Exploring dynamical CP violation induced baryogenesis by gravitational waves and colliders

By assuming a dynamical source of CP violation, the tension between sufficient CP violation for successful electroweak baryogenesis and strong constraints from current electric dipole moment measurements could be alleviated. We study how to explore such scenarios through gravitational wave detection, collider experiments, and their possible synergies with a well-studied example.


I. INTRODUCTION
Electroweak (EW) baryogenesis becomes a promising and testable mechanism both at particle colliders and gravitational wave (GW) detectors to explain the observed baryon asymmetry of the Universe (BAU), especially after the discovery of the 125 GeV Higgs boson at the LHC [1,2] and the first detection of GWs by Advanced LIGO [3]. The longstanding puzzle of BAU in particle cosmology is quantified by the baryon-to-photon ratio η B = n B /n γ = 5.8 − 6.6 × 10 −10 [4] at 95% confidence level (C.L.), which is determined from the data of the cosmic microwave background radiation or the big bang nucleosynthesis. It is well known that to generate the observed BAU, Sakharov's three conditions (baryon number violation, C and CP violation, and departure from thermal equilibrium or CPT violation) [5] need to be satisfied, and various baryogenesis mechanisms have been proposed [6]. Among them, EW baryogenesis [7][8][9] may potentially relate the nature of the Higgs boson and phase transition GWs. An important ingredient for successful EW baryogenesis is the existence of a strong first-order phase transition (SFOPT) which can achieve departure from thermal equilibrium. The lattice simulation shows that the 125 GeV Higgs boson is too heavy for efficient SFOPT [9], nevertheless, there exist already in the literature four types of extensions of the standard model (SM) Higgs sector to produce SFOPT [10]. Another important ingredient is sufficient source of CP violation, which is too weak in the SM. One needs to introduce large enough CP violation, which also needs to escape the severe constraints from the electric dipole moment (EDM) measurement.
Thus, in this work, we study the dynamic source of CP violation 1 , which depends on the cosmological evolution of a scalar field. For example, this can be realized by the two-step phase transition, where sufficient CP violation and SFOPT can be satisfied simultaneously to make the EW baryogenesis work. The studied scenario could explain the observed BAU while satisfying all the constraints from EDM measurement and collider data.
As a well-studied example, the SM is extended with a real scalar field S and a dimensionfive operator y t η Λ SQ LΦ t R + H.c. to provide the SFOPT and sufficient CP violation for EW baryogenesis, which was firstly proposed in Refs. [15,16]. This dimension-five operator actually appears in many composite models and this source of CP violation for BAU evolves 1 In recent years, inspiring works on the dynamical CP violation appears in Refs. [11][12][13][14].
with the scalar field S. At the very early universe, S = 0 2 , then the value evolves to S = σ through a second-order phase transition. The CP violating top quark Yukawa coupling is then obtained and can source the BAU in the following SFOPT [17]. After that, S evolves to zero again, and the CP violating top quark Yukawa coupling vanishes at tree level. This evolution of the coupling naturally avoids the strong constraints from the EDM measurements, and yields distinctive signals at hadron colliders and lepton colliders, such as the LHC, the Circular Electron-Positron Collider (CEPC) [18] , the International Linear Collider (ILC) [19], and the Future Circular Collider (FCC-ee) [20]. We discuss the constraints on the parameters of the effective Lagrangian from both particle physics experiments and cosmology. Since probing the nature of the EW phase transition or EW baryogenesis is one important scientific goal for fundamental physics after the discovery of the Higgs boson [18,21]. This EW baryogenesis scenario with dynamical CP violation should and could be probed by future colliders and help us to unravel the nature of the Higgs potential and the dynamics of EW symmetry breaking [18,21]. Especially, the collider signals when we include the dynamical source of CP violation are quite distinctive from the collider signals when only SFOPT is considered [22].
Thus, after considering the GW signals from SFOPT, we report on a joint analysis of observational signatures from the EW baryogenesis under our scenario, correlating the GW and collider signals. This type of two-step phase transition with its GW signals, and the EW baryogenesis in this scenario are well-studied in the previous study. In this work, we recalculate and describe this scenario from the dynamical CP violation perspective, and firstly investigate how to explore this scenario by collider signals and their correlations with the GW signals. The structure of the paper is as follows, in section II, we describe the effective model of the dynamical CP violation for successful baryogenesis. In section III, we discuss the dynamics of the phase transition in detail. In section IV, size of the dynamical CP violation and the BAU are estimated. In section V, the constraints and predictions from the EDM measurements and colliders are given. In section VI, we investigate the GW signal and its correlation to the collider signals. Finally, we conclude in section VII.

BARYOGENESIS
Based on the fact that sufficient source of CP violation for successful baryogenesis are typically severely constrained by EDM measurement, there is a possibility that the CP violating coupling depends on the cosmological evolution history. During the early universe, there exists large CP violation for successful baryogenesis. When the universe evolves to the current time, the source of CP violation evolves to zero at tree level. In this work, we study the CP violating Yukawa coupling which evolves from a sufficiently large value to a loop-suppressed small value at current time, by assuming it depends on a dynamical scalar field, i.e. the phase transition process can make the CP violating Yukawa coupling transit from a large value to zero at tree level. A well-studied example is the CP violating top Yukawa coupling scenario as proposed in Refs. [15,16]. Namely, there exists extra terms to the SM top quark Yukawa coupling which reads: where y t = √ 2m t /v is the SM top quark Yukawa coupling, η = a+ib is a complex parameter, Λ is the new physics (NP) scale, Φ is the SM Higgs doublet field, Q L is the third generation SU (2) L quark doublet, t R is the right-handed top quark and S is a real singlet scalar field beyond the SM. During the phase transition process in the early universe, the scalar field S acquires a VEV σ, then a sizable CP violating top Yukawa coupling can be obtained and contribute to the EW baryogenesis for BAU. After the phase transition finishes, the VEV of S vanishes and the Higgs field acquires a VEV v, meaning that the CP violating top quark Yukawa coupling vanishes at tree-level and evades the strong EDM constraints naturally.
More generally, we can assume that the top quark Yukawa coupling depends on a scalar field or its VEV, which changes during the cosmological evolution. For the phase transition case, the CP violating top Yukawa coupling simply depends on the phase transition dynamics.
We take the n = 1 as a simple but representative example to show how it gives successful baryogenesis and how it is detected with the interplay of collider experiments and gravitational wave detectors. The corresponding effective Lagrangian [15,16,38] can be written as: Based on this Lagrangian, we study the collider constraints, predictions, GW signals and EDM constraints in detail. For simplicity, we choose the default values as a = b = 1, namely, η = 1 + i. We can of course rescale η and Λ simultaneously to keep the effective field theory valid up the interested energy scales. It is not necessary to consider the domain wall problem here as shown in Refs. [15,39]. The coefficients µ 2 , λ, κ are assumed to be positive in this work. It worth noticing that we just use the same Lagrangian in Ref. [15,16] to realize the two-step phase transition and do not consider other possible operators, which may make the two-step phase transition difficult to realize. If we neglect the dimension-five operator, there is a Z 2 symmetry in the potential, which makes the two-step phase transition more available.
For the above effective Lagrangian, a second-order and first-order phase transition could occur in orders. Firstly, a second-order phase transition happens, the scalar field S acquire a VEV and the dimension-five operator generates a sizable CP violating top Yukawa coupling, which provides the source of CP violation needed for BAU. Secondly, SFOPT occurs when the vacuum transits from (0, S ) to ( Φ , 0). After the two-step phase transition 3 , the VEV of S vanishes at tree level, which naturally avoids the electron and neutron EDM constraints, and the dimension-five operator induces the interaction term − mt Λ (aStt + ibStγ 5 t), which produces abundant collider phenomenology at the LHC and future lepton colliders, such as CEPC, ILC, FCC-ee.

III. PHASE TRANSITION DYNAMICS
In this section we discuss the phase transition dynamics, which provides the necessary conditions for EW baryogenesis and produces detectable GWs during a SFOPT. To study phase transition dynamics, we use the the methods in Refs. [54][55][56], and write the effective potential as a function of spatially homogeneous background scalar fields, i.e., S(x) → σ(x) and Φ(x) → 1 √ 2 (0, H(x)) T . Thus, the full finite-temperature effective potential up to oneloop level can be written as where V tree (H, σ) is the tree-level potential at zero temperature as defined below in Eq.(4), (H, σ, T ) is the one-loop thermal corrections including the daisy resummation, and is the Coleman-Weinberg potential at zero temperature.
The tree-level potential at zero temperature in Eq. (3) is We can see that there are four distinct extremal points 4 , and requiring only two global λ , the two minima at tree level degenerates, and if becomes the only global minimum. The SFOPT can be realized easily since the potential barrier height appears at tree level and is not suppressed by loops or thermal factors. Based on these properties, it is convenient to parameterize λ and µ 2 as Later on we use the full effective potential in Eq.(3) to numerically calculate the phase transition dynamics and GW signal, but first we can qualitatively understand the phase transition dynamics using the tree-level potential and leading-order temperature correction, since the full one-loop effective potential only sightly modifies the values of the parameter space. Thus, using the high-temperature expansion up to leading order O(T 2 ), the effective thermal potential in Eq.(3) can be approximated as with where the SM U (1) gauge coupling g = 0.34972, SU (2) gauge coupling g = 0.65294 and top quark Yukawa y t = 0.99561 [57]. The terms D H T 2 and D σ T 2 represent the leading-order thermal corrections to the fields of H and σ, respectively. Here, the contributions from the dimension-five operator are omitted as similarly argued and dealt with in Refs. [15,16,38].
Thus, the washout parameter can be approximated as Numerically, the allowed parameter space for large washout parameter v(T c )/T c is shown in Fig.1 for κ = 1.0 and κ = 2.0 cases, respectively. We use the washout parameter to  qualitatively see the SFOPT favored parameter region. Generally speaking, larger washout parameter represents stronger first-order phase transition. For the quantitative determination of the SFOPT, we need to calculate the nucleation temperature T N as discussed below.
Eventually, some typical parameter sets that give two-step phase transition (the phase tran- where SFOPT occurs during the second step) and produce SFOPT are shown in Tab. I.  We now describe the methods used to obtain the values in Tab. I. We first introduce two important quantities α andβ, which can precisely describe the dynamical properties of the phase transition [58]. The key quantity to obtain α andβ is the bubble nucleation rate per unit volume per unit time Γ = Γ 0 exp[−S E ], where S E (T ) = S 3 (T )/T is the Euclidean action and Γ 0 ∝ T 4 . And S 3 is the three dimensional Euclidean action, which can be expressed as To calculate the nucleation rate, we need to obtain the profiles of the two scalar fields.
Here, we need to deal with phase transition dynamics involving two fields using the method in Refs. [59][60][61] by choosing a path ϕ(t) = (H(t), σ(t)) that connects the initial and final vacuum. Then, we can get the bounce solution ϕ b by solving the following differential bounce equation with the boundary conditions After we obtain the nucleation rate, the parameterβ can be defined as Another important quantity α parametrizes the ratio between the false-vacuum energy density ε(T N ) and the thermal energy density ρ rad (T N ) in the symmetric phase at the nucleation temperature T N . It is defined as where the thermal energy density is given by ρ rad (T ) = (π 2 /30)g * (T )T 4 with g * being the relativistic degrees of freedom in the thermal plasma. And ε(T ) is defined as where ϕ B (T ) is the VEV of the broken phase minimum at temperature T .
To calculate the parameters α andβ, it is necessary to determine the nucleation temperature T N where the nucleation rate per Hubble volume per Hubble time reaches unity as Γ/H 4 | T =T N 1, here H is the Hubble parameter. Thus the condition can be simplified as Using the method above, we are able to numerically calculate T N , α andβ. For the following discussion, we pick two benchmark sets which can produce two-step phase transition and SFOPT, and the parameters α,β, m S , and T N are listed in Table II, Usually, a larger α and smallerβ give stronger first-order phase transition and stronger GWs.  [15,16]. After the first step of phase transition, S field obtains a VEV, and then the CP violating top quark Yukawa coupling is obtained. Thus, during the SFOPT, the top quark in the bubble wall has a spatially varying complex mass, which is given by [15,16] where z is the coordinate perpendicular to the bubble wall. The CP violating phase Θ will provide the necessary CP violation for the BAU. Taking the transport equations in Refs. [16,[62][63][64], one can estimate the BAU as whereṽ b is the relative velocity between the bubble wall and plasma front in the deflagration case (the bubble wall velocity v b is smaller than the sound velocity c s = √ 3/3 ∼ 0.57 in the plasma). Here, µ B L is the left-handed baryon chemical potential, Γ sph is the sphaleron rate, and f sph is a function that turns off quickly in the broken phase. The position dependent Θ(z) can provide the CP violating source in the transport equations and contribute to net left-handed baryon µ B L . Here, we chooseṽ b ∼ 0.2, which is smaller than the bubble wall velocity v b [34]. It is because the EW baryogenesis usually favors the deflagration bubble case, and the BAU depends on the relative velocity between the bubble wall and the plasma front. Thus, we have reasonably small relative velocityṽ b , which is favored by the EW baryogenesis to guarantee sufficient diffusion process in front of the bubble wall, and large enough bubble wall velocity v b to produce stronger phase transition GWs (In the deflagration case, a larger bubble wall velocity gives stronger GWs [33,34]). We take the default value of the bubble wall velocity v b ∼ 0.5, which is reasonable since the difference betweenṽ b and v b can be large for SFOPT with large washout parameter in the deflagration case.
From the roughly numerical estimation, we see that the observed BAU can be obtained as long as ∆σ/Λ ∼ 0.1 − 0.3, where ∆σ is the change of σ during the phase transition and is determined by the phase transition dynamics. For the two benchmark sets given in Tab. II, the needed Λ should be around 1 TeV. Larger Λ gives smaller baryon density, and smaller Λ produces over density. The exact calculation of η B would need improvements from the non-perturbative dynamics of the phase transition, and higher order calculations. In the following, we discuss how to explore the parameters from the GWs, EDM data and collider data, which offer accurate constraints or predictions.

MENTS
After the SM Higgs obtains a VEV v at the end of the SFOPT, the SM Higgs doublet field can be expanded around the VEV as Φ(x) → 1 √ 2 (0, v + H(x)) T . Thus, the interaction between S and top quark becomes Detailed calculations can be referred in the Appendix.
Another effect that needs to be considered here is the one-loop mixing effect between the particle S and H. In our tree level Lagrangian, there is no mixing term between the S and H, but such mixing term will be induced by the top quark loop. Considering the one-loop correction, the (squared) mass matrix terms of the scalar fields can be written as: Those corrections are: The calculation details can also be found in the Appendix. This mass matrix can be diago- From now on we neglect the subscript " phy " and all the fields and masses are physical by default. with The numerical results are shown in Fig. 3, where the region below the dotted blue lines are excluded by the EDM experiments.
But through our calculation, we find that limits from current neutron and mercury EDM experiments are weaker than electron EDM. However, the expected future neutron EDM measurement [77] with a much enhanced precision could have the capability to detect this type of CP violation.

B. Collider direct search and Higgs data
Production and decay patterns of both the Higgs boson and S particle are modified by the loop induced mixing, see Fig. 2 for an illustration. In Fig. 2, the mass gap around 125 GeV comes from the mass mixing term ∆m 2 HS = a 3m 4 t 2π 2 Λv , which is fixed by Λ rather than a free parameter. This feature is shown more clearly in Fig. 3, where the mass region between  Λv ; regions below dotted blue lines have been excluded by EDM measurement; regions below dashed red lines have been excluded by collider scalar searches and Higgs data. In the left plot, regions below dash dotted olive lines can be observed from ZS production at 5 ab −1 CEPC with a C.L. higher than 5σ. In the right plot, we show the ratio σ(HZ) σ SM (HZ) with purple dash dotted contour lines. In the plot we set a = b = 1, and κ = 2.
black dashed lines is forbidden by this mass mixing term. Fortran code eHDECAY [78][79][80][81] is used here to do precise calculations. Fig. 2 shows that the branching ratios of S is quite SM-like near the Higgs mass due to a large mixing with H. While in the region away from 125 GeV, i.e. the region with a smaller mixing, top loop induced γγ and gg channels are enhanced. Our scenario get constraints from the SM and non-SM Higgs searches in various channels at LEP, Tevatron, and LHC experiments, and the observed 125 GeV Higgs signal strengths. We apply cross section upper limits on relevant channels from these collider searches as included in the package HiggsBounds-5 [82][83][84][85]. Besides, we use the framework implemented in HiggsSignals-2 [86] to perform a Higgs data fitting. Experimental data from 7 + 8 TeV ATLAS and CMS combined Higgs measurements [87], and two 13 TeV Higgs measurements with a higher precision [88,89] Fig. 3 also shows that the limits from the colliders are much stronger than EDM in our scenario.

C. Collider signals in the future
There are several channels in our model that may produce observable signals at high luminosity LHC. For example: pp → S → jj, pp → S → γγ, pp → S → ZZ * → l + l − l + l − , and pp → SH. Light di-jet resonance search suffers from huge QCD background [92], and remains difficult even at future LHC run. While due to a much less background, previous di-photon and 4-lepton search results, as shown earlier, already excluded some parameter space of our model. So di-photon and 4-lepton channels would continue to exclude parameter space or give the first hint of signals as the LHC continues accumulating data. In Tab. III we give the production cross sections times branching ratios at 14 TeV LHC of these two channels for the two benchmark points. A concrete analysis relies on detailed simulation and dedicated final state studies, which is beyond the scope of the current paper, and could be interesting future work. The pp → SH process is mostly through the one loop gg → SH contribution, and an exact calculation at the leading order is performed. There are three types of Feynman diagrams as shown in Fig.4. The second (tri4) and third (box) diagrams are proportional to the contribution of the dimension-five effective operator, and thus interfere destructively according to the low energy theorem [93]. Their contributions nearly cancel out at low energy scale, just above the m S + m H threshold. The first diagram (tris), however, is proportional to κ, and contributes dominantly when κ becomes large.
The leading order total cross section of pp → SH is around 25 fb with κ = 2, m S = 115 GeV, Λ = 1 TeV and √ s = 14 TeV, and roughly scales with κ 2 for even larger κ values.
We illustrate the separate contributions to the leading order differential cross section as a function of m SH from the different diagrams in Fig. 5. As seen in the figure, the total contribution is indeed dominated by the "tris", or κ term at low energy scale, and dominated by the "tri4+box", or the dimension-five term proportional to η at high energy scale. Thus by probing this process, we obtain complimentary information on the model parameters compared to the di-photon and 4-lepton search. Multiplied by a k-factor of around two for typical gg to scalar(s) processes, this gg → SH process becomes comparable or even larger than the SM pp → HH total cross section, which is about 40 fb at 14 TeV. In our scenario, the S decays dominantly to a pair of gluons and by a small fraction to a pair of photons.
A study that is similar to the di-Higgs search at the high luminosity LHC, while with one scalar at a different mass, in the γγbb and jjbb final states, becomes another interesting future work. The pp → SH study would benefit from a future hadron collider with higher center of mass energy, for example at a 27 TeV HE-LHC and a 100 TeV FCC-hh, SPPC.
Very similar to the study of di-Higgs production, the cross section of the gg → SH increase from 25 fb to 92 fb and 770 fb at 27 TeV and 100 TeV center of mass energy respectively, with our leading order calculation.
Note here that the scalar S is larger than half the Higgs mass in our benchmark scenarios, and can not be produced or probed through Higgs decay, the 1 2 κS 2 Φ 2 term with large κ could as well be indirectly probed at the off-shell Higgs region, for example as discussed in Ref. [94].  with CEPC group report [95]. The signal is a scalar-strahlung process e + e − → Z * → ZS, with a total cross section [96] σ(e + e − → ZS) Here v e = −1 + 4s 2 w , a e = −1, andλ = (s 2 + m 4 GeV, s w is sine of the Weinberg angle. Shape of the signal peak is estimated and obtained by a rescaling and shifting from the fitted SM Higgs shape. Fig. 6 shows the recoil mass distribution. Then we count the number of SM background and signal events in [m S − 1GeV, m S + 1GeV] mass window, noted as B and S respectively. So the significance can be written as S/ √ B + 2 B 2 , with = 1.0% being the dominant systematic uncertainty.
The region with S/ √ B + 2 B 2 > 5 can be observed at 5 ab −1 CEPC with a significance higher than 5σ, and the curve is shown as well in Fig. 3. It is clear from Fig. 3 that there is a large currently-allowed parameter space that can be covered by High Luminosity LHC or CEPC. We are especially sensitive to regions with m S closer to 125 GeV, which corresponds to an increasing S-H mixing.
In addition, S-H mixing could also be detected through a potentially visible deviation of σ(e + e − → HZ) measurement, which can be an indirect signal of our model [98]. Further more, wave function renormalization of the Higgs field which comes from 1 2 κS 2 (Φ † Φ) reduces σ(e + e − → HZ) by a global rescaling factor: As a results, the total cross section σ(e + e − → HZ) will be rescaled by a factor of |O 22 | 2 Z.
Quoting from the proposed precision of CEPC with 5 ab −1 data, it is capable to measure the inclusive HZ cross section to about 1.0% sensitivity. In Fig. 3 we draw contour lines for color schemes. Title "tris"(magenta) represents the cross section considering only the first type of diagrams as in Fig.4, "tri4"(red) represents the second, and "box"(green) represents the last. The blue curve shows the cross section including the "tri4" and the "box" contributions. The black curve is the total cross section including all diagrams and their interference, which is dominated by the "tris", or κ term at low energy scale, and by the dimension-five η term and interference at high energy scale.
different ratio σ(HZ) σ SM (HZ) . Unlike the nearly symmetric shape the direct search lines, σ(HZ) shows a larger deviation in the lighter m S region. This effect comes from the Higgs field wave function renormalization, which is more sensitive to a lighter m S . This indirect detection method shows good sensitivity, and gives complimentary information on the model parameters in addition to our direct search. Luminosity is taken at 5 ab −1 following the CEPC report. The y-axis represents the number of events per bin, which is taken to be 0.25 GeV.

COLLIDER SIGNALS
The key point to predict the phase transition GW signal is to calculate the two parameters α andβ from the finite temperature effective potential in Eq. (3) using the method described in Sec. III. The two parameters are related to the phase transition strength and the inverse of time duration, respectively. The GWs also depend on the energy efficiency factors λ i (i=col, turb, sw, denoting bubble collision, turbulence, and sound waves, respectively) and bubble wall velocity v b . For the GW spectrum from bubble collisions, we use the formulae from the envelope approximations [31,97]: at the peak frequencỹ The efficiency factor λ col is a function of α and v b , and we use the results for the deflagration case as obtained in Ref. [33]. As for GW spectrum from sound waves, numerical simulations give [35,37] with the peak frequencỹ The turbulence contribution to the GW spectrum is [32,99] with the peak frequencỹ We now show our numerical results of the total GW spectrum from the three contributions in the concerned scenario with the benchmark parameter sets. From Fig. 7, we can see that the GWs produced in this EW baryogenesis scenario can be detected marginally by LISA, BBO and certainly by U-DECIGO. We also show the corresponding CEPC cross sections as a double test on this scenario, and vice versa. For example taking benchmark set I, the GW spectrum is represented by the black line in Fig. 7, which can be detected by LISA and U-DECIGO. The black line also corresponds to 0.9339 σ SM (HZ) of the HZ cross section for e + e − → HZ process and 115 GeV recoil mass with 13.6 fb cross section for the e + e − → SZ process, which has a 5σ discovery potential with 5 ab −1 luminosity at CEPC. Other lepton colliders are also capable to detect this collider signals, such as ILC and FCC-ee. The

VII. CONCLUSION
We have studied the collider search and GW detection of the EW baryogenesis scenario with a dynamical source of CP violation realized by a two-step phase transition. The VEV of a new scalar field S evolves with the two-step phase transition, and provides both the SFOPT and sufficient CP violation at early universe. At current time, the VEV of S becomes zero at tree level, which makes it easy to evade the severe EDM constraints. Nevertheless, the loop-induced mixing between the scalars S and H can produce abundant collider signals.
We have shown the possible collider signals at future collider experiments, especially at the lepton colliders. Meanwhile, collider signals and GW surveys could cross check this EW baryogenesis scenario. As a by product, the discussion here suggests potentially interesting collider signals for additional generic light scalar searches near the Higgs mass. The analysis in this work may help to understand the origin of CP violation and EW baryogenesis, furthering the connection between cosmology and particle physics. More systematical study is left to our future study.

Acknowledgments
We deeply appreciate Ryusuke Jinno's comments concerning gravitation wave generation from phase transition, and Eibun Senaha's helpful discussion on the CP violation source in EW baryogenesis. We also thank Takumi Kuwahara for his useful suggestion on accounting for the EDM contribution, and Taehyun Jung on wave function renormalization. This work is supported by IBS under the project code IBS-R018-D1.
Appendix: Effective Lagrangian calculation through covariant derivative expansion Ccovariant derivative expansion (CDE) is a convenient method to calculate one-loop effective Lagrangian [65][66][67]. In this appendix we use CDE to calculate several most important operators in our work. Operators we calculated here are those connecting gluon or photon pairs to scalars and induced by top quark loop, which are most relevant to the phenomenology we want to study at hadron colliders. In order to make things clear and easy to check, we write down our calculation procedure in detail. For notation and convention, we follow Ref. [66].
The particle being integrated out here the is top quark, so the corresponding one-loop contribution to the effective action is: with P µ ≡ iD µ , and M is the bilinear coefficient of the top quark field: S eff,1-loop can thus be rewritten as: ≡ − i 2 Tr log −P 2 + m 2 t + U , where G µν = [D µ , D ν ],and σ µν = i 2 [γ µ , γ ν ]. After separating the covariant derivatives and the loop momentum, one-loop effective Lagrangian can be written as: Here dq ≡ d 4 q/(2π) 4 , ∆ ≡ 1/(q 2 − m 2 )(for our case, m = m t ), and: The trace "tr" here only acts on indices like the spin, generation, flavor, but not momentum.
Then ∆L eff,1-loop can be expanded by a series of integral: In most cases one does not need all the terms in Eqs. (36) and (37), and only the first few terms are important. In our case, the relevant terms we need to calculate contain at least two vector field strength G µν and at least one scalar S or H, and don't contain the derivatives of these field. So the sum of all the relevant terms, after loop momentum integral, are: Then we calculate the trace and express the effective Lagrangian by S, H, gluon field strength G a µν , and photon field strength F µν . In order to make the calculation clear and get a concise expression, we introduce some useful notations.
Now we define: Using these replacement, U can be rewritten in the simple form: Then we can easily express those traces (here we only show those relevant terms): tr U G µν G µν ⊃ 4I A tr G µν G µν (50) tr Then the top-quark loop-induced effective coupling between vector pairs and the scalars can be obtained as: Calculation of scalar mass corrections is much easier. Scalar mass corrections come from terms with no field derivatives: Expand U and set renormalisation scale µ R as m t , we obtain: