New Physics Scale from Higgs Observables with Effective Dimension-6 Operators

No matter what the scale of new physics is, deviations from the Standard Model for the Higgs observables will indicate the existence of such a scale. We consider effective six dimensional operators, and their effects on the Higgs productions and decays to estimate this new scale. We analyze and identify the parameter space consistent with known properties of the Higgs boson using recent Run II results from ATLAS and CMS experiments corresponding to $\sim$ 37 fb$^{-1}$ of data. We then calculate the $\bar{t} t h$ productions , as well as double Higgs production at the LHC using the effective couplings, and show that these can be much different than those predicted by the Standard Model, for a wide region of allowed parameters space. These predictions can be tested in the current or the future runs of the LHC. We find that the data are consistent with the existence of a new physics scale as low as 500 GeV for a significant region of parameter space of this six-dimensional couplings with these new physics effects at the LHC. We also find that for some region of the parameter space, di-Higgs production can be much larger than that predicted by the Standard Model giving rise to the prospect of its observation even in the current run II of the LHC.


I. INTRODUCTION
There have been several major discoveries in the past few decades culminating with the observation of the Higgs boson in 2012 [1,2]. This is a tremendous success of the SM. However, as most of us agree, SM can not be the whole story. The Higgs production in various modes and its decays into various final states so far agrees with the SM. But uncertainties with the SM predictions still remain in some of the observables of these measurements. This encourages us to venture into the possibility of a new physics scale that might be estimated from the uncertainty in these measurements. Also, using this approach, we might be able to make predictions which can be tested at the LHC. With this aim in mind, we consider the effect of a selected set of dimension six operators relevant for the Higgs Physics, in addition to the contribution from the SM. The dimension six operators related to the Higgs physics can be introduced both in the strong sector, as well as in the electroweak sector. Such operators will make extra contributions for the Higgs productions, as well as for its various decay modes. In the most general case, for the effective dimension six operators, there are many operators, and involve large number of parameters. In order to reduce the number of parameters, we only consider a selected set of such operators in the gauge sector (both strong and electroweak (EW)), as well as in the Yukawa sector. In particular, we include only those operators which are responsible for larger effects, and do not affect the constraints from the EW precision tests in a significant way.
The effective field theory provides a model independent framework for interpreting precision measurements connecting to specific UV models systematically [3]. Constraints on these operators have been derived from electroweak (EW) precision measurements [4][5][6], Higgs sector measurements [7][8][9][10] and from the triple gauge couplings [11,12]. Using EW data, global fits incorporating various searches have been performed in [13]. Subsequently fits have been performed including Higgs sector constraints [14][15][16][17]. In this context, di-Higgs production also has been studied here [18][19][20][21]. When the Standard Model is considered as an effective low-energy theory, higher dimensional interaction terms appear in the Lagrangian. Dimension-six terms have been enumerated [22,23] and there are 15 + 19 + 25 = 59 independent operators (barring flavour structure and Hermitian conjugations). However, many of these operators affect processes that are well measured, e.g. flavor physics or electroweak precision observables set strong constraints on subsets of those operators. Some of them are also not relevant for the Higgs physics observables, which is the main emphasis of this work. Here, we focus on the effective operators that focus on the Higgs physics, and nothing else. This is in the spirit of reference [24]. At the LHC, SM Higgs boson (h) can be produced 1 significantly via gluon gluon fusion (ggF), vector boson fusion (VBF), associated production with W and Z bosons (Vh) or in association with tt (tth). Due to insertion of the dimension-6 terms, SM Higgs production as well as decay branching ratios can be largely affected in these production modes. (1) In the single Higgs production , the most important is the coupling of the gluon pairs to the Higgs boson. Here we have the contribution from the SM dimension-4 operators contributing via the top quark loop. There may exist effective dimension 6 operator (contact interaction) emerging from new physics contributing to this production. (2) The Yukawa coupling of the top quark to the Higgs boson is most important in single Higgs production. Here also, there may exist dimension-6 operator (in addition to the dimension 4 present in the SM) emerging again from the new physics. This will also affect the tth production, as well as the double Higgs productions, which are of great importance in the upcoming LHC runs. (3) In the production of the Higgs boson in association with W or Z, the important contribution of dimension-6 operator will be the hZZ or hW W couplings, which will further effect the decays of the Higgs to W W * and ZZ * . Thus, in addition to the contribution from the usual SM, the contribution of the effective dimension six opera-tors will be important here. (4) The dominant decay mode of the Higgs boson is to bb, the branching ratio being 60%. Thus the dimension-6 contribution to the Yukawa coupling of the Higgs to the bottom pairs will also be very important to look for a new physics scale in the Higgs observables. (5) We have also included dimension-6 operator in the Higgs potential. This has the largest effect on our results on the di-Higgs productions, since it changes the effective triple Higgs coupling in a major way. Using the above five criteria, we narrow down our analysis to include five new parameters and these are g (6) , y (6) t , y (6) b , y (6) g , λ (6) and the new physics scale , M. We have done the analysis also including dimension-6 tau Yukawa term y (6) τ . As the branching ratio of the Higgs in the τ τ mode is 6%, it does not significantly affect the major Higgs observables and hence, the phenomenology we are concentrating. We ignore its contribution for rest of our analysis. A complete list of all effective dimension-6 operators can be found in [22,23].
With these above assumptions, we first identify the parameter space consistent with the Higgs observables and then we find two important results. (1) The tth coupling can be much larger or smaller than that predicted by the SM, and thus giving rise to significantly different rate of tth productions. (2) Double Higgs productions can be much larger than that predicted by the SM.
Very recently, the CMS collaboration has reported a search for the production of a Standard Model (SM) Higgs boson in association with a top quark pair (tth) at the LHC Run-2 and a best fit tth yield of 1.5 ± 0.5 times the SM prediction with an observed significance of 3.3σ [25], whereas ATLAS reported limit is 1.8 ± 0.7 [26] on tth production. ATLAS and CMS reported signal strength values are consistent with the SM. However, the central values of the signal strength µ tth is significantly different from one. There are several literatures [27] attempting to explain the issue for the enhanced tth production. As we shall see, in our framework, the signal strength µ tth can be as large as 2.4 and also as low as 0.5. There are still large uncertainties in the tth measurements. If any significant deviation (enhancement or suppression) arises in tth production rate at the LHC, this is the best model independent approach to explain the scenario. On the other hand, ATLAS and CMS collaborations have reported the new results on di-Higgs boson searches [28][29][30][31][32] looking at the different final states (bbγγ, bbτ + τ − , bbbb and bbW + W − ), using 36 fb −1 data from Run II of LHC at 13 TeV. No signal has been observed and the stringent limit of 646 fb on the di-Higgs production cross section is reported [28][29][30][31][32]. In SM, hh production cross-section is about 33.45 fb. After considering effective dimension six couplings, according to our analysis, the di-Higgs production can be as large as about ∼ 636 fb, which is 19 times of the SM predicted cross-section for some region of the six dimensional parameter space. If nature does realize this parameter space, di-Higgs production may be observable even at the current run 2 of the LHC as more data are accumulated.
The paper is organized as follows: In Sec. II, we discuss the formalism and analyze the dimension-6 operators. Thereafter in Sec. III, we perform the numerical simulations for collider signatures. Finally we conclude.

II. FORMALISM
Our gauge symmetry is the same as the SM. We are introducing a selected set of additional dimension six operators which can affect the Higgs observables in a major way. These operators are all invariant under the SM gauge symmetry.
• EW Yukawa sector: We have included the dimension-6 terms for third generation fermions only. For simplicity, we have included only the flavor diagonal dimension six Yukawa couplings. Similarly, we can extend it for first and second generation fermions also. But, since we are interested in new physics affecting Higgs rates in a major way, we ignore the negligible effects originating from dimension-6 Yukawa terms for first and second generation fermions. We will also ignore the dimension six operator for the τ lepton. The Higgs branching ratio to τ pair is very small 6%, and its inclusion does not affect the phenomenology we are concentrating.
• Strong sector: This operator will contribute to the Higgs production , as well as its decay to two gluons. g (6) is an unknown parameter, and M is the new physics scale. This operator (the contact term) will significantly contribute, in addition to the SM contribution via the top quark loop, in single Higgs production via gluon gluon fusion process.
• EW gauge sector: where the coupling y (6) g is an arbitrary coefficient 2 . There are several other dimension six operators which we neglect. The reason is that they do not contribute in a significant way to the processes we are emphasizing on this work, and some of them, if the coefficients are not very small, may mess up the EW precision test. We discuss briefly the effect of this operator above for the processes of interest. This operator contributes to the decays of h → W W , ZZ as well as to the production through VBF and associated Higgs production with W or Z boson.
• Scalar Potential: This operator will modify the Higgs trilinear coupling, and hence, contribute significantly to the di-Higgs production.
Note that in Eq.3, when we put the VEV of the Higgs boson , this operator modifies [57] the Higgs (Throughout our analysis, we use the convention H = 0 h+v √ 2 in unitary gauge). Hence, we need to redefine the Higgs field by dividing out with the factor to get the canonically normalized form for the kinetic term 1 2 ∂ µ h∂ µ h. This modifies the usual couplings of the Higgs field to the gauge bosons , the fermions and the Higgs bosons as given below.
Loop functions used in this paper are defined as follows: For a Higgs mass below the kinematic threshold of the loop particle, m h < 2 m loop , we have . We now calculate the partial decay widths for various SM Higgs decay modes : where the partial decay widths in the SM can be found in [33].

III. COLLIDER PHENOMENOLOGY
In this section, we study the collider phenomenology of the Higgs sector. In particular, we discuss the possibility if the effective dimension-6 operators within this framework can explain the significant deviation in tth production cross section, as recently indicated by CMS [25] and ATLAS collaboration [26], along with the other Higgs boson properties. We also want to investigate if the di-Higgs production may be observable at the current or future runs of the LHC.
To start this effort, we first numerically analyze the effects of dimension-6 terms on the tth production as well as the signal strengths of Higgs boson decay modes for h → γγ, W W, ZZ, bb, ττ , Zγ. Then, we identify a parameter space which is consistent with both the recent ATLAS and CMS results on the LHC Run-1 and Run-2 (37 f b −1 ) data. Then remaining within the allowed parameter space, we analyze the possible signals, such as the enhanced di-Higgs boson production that may be observable at the current or future run of the LHC. The relevant parameter space of this model is spanned by the three new dimension-6 Yukawa terms, dimension-6 term from electroweak gauge sector, dimension-6 term from strong sector, dimension-6 term from scalar potential and the mass of the new physics scale : y (6) t , y (6) b , y (6) g , g (6) , λ (6) , M In the LHC Higgs observable analysis 3 [34], the searches for Higgs boson at ATLAS and CMS can give strong bounds on these free parameters. The signal strength µ, defined as the ratio of the measured Higgs boson rate to its SM prediction, is used to characterize the Higgs boson yields and it is given by : Here σ i (i = ggF, V BF, W h, Zh, tth) and BR f (f = ZZ , W W , γγ, τ + τ − , bb, µ + µ − ) are respectively the SM Higgs production cross section for different production mechanism (i → h) and the branching fraction for different decay modes of SM Higgs (h → f ).
The ATLAS and CMS run 1 data are combined and analyzed using the signal strength formalism and the results are presented in [34]. Recently, ATLAS and CMS collaborations have updated the results [28] on Higgs searches based on 37 f b −1 data at 13 TeV LHC. The individual analyses examine a specific Higgs boson decay mode, with categories related to the various production processes and they are h → γγ [36][37][38][39], h → ZZ [40][41][42][43], h → W W [44][45][46], h → τ τ [47,48], h → bb [49,50] and h → Zγ [51,52]. Throughout our study, we have used the most updated ATLAS and CMS reported results on 125 GeV Higgs boson searches to impose constraints on signal strengths for various decay modes at 95% confidence level and which is summarized in Table I.
For our analysis, we adopt the following strategy. 3 In our analysis, we employ the center value of the Higgs boson mass m h = 125.09 GeV [34] and the center value of the combination of Tevatron and LHC measurements of the top quark mass mt = 173.34 in GeV [35].    (1) First, we introduce dimension-6 operator in the Yukawa sector and try to explore whether any new physics effect (enhanced/suppressed couplings of Higgs to fermions) can be achieved satisfying all Higgs physics constraints and try to identify the six dimensional parameter space where these effects can arise.
(2) Then we introduce dimension-6 operator in the EW gauge sector and discuss its effect following the previous effects from the Yukawa sector.
(3) After that we introduce dimension-6 term in strong sector and analyze both individual and combined effects of all of these dimension-6 operators and discuss the new physics effects.
(4) Then, we introduce dimension-6 operator in the scalar potential and analyze its effect in di-Higgs production.
(5)Finally, we discuss about two correlated new physics signatures : enhanced (or suppressed) tth and enhanced hh production.
Since the gauge structure of the SM has been very well established from the precision measurements, as mentioned above, we first concentrate on the Yukawa sector, in particular, the effects coming from the six dimensional Yukawa couplings for the third generation fermions. The top and bottom Yukawas (y (6) t and y (6) b ) play key roles in Higgs observable. The top Yukawa dictates the production of SM Higgs mostly, whereas the bottom Yukawa guides the branching ratio for different decay modes of SM Higgs h. Since the partial decay width for h → bb mostly contributes ∼ 58% to the total Higgs decay width, any slight deviation in bottom Yukawa will change the total decay width and hence the branching ratio to other decay modes. We analyze the full parameter space of extra Yukawa terms and new physics scale affecting the SM Higgs physics and impose constraints from the signal strength limits [cf. Table I] for various decay modes (γγ, τ τ, bb, ZZ , W W ) at 95% confidence level. The effect is displayed in Fig.1. The white shaded region simultaneously satisfies all the experimental constraints. Since y (6) τ has no significant contribution to the total decay width of SM Higgs compared to y (6) b , as mentioned before, we have ignored y (6) τ for our analysis regarding the effect of dimension six operators. It does not affect the phenomenology we are concentrating.
Next, we evaluate the signal strength µ tth (= κ 2 t ) for the production of SM Higgs associated with the top quark pair. Upper left segment of Fig. 1 shows the contour plot of µ tth in {y (6) t , M } plane for a fixed value of y      Fig. 1 clearly indicates that within this framework, tth can be produced up to 2 times of the SM predicted cross-section at the LHC satisfying all the current experimental constraints from 125 GeV Higgs boson searches while we allow a variation of y (6) t between -3 to 3. On the other hand, tth production rate can also be as low as 0.5 times weaker than the SM predicted value. This enhanced or suppressed tth production can be the new physics signature and it can be tested at the LHC. We mention that, although SM Higgs h is resonantly produced in gluon gluon fusion via triangular loop circulated by top quarks mainly, there is small effect (∼ 7%) due to the bottom    quark circulated loop. When bottom Yukawa comes up with negative sign, its effect becomes larger (15%) and we consider that effect too. Due to the different interference pattern between Yukawas (y (6) t , y (6) b ) in production as well as in decay modes, these plots are not symmetric about the central axes. We have also calculated the signal strength for Zγ channel and which is consistent with the available experimental data [51,52]. The signal strength in Zγ channel can be achieved from 0.6 to 1.5 satisfying all the constraints. There are models beyond the SM which predict this type of anomalous Yukawa couplings of the physical Higgs boson, such as the two Higgs doublet model [53,54]. Recently, enhanced tth production and flavor constraints are extensively studied in most general 2HDM [54]. Although we analyze in an effective operator approach, the effect of anomalous Yukawa couplings due to dimension-6 terms is reflected in anomalous Yukawa couplings of SM Higgs due to mixing between two Higgs in 2HDM [54]. Now, we introduce dimension 6 operator in EW gauge sector and analyze its effect on Higgs observable. We found that the dimension-6 term, y (6) g in EW gauge sector, is less influential than the dimension-6 terms in Yukawa and strong sectors. In SM Higgs production via ggF process, y (6) g plays no role, whereas branching ratio for h → W W, ZZ can be modified due to inclusion of y (6) g . Fig. 2 depicts the constraints from the signal strength limits [cf. Table I] for various decay modes (γγ, τ τ, bb, ZZ , W W ) at 95% confidence level in {y (6) t , y (6) g } plane. We choose y   [21] contributing to double Higgs production via gluon fusion. The first two diagrams are present in the SM, while the next three arise due to dimension 6 operators. expected and as can be seen from Fig. 2 that as bottom Yukawa y gets larger value to enhance overall bbh coupling, y (6) g has to have larger value to satisfy the constraints from Higgs observables. This is due to the fact that, whenever y (6) b is large, the partial decay width for h → bb mode gets enhanced and hence, total decay width becomes larger suppressing branching ratio for h → W W, ZZ decay modes. Since y (6) g has no impact on production via ggF process, y (6) g has to be larger to enhance the partial decay width for h → W W, ZZ decay modes making branching ratio almost unaffected to satisfy the correct signal strength limits on ZZ, WW channels. From upper left segment of Fig. 2, we can see that if dimension-6 terms in Yukawa sector are not introduced and only the effect of y (6) g is considered, we can still get enhanced tth production rate which is almost 1.3 times of the SM predicted value. After inclusion of y (6) t and y (6) b , this effect can be much larger and the signal strength for tth production can become as large as 2.4 and as low as 0.5. It is important to mention that whenever tth production is getting enhanced making single Higgs production rate via ggF process larger, overall branching ratios for h → W W or h → ZZ modes has to be suppressed to satisfy correct limits. This also indirectly suppresses the Higgs production in VBF, Wh and Zh processes. Our scenario predicts enhanced tth production and simultaneously suppressed production of SM Higgs boson in VBF, Wh or Zh processes and this can be tested in the upcoming runs of the LHC. However, there are still large uncertainties in these channels [cf. Next, we introduce the dimension-6 term (g (6) ) in the strong sector and investigate its effect. Deviation in di-Higgs production compared to the SM can be one of the new physics effect due to this term. The di-Higgs boson production has drawn a lot of attentions since it is the golden channel to test the EW symmetry breaking mechanism. Since the SM Higgs boson (h) does not carry any color, they are produced in pair through the triangle loop and box loop in SM. The di-Higgs production rate in the SM is small mainly due to the large destructive interference between the triangle and box loop diagrams. At the LHC with a center of mass energy of 13 TeV, the production cross section is about 33.45 fb, which can not be measured owing to the small branching ratio of the Higgs boson decay and large SM backgrounds. The detailed study of SM di-Higgs production can be found in ref. [55]. However, t , g (6) } plane (top right), {y (6) b , g (6) } plane (bottom left) and {y (6) g , g (6) } plane (bottom right) from the signal strength limits [cf. Table I] for various decay modes of SM Higgs (γγ (yellow), τ τ (cyan), bb (green), ZZ (red), W W (blue)) at 95% confidence level. The white shaded region simultaneously satisfies all the experimental constraints.
in new physics models, the di-Higgs production crosssection can significantly deviate from the SM value. Due to insertion of the dimension-6 term in strong sector, there will be additional diagrams contributing to the di-Higgs production in addition to the SM contribution and as shown in Fig. 3. Also, change in SM tth and hhh couplings could give a significant deviation on di-Higgs production cross-section. These two effects could enhance the di-Higgs production and make it testable at the LHC. Therefore, it is important to study how large can the cross section of the double Higgs boson production be considering all the constraints from the single Higgs boson measurements. The bbγγ final state is particularly promising for this search, as it benefits from the clean diphoton signal due to high m γγ resolution and the large branching fraction of the h → bb decay (∼ 58%). We consider the signal strength relative to the SM expectation µ hh as µ hh = σ(pp→hh) N ewP hysics σ(pp→hh) SM . First, we turn off all the dimension 6 operators (in Yukawa sector or EW gauge sector) and explore the effect of g (6) only. This is shown in upper left segment of Fig. 4. We find that to satisfy the constraints from Higgs observables, either g (6) has to be very small ∼ 0 or new physics scale has to be very large. For an example, g (6) can be as large as 0.06 and as low as -0.06 for the new physics scale, M, to be 2 TeV. Since g (6) is responsible for both single Higgs and di-Higgs boson production simultaneously, dimension-6 term (g (6) ) is highly constrained to give large di-Higgs production. For three of the benchmark points (BP1 and BP2), noted in upper left segment of Fig. 4, µ hh and µ tth is almost 1 and there is no significant deviation from SM prediction. Then, we add the con- tribution from dimension-6 Yukawa terms and we get a large region of the parameter space which is consistent with the Higgs observables and also gives significant deviation in tth and hh production. Due to the g (6) term, there will be two dominant processes for single Higgs production via ggF mode, one is due to the triangular loop circulated by top quark and the other one due to contact interaction term (ggh) and there will be large interference between these two diagrams. Upper right segment of Fig. 4 depicts the constraints in {y (6) t , g (6) } plane from the signal strength limits [cf. Table I] for various decay modes of SM Higgs at 95% confidence level. It is clear that when y (6) t gets positive values, g (6) prefers negative values to compensate the overall enhancement effect in single Higgs production and vice versa. For two of the benchmark points (BP3 and BP4), the signal strength µ tth becomes 2.0 and 0.7 and di-Higgs production cross-section becomes 64 fb and 41.8 fb respectively. Similarly, Lower left segment of Fig. 4 depicts the constraints in {y (6) b , g (6) } plane from the signal strength limits. Here we have fixed the value of y (6) t (=1) and new physics scale M (=500 GeV). In the survived parameter space, we choose three benchmark points (BP5 and BP6 as noted in this fig.) and calculate the tth and hh production rate. For benchmark points (BP5 and BP6), µ tth equals 1.55 and di-Higgs production cross-sections are 31 fb and 81 fb respectively. Similarly, Lower Right segment of Fig. 4 shows the constraints from the signal strength limits in {y (6) g , g (6) } plane, where we have kept a fixed value of , y For two of the benchmark points (BP7 and BP8), signal strengths (µ tth ) become 1.3 and 1.6 and di-Higgs production cross-sections become 221 fb and 52 fb respectively. As we already mentioned, since dimension 6 term in the strong sector, responsible for di-Higgs production, is not decoupled from the term responsible for the single Higgs production, di-Higgs production rate can not be enormously large, but it can be as large as 6 times of the SM predicted cross-section. Now, we try to emphasize on the determination of the mass scale where these dimension-6 operators are generated and which is consistent with the measurement of Higgs observable. We mention that the contribution of any effective operator is only sensitive to the ratio g ef f ective /M 2 , and hence, new physics scale M is not observable without extra assumptions on the strength of the couplings g ef f ective {y (6) t,b,g , g (6) , λ (6) }. In order to set limit on the new physics scale M, we have assumed g ef f ective to be less than 3.5 to satisfy the perturbativity constraint. In Fig. 5 On the other hand, if we turn off dimension-6 term in strong sector and in EW gauge sector, the new physics scale can be much lower (3.7 TeV), setting the dimension-6 term in yukawa sector ∼ O(1). Now we numerically scan the whole parameter space and we find that for a judicious choice of parameter space (g (6) = −0.01, y , the new physics scale M can be as low as 478 GeV satisfying all the Higgs physics constraints and giving new physics effect of enhanced tth production. The reason is that there is a negative interference effect between two diagrams contributing to the single Higgs production in ggF process (one is due to effective ggh coupling via triangular loop circulated by top quark and other one is the contact interaction term ggh due to dimension-6 operator), if g (6) has negative values. Hence, enhanced tth coupling compensate that factor satisfying all the Higgs constraints and as a result, we can get enhanced tth production. This scenario can be realized, if any colored particle contributes to the triangular loop in addition to the top and bottom quarks.
We now clarify some points regarding the mass scale, and the limitation used for the six dimensional couplings. Current LHC Higgs observables data are in agreement with the SM, so in principle all the 6dimensional couplings can be zero. In that case, it is not possible to say anything about the scale of new physics. However, the Higgs observables still have large errors, and hence gives the possibility of the existence of new physics. The questions we have addressed is whether in this effective coupling parameter space, there are regions which are allowed by the data, and allow low scale of new physics as well as giving some new physics signatures such as enhanced tth and hh predictions. For example, regarding the new physics scale, we want to mean that the new physics scale can be as low as 478 GeV making consistent with Higgs properties and also giving associated new physics signals like enhanced tth or hh predictions which can be testable at the current or upcoming run of LHC. Regarding the restriction on the effective couplings, since we consider the lowest order contributions, the higher order contributions will no longer be small, if the values of the couplings exceeds the perturbativity limit. This gives a reasonable justification to our assumption.
Next, we analyze the new physics contributions of the dimension-6 operator in the Higgs potential which contributes to the cubic Higgs coupling. Due to the addition of the effective dimension six operator in the Higgs potential, the effective triple Higgs coupling is modified significantly as shown in Eq. 22. As a result, this has the most major effect on the di-Higgs production at the LHC. We take two set of benchmark points which allow enhanced tth production rate (1.5 times of the SM predicted value) at the LHC making consistent with the Higgs properties. It is quite interesting to see from Fig. 6 that we can get the signal strength µ hh as big as 19 which means that the di-Higgs production cross-section can be as big as 636 fb which is 19 times of the SM predicted cross-section. We mention that the di-higgs production cross-section can be even larger than 636 fb for a certain region of parameter space as we can see from Fig. 6. But, ATLAS and CMS collaborations have analyzed and reported the new results on di-Higgs boson searches [28-32] looking at the different final states (bbγγ, bbτ + τ − , bbbb and bbW + W − ), using 36 fb −1 data from Run II of LHC at 13 TeV. Due to non-observation of any signal, the stringent limit of 636 fb on di-Higgs production cross section is reported [28][29][30][31][32]. The black meshed zone in Fig. 6 is excluded from this current di-Higgs searches. If LHC luminosity is upgraded to 3 ab −1 , SM like double Higgs production (33.45 fb) can be observed with 3.6σ significance [62]. On the other hand, in our scenario, the enhanced di-Higgs production can be even sensitive to the 50 fb −1 LHC luminosity which is close to the data set currently analyzed. We think this a very interesting scenario which simultaneously provides a testable smoking gun signal for the di-Higgs production and enhanced tth production at the LHC. The future hadron-hadron circular collider (FCC-hh) or the super proton-proton collider (SppC), designed to operate at the energy of 100 TeV, can easily probe most of the parameter space in our scenario through the hh pair production [59][60][61][62]. As mentioned, the di-Higgs production in some sets of the six dimensional parameter space can be large enough to be observable even in this run of the LHC.

IV. CONCLUSION
In this work, we have made an investigation on the effect of the effective dimension six operators for the single Higgs productions, and the corresponding µ tth , as well as di-Higgs signals at the LHC. Since the number of the effective dimension six operators are too many, we have made a judicious choice of few operators which has the maximum impact for these observable. Using the experimental data at the LHC, we have analyzed in some detail the effects of these operators, how large or small the µ tth , and di-Higgs signals can be, and how small the new physics scale can be satisfying all the available experimental constraints. We find the the µ tth signal can be as large as two times of that in the SM, while the di-higgs production cross section can be as large as 19 times of that in the SM at the 13 TeV LHC with a new physics scale, M equal to 478 GeV. These predictions can be tested as more data accumulates at the current and the future runs at the LHC. The results presented here can be taken as an initial guide in the exploration of the enhanced tth and hh signal at the LHC via dimension-6 operators.