [23]), then solve for the X t values which are consistent with a SM like Higgs mass of 122 ∼ 128 GeV.
In Fig. 2 and 3 we show such numerical constraints for three future experiments and a range of SM like Higgs masses at tan β = 20 and decoupling large CP odd Higgs mass. Again we use the same notations and methods for the two cases of Mq < Mt and Mq > Mt, and the gray shaded theoretically inaccessible region. The green 1σ region and yellow 2σ region are determined by fitting to all the EWPT and Higgs precision observables in [13,[15][16][17][18] (including the hW W , hZZ and Higgs to fermion pairs expectations ignored in Table III), and four individual 2σ constraints corresponding to Table III are shown as well. For a fixed SM like Higgs mass value there are in general two X t solutions as shown respectively in the two sets of plots, and the most effective constraints are seen in the large X t branch. While the CEPC can probe most low stop mass (< 2 TeV) region except for two corners, the FCC-ee can completely cover the whole region, even with a 3 GeV SM like Higgs mass uncertainty. On the large X t branch one also need to worry about the possibility of other vacuum deeper than our electroweak symmetry breaking one with nonzero VEV of stops scalars, which may be induced by the large non-diagonal X t . We show the bound of Fig. 3 for comparison.
At last we check the tan β dependence, which has a large effect on the SM like Higgs mass. As can be seen from Fig. 4, smaller tan β will allow a smaller X t on the large X t branch, so that weaken the constraints compared with the large tan β case. Still we can see that even if tan β is as low as 5, the low stop mass (< 2 TeV) region on the large X t branch can be almost completely probed.  Table III) for fixed SM like Higgs mass value of M h = 122 GeV (the first row), M h = 125 GeV (the second row) and M h = 128 GeV (the third row). The 1σ and 2σ allowed region are green and yellow hatched. The individual 2σ constraint from T , S parameters and hgg, hγγ couplings are shown for each experiment again as blue, cyan, red and magenta curves respectively, if strong enough to be shown. There are still theoretically inaccessible region as hatched in gray. Here we choose tan β = 20, and the small Xt solution to reproduce the required SM like Higgs mass.

V. CONCLUSION
In this paper as an example of the CDE generalization, we perform the one-loop integration out of the sfermion sector in the MSSM, with full respect to all the coupling constants and nondegeneracy of the soft mass squares, and matching it to a basis of dimension-6 pure bosonic operators. Analytical expressions are given for each Wilson coefficient, with in general non-logarithmic contributions and logarithmic contributions.
Numerically in the language of EFT, the most constraining T parameter are taken into account in a general way, and comparison is made among all the most stringent operators. Assuming the SM like Higgs mass relation in the MSSM, the probed region for each future experiments are shown. In particular in the large X t branch the constraints can be pushed to very high values, and probably rule out the low scale stop sector, by precisions provided by, e.g., the FCC-ee experiment. 3, and the region below and on the left are allowed.  Table III) Here X i generally indicates some dimension-1 quantity, which could be the X t , X b terms in our sfermion sector, or the large scale itself in the vector-like fermion model. To see this is the case we should at first go through the following calculation steps. The standard textbook trick of Feynman parameterization allows the loop integration over d d p E , with dimensional regularization/reduction which makes no difference at one loop level. Then the integration over du can always be performed, with the the boundary u → ∞ dropped as MS subtraction. The first integration increase the dimension by 4 and the second integration by 2, and all the above categories eventually give correct dimension of −2, as required for the dimension-6 operators. Note that each p E on the numerator corresponds to a ∂ ∂p and so that a covariant derivative in the δṼ ′′ +G, checking with all possible combinatorics for each operator in Table I (see the appendix of [6]), the only p 4 E term arise from the O D calculation 2 and no term falls into the category of p 4 E / ni=6 (p 2 E + M 2 i + u) ni . 2 The field strength related p 4 E term such as in the combinatoric of 4 3! gp µ t a DρF a νµ ∂ ∂p ρ ∂ ∂p ν always cancels due to the antisymmetric property of F a νµ .
After finishing the loop integration and du, the Feynman parameter integration we actually need to do is F ((p 2 ) np , (M 2 1 ) n1 , (M 2 2 ) n2 , · · · ) = ( n i − 1)! (n i − 1)! (A2) Here we do not keep the factors from loop integration, and the n p and n 1 , n 2 , · · · corresponds to categories in Eq. (A1). In our model of sfermion there are at most three different large mass scales. For the case of two large mass squares we list them all