Dr Paul-Henri Bauwens
email: bauwens.paulhenri@gmail.com

Matthieu Malatray

Gaspard Fournier

Frédéric Rongieras

Antoine Bertani

Risk Factors for Complications after Primary Intramedullary Nailing to Treat Tibial Shaft Fractures: A Cohort Study of 184 Consecutive Patients

Keywords: Tibia, fracture, intramedullary nail, complications, nonunion

Background: Intramedullary nailing is the standard of care for tibial shaft fractures. The risk factors for infectious and/or mechanical complications, notably non-union, remain incompletely understood. The objective of this study was to evaluate risk factors for complications, notably non-union. Hypothesis: Active smoking and an initial open wound are independent risk factors for complications.

INTRODUCTION

Tibial shaft fractures account for 36.7% of fractures of long bones and over 2% of all fractures [START_REF] Court-Brown | Epidemiology of adult fractures: A review[END_REF][START_REF] Papakostidis | Prevalence of complications of open tibial shaft fractures stratified as per the Gustilo-Anderson classification[END_REF]. Because the tibia is covered by only a thin layer of soft tissues, over 15% of tibial fractures are classified as open [START_REF] Papakostidis | Prevalence of complications of open tibial shaft fractures stratified as per the Gustilo-Anderson classification[END_REF]. Intramedullary nailing of the tibia is the standard of care for tibial fractures as it ensures rapid bone healing and an expedited recovery [START_REF] Court-Brown | Epidemiology of adult fractures: A review[END_REF][START_REF] Xia | A meta-analysis of reamed versus unreamed intramedullary nailing for the treatment of closed tibial fractures[END_REF][START_REF] Duan | Intramedullary nailing for tibial shaft fractures in adults[END_REF].

Nevertheless, leg injuries remain the source of a non-negligible number of complications, such as non-union, which may require further surgery. A meta-analysis by Duan et al. indicated that the frequency of tibial non-union after intramedullary nailing ranged from 3% to 48% [START_REF] Duan | Intramedullary nailing for tibial shaft fractures in adults[END_REF]. This variability is probably ascribable to differences in study populations, the diversity of post-operative measures and, potentially, to factors that remain poorly investigated and that relate to the patient, the fracture, and the management strategy.

Improved knowledge of the risk factors for complications would help to improve the information given to patients [START_REF] Jaeger | Antibiotics in trauma and orthopedic surgery -a primer of evidence-based recommendations[END_REF] and to implement preventive strategies in high-risk patients. These factors have not been thoroughly analysed in the literature.

The objective of our single-centre retrospective study was to evaluate the preoperative, intra-operative, and post-operative risk factors for complications after intramedullary nailing of a tibial shaft fracture. Our working hypothesis was that active smoking and an initial open wound were independent risk factors for complications, notably non-union.

MATERIALS AND METHODS

Patients

For this single-centre retrospective study, we identified consecutive patients who underwent surgery at a level I trauma centre between January 2013 and June 2018. The study protocol was approved by our institutional review board (N°20-68).

The inclusion criteria were an open or closed fracture of the tibial shaft treated by primary intramedullary nailing within 6 hours after the injury. The exclusion criteria were age younger than 15 years, previous internal fixation using another method, initial cast immobilisation, and follow-up shorter than 12 months. For intramedullary nailing, an S2 stainless steel nail was used in 164 patients and a T2 titanium nail in 20 patients (Stryker™, Mahwah, NJ, USA). Gradual reaming of the intramedullary cavity was performed routinely and locking was static. Weight-bearing was started in the immediate post-operative period if allowed by the quality of the reduction and the adherence of the patient. Follow-up visits were scheduled 21 days, 45 days, 3 months, 6 months, and 12 months at least after the operation. No change was made to our usual followup schedule for the study. When healing was insufficient after 3 months, unlocking was performed to allow dynamisation of the nail, as far as possible from the fracture site. In the absence of any complications, the material was removed 18 to 24 months after the initial surgery; this procedure was not studied here.

Operative technique

Evaluation

The potential risk factors selected for this study fell into three groups: pre-operative or extrinsic, intra-operative or intrinsic (quality of the reduction), and post-operative (early weight-bearing). We classified the energy of the trauma as low or high. High-energy traumas were traffic accidents, gunshot wounds, penetrating wounds from the outside in, and falls from heights [START_REF] Dubrana | Fractures ouvertes de jambe[END_REF]. Moderate-energy traumas were classified in the low-energy group. Followup was at least 12 months but ideally was continued until fracture healing was complete.

Fracture healing was considered complete when follow-up radiographs showed a callus involving at least three of the four cortices (Figure 1). The nature and timing of complications were recorded. Non-union was defined as failure of healing after 6 months with no changes in the callus between two radiographs taken at least 1 month apart (Figure 2). Computed tomography was performed to confirm the diagnosis. Malunion was defined as malalignment of more than 10° along any of the axes on the anteroposterior and lateral radiographs. To measure the residual fragment gap, the image was calibrated using Centricity PACS Software (General Electric Healthcare, Chicago, IL, USA). The fragment gap was defined as the largest distance measured between two bony fragments on the anteroposterior or lateral post-operative radiograph. Infection was defined as the presence of clinical signs (purulent discharge and/or evidence of skin inflammation) combined with an elevation of the C-reactive protein level and/or hyperleukocytosis and/or a positive bacteriological test, with (deep surgical-site infection) or without (septic non-union) bone healing. Delayed wound healing was defined as absence of skin healing after 21 days requiring specific and closely spaced wound care. Secondary dynamisation was not counted as a complication.

Statistical methods

The data were entered into an Excel ® (Microsoft Corporation, Redmond, WA, USA) spreadsheet then analysed using JMP Pro 14 software (JMP ® , Version JMP Pro 14.0, SAS Institute Inc., Cary, NC, USA; 1989-2019). A descriptive analysis was performed, followed by a univariate analysis using tests appropriate for the nature and distribution of the study variables to compare patients with and without complications then with and without nonunion. A multivariate analysis was then performed and the odds ratios (ORs) with their 95% confidence intervals (95%CIs) were computed. Values of p below 0.05 were considered significant.

RESULTS

Population

During the inclusion period, 322 tibial shaft fractures were managed consecutively at our centre. Among them, 182 were open and 140 were closed. Treatment was by external fixation in 127 patients, plate fixation in 4 patients, and a plaster cast in 4 patients (Table 1).

The remaining 187 patients were managed by primary intramedullary nailing (S2/T2 nail, Stryker™, Mahwah, NJ, USA) and were therefore included in our study. Among them, 3

(1.6%) patients were lost to follow-up and were therefore excluded from the analysis (Figure 3), which therefore included 184 patients. There were 133 males and 51 females with a mean age of 38.5±17.6 years (range, 15-91 years). Table 1 shows the distribution of the 62 open fractures according to the Gustilo classification: 38 fractures were Gustilo I, 18 were Gustilo II, and 6 were Gustilo 3A.

Complications

Complications developed in 28 (15.2%) patients. They consisted in non-union in 15 patients, delayed wound healing beyond 21 days in 6 patients, deep surgical-site infection in 2 patients, post-operative compartment syndrome on the day of surgery in 4 patients, and malunion with more than 10° of malalignment in valgus in 1 patient (Table 2). We categorised the complications into two groups, all complications combined and non-union.

Table 3 reports the management of the complications. Of the 28 complications, 23 (12.5%) required revision surgery. Of the 6 patients with delayed wound healing, only 1 required revision surgery in the operating theatre. No complications developed in the 6 patients with Gustilo 3A open fractures. None of the patients managed with primary intramedullary nailing required a wound coverage procedure such as split-thickness skin grafting.

Risk factors for all complications combined

Risk factors for non-union

Table 4 reports the data for the risk factors associated with non-union. Again, three independent significant risk factors were identified: smoking (OR, 5.1, 95%CI, 1.62-16.43), a residual interfragmentary gap >5 mm (OR, 4.92; 95%CI, 1.72-14.02), and an initial open wound (OR, 7.9; 95%CI, 2.76-16.43) (p<0.05). Neither early nor delayed weight-bearing was associated with an excess risk of non-union.

DISCUSSION

The main finding from this study is the identification of risk factors for complications after primary intramedullary nailing to treat tibial shaft fractures. Two of the risk factors are intrinsic, i.e., detectable before surgery: active smoking and an initial open wound. The other factor, namely, a fragment gap greater than 5 mm after nailing, is intrinsic to the surgery, is determined intra-operatively, and reflects the quality of the reduction.

Active smoking is a well-known risk factor for post-operative complications [START_REF] Schmitz | Effect of smoking on tibial shaft fracture healing[END_REF] after any type of surgery. In our study, the relative risk of non-union was increased 5.1-fold (p<0.05) and the relative risk of any complication 7.93-fold (p<0.05) in active smokers. Furthermore, of the 6 patients with delayed wound healing, 5 were active smokers. These data are consistent with previous publications. Thus, in one study, the risk of delayed bone healing of tibial shaft fractures was increased 3-to 18-fold in young smokers [START_REF] Moghaddam | Cigarette smoking influences the clinical and occupational outcome of patients with tibial shaft fractures[END_REF]. In several studies, the risk of complications was increased at least 3-fold in smokers [START_REF] Sloan | The effects of smoking on fracture healing[END_REF][START_REF] Scolaro | Cigarette smoking increases complications following fracture: a systematic review[END_REF][START_REF] Adams | Cigarette smoking and open tibial fractures[END_REF]. Smokers had significantly lower levels of TGF-1 compared to non-smokers during the first 4 weeks after surgery [START_REF] Moghaddam | Cigarette smoking decreases TGF-b1 serum concentrations after long bone fracture[END_REF]; TGF1 is a polypeptide that increases in patients after surgery and serves as a marker for recovery. Although no study has proven that smoking cessation after an injury decreases the risk of complications, it seems reasonable to advise trauma patients to stop smoking in order to improve their global prognosis.

We demonstrated that a residual interfragmentary gap greater than 5 mm was a significant risk factor for complications. This finding, which agrees with data reported by Bhandari and Claes [START_REF] Bhandari | Treatment of open fractures of the shaft of the tibia[END_REF][START_REF] Claes | Influence of size and stability of the osteotomy gap on the success of fracture healing[END_REF], underlines the importance of achieving satisfactory reduction during surgery. Although many studies used the AO classification to predict the risk of nonunion [START_REF] Laigle | Intramedullary nailing for adult open tibial shaft fracture. An 85-case series[END_REF][START_REF] Manon | Predictors of mechanical complications after intramedullary nailing of tibial fractures[END_REF], we preferred to focus on the quality of the final reduction, as this is the only factor that can be modified during surgery [START_REF] Bhandari | Treatment of open fractures of the shaft of the tibia[END_REF][START_REF] Claes | Influence of size and stability of the osteotomy gap on the success of fracture healing[END_REF]. This factor has already been studied for the humerus [START_REF] Ferrara | Residual interfragmentary gap after intramedullary nailing of fragility fractures of the humeral diaphysis: short and midterm term results[END_REF], the femur [START_REF] Gabarre | Influence of gap size, screw configuration, and nail materials in the stability of anterograde reamed intramedullary nail in femoral transverse fractures[END_REF], and for animal bones in vivo by Claes et al. [START_REF] Claes | Influence of size and stability of the osteotomy gap on the success of fracture healing[END_REF]. To our knowledge, ours is the first study to establish this risk factor for intramedullary nailing of the tibial shaft.

An open wound multiplied the risk of complications by 5.16 (p<0.05, Table 3) and the risk of non-union by 7.9 (p<0.05, Table 4). The impact of an open wound is controversial [START_REF] Bhandari | Treatment of open fractures of the shaft of the tibia[END_REF][START_REF] Smith | Comparing hospital outcomes between open and closed tibia fractures treated with intramedullary fixation[END_REF]. Smith et al. [START_REF] Smith | Comparing hospital outcomes between open and closed tibia fractures treated with intramedullary fixation[END_REF] reported that having an open fracture did not increase the infection rate, the risk of complications within 30 days, the risk of further surgery, or the length of stay. Laigle et al. [START_REF] Laigle | Intramedullary nailing for adult open tibial shaft fracture. An 85-case series[END_REF] stated that intramedullary nailing of open fractures is a reliable technique provided the procedure is performed rapidly and combined with scrupulous debridement. Although the rule that management must be initiated within the first 6 hours has been challenged by some authors [START_REF] Greenhill | Does weight-bearing assignment after intramedullary nail placement alter healing of tibial shaft fractures?[END_REF], others have recently demonstrated its importance.

Laigle et al. [START_REF] Laigle | Intramedullary nailing for adult open tibial shaft fracture. An 85-case series[END_REF] reported that 75% of patients who developed deep surgical-site infection after tibial shaft intramedullary nailing for an open fracture had waited more than 6 hours before being treated. In our study, all the patients with open fractures were managed within 6 hours. To avoid combining two risk factors in patients with open fractures, an extensive approach of the fracture site that allows better quality of reduction might have benefits. This point could be the focus of future studies. In patients with open leg fractures, achieving the best possible reduction is of paramount importance. An extensive approach can eliminate the risk factor consisting in poor quality reduction. It would be of interest to plan a randomised controlled trial on the approach to open fractures.

Early or delayed weight-bearing after surgery was not a significant risk factor in our study (p=0.12). Displacement of the material was not seen in any patient. Early weightbearing after internal fixation of lower limb fractures is controversial. Houben et al. [START_REF] Houben | Delay in weight bearing in surgically treated tibial shaft fractures is associated with impaired healing: a cohort analysis of 166 tibial fractures[END_REF] found that delayed weight-bearing after intramedullary nailing of tibial fractures was associated with poorer outcomes. In another study, immediate weight-bearing after statically locked intramedullary nailing of simple tibial shaft fractures did not affect time to bone healing [START_REF] Greenhill | Does weight-bearing assignment after intramedullary nail placement alter healing of tibial shaft fractures?[END_REF]. In contrast, Bhandari et al. [START_REF] Sprint Investigators | Study to prospectively evaluate reamed intramedually nails in patients with tibial fractures (S.P.R.I.N.T.): study rationale and design[END_REF] found that immediate weight-bearing after tibial fracture nailing increased the risk of adverse events (p=0.048). However, secondary dynamisation was counted as an adverse event in this study [START_REF] Sprint Investigators | Study to prospectively evaluate reamed intramedually nails in patients with tibial fractures (S.P.R.I.N.T.): study rationale and design[END_REF]. Thus, if dynamisation is not considered to be a complication, immediate weight-bearing is no longer associated with adverse events. In addition, in this multicentre study including 1226 tibial fractures managed at 29 trauma centres, over 90% of surgeons prescribed delayed weight-bearing, which they viewed as the preferential method of fracture site loading.

High-energy trauma was not a risk factor for complications in our study. This finding is consistent with the literature. Bhandari et al. [START_REF] Bhandari | Treatment of open fractures of the shaft of the tibia[END_REF] showed that the apparent association between high-energy trauma and complications disappeared when the soft tissue lesions (skin breached or intact) were factored into the analysis. These lesions are strongly associated with high-energy trauma, revealing selection bias. Sarmiento et al. [START_REF] Sarmiento | Factors influencing the outcome of closed tibial fractures treated with functional bracing[END_REF] obtained similar results. In their study, male sex was closely associated with high-energy trauma, which was itself associated with an initial open wound.

Our study has several limitations. The first is the retrospective design. Second, the analysis of the initial skin wound can be biased by inter-observer variability [START_REF] Brumback | Interobserver agreement in the classification of open fractures of the tibia. The results of a survey of two hundred and forty-five orthopaedic surgeons[END_REF]. The same applies to the measurement of the fragment gap. Nevertheless, the detailed review of the radiographs at a distance of the injury, by two observers,, decreases this risk. Other factors that were not considered in our study, such as the degree of initial contamination of the wound or the degree of fracture comminution, may have influenced the development of septic complications [START_REF] Hernandez | Patient-related risk factors for fracture-healing complications in the United Kingdom General Practice Research Database[END_REF][START_REF] Yokoyama | Immediate versus delayed intramedullary nailing for open fractures of the tibial shaft: a multivariate analysis of factors affecting deep infection and fracture healing[END_REF]. Finally, we did not study factors such as diabetes, overweight, or obesity. Studies have clearly shown that these factors have an important place in patient management [START_REF] Smith | Comparing hospital outcomes between open and closed tibia fractures treated with intramedullary fixation[END_REF][START_REF] Greenhill | Does weight-bearing assignment after intramedullary nail placement alter healing of tibial shaft fractures?[END_REF][START_REF] Houben | Delay in weight bearing in surgically treated tibial shaft fractures is associated with impaired healing: a cohort analysis of 166 tibial fractures[END_REF][START_REF] Sprint Investigators | Study to prospectively evaluate reamed intramedually nails in patients with tibial fractures (S.P.R.I.N.T.): study rationale and design[END_REF][START_REF] Fong | Predictors of nonunion and reoperation in patients with fractures of the tibia: an observational study[END_REF].

CONCLUSION

Conflicts of interest

None of the authors has any conflicts of interest to declare.

Within 6

 6 hours after the injury, the first step was a methodical examination of the soft tissues in the operating theatre. Lavage and debridement were performed in patients with an open wound. The objective was to achieve one-stage skin closure. The Gustilo classification was used to characterise open fractures. If the skin wound seemed to carry a high risk of secondary complications, external fixation was used to treat the fracture. Patients with open fractures were started on antibiotics as soon as possible [5] as recommended by the French Society for Anaesthesiology and Intensive care (Société Française d'Anesthésie et Réanimation, SFAR) [6].

  Primary intramedullary nailing of tibial shaft fractures seems to us to be a reliable technique. Significant risk factors for complications are active smoking, an initial open wound, and a residual interfragmentary gap greater than 5 mm.It is therefore crucial to closely monitor patients with initially open fractures, patientswho smoke, and the quality of the intra-operative reduction, notably in smokers. Of the three identified risk factors, only the quality of reduction can be modified by the surgical procedure.

Figure 1 .Figure 2 .Figure 3 .

 123 Figure 1. Radiograph showing complete fracture healing 2 years after surgery

Figure 1 .

 1 Figure 1.

Figure 2 .

 2 Figure 2.

Figure 3 .

 3 Figure 3.

Table 3

 3 reports the results of the multivariate analysis for each potential risk factor.

	Three factors were independently and significantly associated with the occurrence of
	complications: smoking (OR, 7.93, 95%CI, 2.76-22.7), a residual interfragmentary gap >5
	mm (OR, 4.92; 95%CI, 1.72-14.02), and an initial open wound (OR, 5.16; 95%CI, 1.62-
	16.43) (p<0.05). Early or delayed weight-bearing was not associated with an excess risk of
	complications.

Table 1 :

 1 Main features of the study patients

		Total	%
	Tibial nailing	184	-
	Open fractures	62	33.70
	Gustilo 1	38	20.65
	Gustilo 2	18	9.78
	Gustilo 3A	6	3.26
	Gustilo 3B	0	0
	Closed fractures	122	66.30
	Males	133	72.28
	Mean age, years	38.58	-
	% high-energy traumas	53	28.8
	Weight-bearing before day 45	85	46.2
	Interfragmentary gap >5 mm	47	25.54

Table 2 :

 2 Complications and final management

		n (%)	Final management
	Complications	28 (15.2)	-
	Surgical revision	23 (12.5)	-
	Type of complications
	Aseptic non-union	13 (7)	Exchange nailing + Autograft (n=2)
			Exchange nailing (n=6)
			Nail removal + plate + autograft (n=4)
			Graft + unlocking (n=1)
	Sepsis	4(2.1)
	Septic non-union	2 (1)	Nail removal + plaster cast + antibiotics (n=2)
	Deep SSI	2 (1)	Nail removal + antibiotics (n=2)
	Delayed wound healing	6 (3.2)	Frequently repeated wound care (n=5)
			Surgical necrosectomy (n=1)
	Compartment syndrome	4 (2.1)	Decompression fasciotomy (n=4)
	Malunion	1 (0.5)	Nail removal + Osteotomy of the bony callus
			(n=1)
	SSI: surgical-site infection

Table 3 :

 3 Multivariate analysis for risk factors for any complication Complication (28) OR (p value)p value

	Pre-operative	Age (years)	37.3	0.99 [0.68;1.028]	0.87
	factors
		Male sex	23 (82.1%)	1.028 [0.28;3.68]	0.96
		High-energy trauma 15 (53.5%)	1.71 [0.59;4.97]	0.31
		Active smoker	18 (64.2%)	7.93 [2.76;22.7]	0.0001
		Initial open wound	14 (50%)	5.16 [1.62;16.43]	0.005
	Intraoperative	Gap a >5 mm	14 (50%)	4.92 [1.72;14.02]	0.003
	factors
	Postoperative	WB<45 days	7 (25%)	0.41 [0.13;1.26]	0.12
	factors	WB>45 days	21 (75%)	2.41 [0.79;7.36]	0.12

a residual interfragmentary gap after nailing OR: odds ratio; WB: weight bearing

Table 4 :

 4 Multivariate analysis for risk factors for non-union

	Non-union (15)	OR	p value

Funding

None

Contributions of each author

PHB, MM, GF, AB, and FR contributed to design the research, to collect and enter the data, and to draft the manuscript.

All authors read and approved the final submitted version.


Giuseppe Degrassi

Biagio Di Micco

Pier Paolo Giardino

Eleonora Rossi

Higgs boson self-coupling constraints from single Higgs, double Higgs and Electroweak measurements

Keywords: Higgs boson self-coupling, Electroweak observables, single-Higgs measurements, double-Higgs measurements

We set constraints on the trilinear Higgs boson self-coupling, λ 3 , by combining the information coming from the W mass and leptonic effective Weinberg angle, electroweak precision observables, with the single Higgs boson analyses targeting the γγ, ZZ * , W W * , τ + τ -and bb decay channels and the double Higgs boson analyses in the b bb b, b bτ + τ -and b bγγ decay channels, performed by the ATLAS collaboration. With the assumption that the new physics affects only the Higgs potential, values outside the interval -1.8 λ SM 3 < λ 3 < 9.2 λ SM 3 are excluded at 95% confidence level. With respect to similar analyses that do not include the information coming from the electroweak precision observables our analysis shows a stronger constraint on both positive and negative values of λ 3 .

Introduction

With the discovery of the Higgs boson [1,2,3,4], the study of the Higgs boson potential and of the Higgs self-interactions [5] has become of great interest in the scientific community [6]. The shape of the Higgs boson energy potential and the value of the Higgs self-couplings have deep implications on cosmology [7,8,9] and on quantum field theory, in particular in connection with gravity [10].

In the Standard Model (SM) the coupling of the quartic term in the Higgs potential, λ, dictates the trilinear and quadrilinear Higgs self-interactions and its value is related to the Higgs field vacuum expectation value, v, and the Higgs boson mass, m H , by λ ≡ λ SM 3 = m 2 H /2v 2 . Thus, it can be expressed as a function of physically measurable quantities in terms of G F and m H as 1 Corresponding author: eleonora.rossi@cern.ch

λ SM 3 = G F m 2 H √ 2
, where G F is the Fermi coupling constant, linked to v via v = ( √ 2 G F ) -1/2 , whose value is obtained from the muon lifetime measurement: G F = 1.1663788×10 -5 GeV -2 [11], while m H is the Higgs boson mass measured from the Higgs boson decay products [12,13,14,15], m H = 125. 14 GeV.

The Higgs self-interactions affect any observable either at the tree-level or via quantum corrections. In particular, the trilinear Higgs coupling, λ 3 , affects the double-Higgs boson production, pp → HH at the tree-level [5,16] while both single-Higgs boson production and decay processes are affected at the oneloop level [17]. Going on in the perturbative expansion, i.e. at the two-loop level, λ 3 affects observables with no Higgs bosons as external states, like the electroweak precision observables (EWPO), in particular the W boson mass, m W , and the leptonic effective Weinberg angle sin 2 θ lep eff [18]. The latter differs from the Weinberg angle θ W defined in terms of the physical W and Z boson masses through the relation cos θ W = m W /m Z , by a renormalization factor κ lep such that sin 2 θ lep eff = κ lep sin 2 θ W , where κ lep includes all higher order corrections affecting the coupling of the Z boson to leptons.

While the couplings of the Higgs boson to vector bosons and fermions have been measured at the 5% level [19,20,21], among the Higgs self-interactions only the trilinear one can be constrained experimentally, although very weakly. Then, it is worth to use all the available experimental information in order to strengthen the constraint on λ 3 , albeit under some assumptions. At present the constraint on λ 3 obtained by the ATLAS collaboration combines the information from double Higgs analyses with an integrated luminosity up to 36.1 fb -1 with single Higgs analyses up to 79.8 fb -1 reporting that values of λ 3 outside the interval -2.3 λ SM 3 < λ 3 < 10.3 λ SM 3 are excluded at 95% confidence level (CL) [22]. Instead the CMS collaboration, using the process pp → HH → b bγγ with an integrated luminosity of 137 fb -1 , is able to exclude λ 3 values outside -3.3 λ SM 3 < λ 3 < 8.5 λ SM 3 at 95 % CL [23]. In this letter we make a further step on the path of strengthening the constraint on λ 3 by combining the public information available on the double and single-Higgs processes from the ATLAS Collaboration with the information coming from the EWPO. We perform a fit to double and single-Higgs production cross sections and Higgs decay channels together with the value of W mass and sin 2 θ lep eff building a likelihood function of one parameter of interest, κ λ , that measures the deformation of the Higgs trilinear coupling with respect to its SM value, or λ 3 = κ λ λ SM 3 . The paper is organized as follows. In section 2 we discuss the theoretical framework in which our analysis is inserted. In section 3 the experimental inputs that enter in our analysis are presented and discussed. Section 4 contains the fit procedure we employed, while the next section contains the results of the various fits we perform. Finally we present our conclusions.

Theoretical framework

In this section we briefly review some of the results presented in refs. [17,18,24] that were used as a basis for our analysis. We are interested in studying a Beyond the Standard Model (BSM) scenario where the dominant effect of an unknown new physics (NP) is concentrated on the modification of the Higgs potential

V (H) BSM = 1 2 m 2 H H 2 + κ λ λ SM 3 vH 3 + κ λ4 λ SM 3 4 H 4 + • • • , (1)
where the dots represent higher orders interactions, while at the same time the effects of NP on the other SM couplings are assumed to be negligible. This scenario can be described by a Lagrangian that differs from the SM one only in the scalar potential part that is modified via an (in)finite tower of (Φ † Φ) n terms or

V N P = N n=1 C 2n (Φ † Φ) n , Φ = 0 1 √ 2 (v + H) , (2)
with Φ the Higgs doublet, as shown in the Unitary gauge. The κ factors in eq.( 1) can be easily related to the coefficient C 2n in eq.( 2). In particular for the trilinear Higgs self-interaction one finds [18]

κ λ = 1 + 2 v 2 m 2 H 1 3 N n=3 C 2n n(n -1)(n -2) v 2 2 n-2 . ( 3
)
We remark that the potential in eq.( 2) is assumed to be general, i.e. the coefficients C 2n are not supposed to obey a hierarchy scaling as C 2n+2 ∼ C 2n v 2 /Λ 2 N P , with Λ N P the scale of NP, like in a well-behaved Effective Field Theory (EFT).

The effects induced on the observables by a modified trilinear Higgs selfinteraction occur at different orders in the perturbative expansion (tree or loop level) depending on the observable under consideration. When these effects appear for the first time at the loop level, i.e. in single-Higgs processes and EWPO, the modification of the observable induced by the lowest-order contribution can be parametrized as

O BSM = O SM 1 + (κ λ -1)C 1 + (κ 2 λ -1)C 2 , ( 4
)
where O is a generic observable defined in the BSM scenario or in the SM respectively, and C 1 and C 2 are finite numerical coefficients, i.e. their values do not depend on Λ N P . The values of the C 1 , C 2 coefficients for single-Higgs observables are reported in ref. [17] while those for the EWPO can be found in ref. [18]. Here we use the latest SM theoretical predictions for m W [25] and sin 2 θ lep eff [26] to refine the calculation of the latter coefficients. We employ as SM predictions m W = 80.359 ± 0.06 GeV and sin 2 θ lep eff = 0.23151 ± 0.00006 where the errors reported are obtained combining in quadrature the parametric uncertainties with our estimate of the missing higher order terms [18]. In Table 1 the updated values of the C 1 , C 2 coefficients are presented.

Before concluding this section we want to comment on the total uncertainties that affect our analysis. For any measurement, besides the experimental uncertainty, we take into account a theory uncertainty that can be divided in a part related to the SM prediction and another κ λ -dependent associated to missing higher order terms. The former is usually already included in the experimental analyses while the latter is actually very difficult to estimate. In ref. [17] the κ λ -dependent uncertainty was estimated in terms of the process-dependent coefficient C 1 , however the result of that analysis showed a very mild dependence on this uncertainty. Concerning the κ λ -dependent uncertainty of the EWPO we used the same kind of estimate of ref. [17] finding also in our case a very mild dependence on this uncertainty.

Data inputs

In this Section we discuss the experimental inputs we use in the fit. The observables we consider are: the pp → HH → b bγγ, the pp → HH → b bb b and the pp → HH → b bτ + τ -production cross sections as measured by the AT-LAS collaboration [27,28,29,30]; the measurements of the single-Higgs boson production cross sections including the gluon fusion (ggF), the vector boson fusion (VBF), the associate production (VH) and the t tH production modes; the branching fractions of the H → γγ, H → ZZ, H → W + W -, H → b b and H → τ + τ -decay channels [19]; the value of the W boson mass from the world average [12,31,32,33,34,35,36,37]; sin 2 θ lep eff as estimated in ref. [25] from the average of the LEP [38,39,40,41], SLD [42,[START_REF] Schael | SLD Electroweak and Heavy Flavour Groups[END_REF], Tevatron [44,45,46] and LHC [47,48,49] data.

The sin 2 θ lep eff measurements are slightly inconsistent due to a discrepancy at the level of 3σ between the LEP and the SLD most accurate measurements, namely the measurement obtained from the forward-backward asymmetry in the e + e -→ Z → b b at LEP and the one obtained from the left-right asymmetry A LR in e + e -→ Z → l + l -at SLD. The χ 2 of the fit is 11.5 with 5 degrees of freedom. In order to not underestimate the error on the average, from combining discrepant measurements, and to be conservative, we assume that the discrepancy is due to an underestimated systematic error that affects all measurements. Therefore all measurement uncertainties are multiplied by a scaling factor κ = 11.5/5 such that the χ 2 of the fit of the combined measurements equals its expectation value, this in turn consists in multiplying by 1.52 the error of the average computed in ref. [25]: sin 2 θ lep eff = 0.23151 ± 0.00014. The single-Higgs boson measurements were taken from the ATLAS collaboration results, nevertheless few measurements were excluded from the fit to extract κ λ . In particular the t tH production mode with the H → γγ decay mode was excluded to avoid double counting between this channel and the pp → HH → b bγγ channel, as discussed in ref. [22]. For the single-Higgs boson production and decay modes, it is conventional to fit data using the production cross section and decay branching fraction signal strengths (µ i ,µ f ), defined as the ratio between the observed values and their SM expectations:

µ i = σ observed i /σ SM i , i = ggF, VBF, W H, ZH, t tH µ f = Br observed H→f /Br SM H→f , f = γγ, ZZ * , W + W -, b b, τ + τ -.
In this work the signal strengths are taken from ref. [19] where the product µ i × µ f is tabulated for each production and decay mode; the values used in this fit are summarised in Table 3. The fit performed in ref. [19] combines the ZH and the W H channel assuming that the ratio of their cross section equals its SM expectation. Such assumption cannot be made in our case because κ λ = 1 affects differently the W H and ZH cross sections. Nevertheless the sensitivity of the V H result is dominated by the H → b b channel where ZH provides the most accurate measurement of the V H signal strength value [50]. In this work the V H signal strength is therefore assigned to ZH, the impact of this assumption has been tested assigning the V H signal strength of the γγ channel (the second most relevant channel after H → b b) to W H and decoupling the W H and ZH signal strengths in the H → b b channel using inputs from ref. [50]. The impact on the result has been found to be negligible. In addition, the signal strength relative to t tH + tH of the original paper has been assigned to t tH being the tH contribution negligible, and the V V channel indicated in ref. [19] has been assigned to W + W -that dominates the sensitivity. Finally the uncertainties on the signal strengths have been symmetrised by averaging the squares of the positive and negative uncertainties.

µ i × µ f ggF VBF ZH t
The measurements shown in Table 3 are correlated due to the cross contamination of signal events among different production and decay channels and the correlation matrix is provided in Figure 6 of ref. [19]; in the present work only correlations larger than 0.05 are taken into account for simplicity; this choice doesn't have impact on the final results. The correlation coefficients are shown in Table 4.

The determination of constraints on κ λ has been performed by the ATLAS collaboration in ref. [27] using data from the pp → HH production measurements, with the HH pair decaying to the final states b bγγ, b bτ + τ -and b bb b, finding -5.0 < κ λ < 12.0 (-5.8 < κ λ < 12.0) at 95% CL in observation (expectation). In that paper only the HH production cross section was parameterised as a function of κ λ while the decay branching fractions were assumed to be independent from κ λ and equal to their SM expectation. This assumption was removed in a conference note of the same collaboration [22] where the HH constraints were combined with single-Higgs differential measurements. In the pp → HH process the κ λ value has a big impact on the dynamic of the decay products affecting strongly the Higgs-pair invariant mass distribution, therefore it is not possible to extract sensible information from the final result without the information on the likelihood of each channel expressed as a function of κ λ . The likelihood shapes have been taken from ref. [51,52]. The shape of the expected and observed likelihood of the combination HH → b bb b, HH → b bγγ and HH → b bτ + τ -has been first scanned with 200 points and then interpolated using a third degree polynomial. Continuity of the first and the second derivative has been imposed at each point of the scan. The resulting likelihood function used in the fit procedure is shown in Figure 1.

Fit procedure

The fit procedure is performed by building up a likelihood function as a product of the likelihood function associated to each experimental measurement:

L = L H × L HH × L m W × L sin 2 θ lep eff
The likelihood L is a function of one parameter of interest κ λ , the ratio: Λ(κ λ ) = L(κ λ )/L(κ λ ) is used to extract the best fit values and confidence intervals on κ λ , where κλ is the value that maximises L.

The likelihoods L H , L HH , L m W and L sin 2 θ lep eff are relative to the single Higgs production and decay measurements, the HH production, the m W and the sin 2 θ lep eff measurements respectively. The minimisation is performed on the quantity -2 lnΛ whose expression is:

-2 lnΛ = -2 lnL H -2 lnL HH -2 lnL m W -2 lnL sin 2 θ lep eff + 2 lnL(κ λ ) .
The -2 lnL HH is obtained directly from Figure 1, -2 lnL m W and -2 lnL sin 2 θ lep eff are χ 2 function built as:

-2 lnL m W = χ 2 m W = [m exp W -m theo W (κ λ )] 2 [σ 2 exp m W + σ 2 theo m W ] 2 -2 lnL sin 2 θ lep eff = χ 2 sin 2 θ lep eff = sin 2 θ lep,exp eff -sin 2 θ lep,theo eff (κ λ ) 2 σ 2 exp sin 2 θ lep eff + σ 2 theo sin 2 θ lep eff 2 .
In this expression the labels "exp" and "theo" refer to the experimental and theoretical quantities respectively, while σ is the uncertainty on the observable denoted at the subscript. The experimental uncertainties are listed in Table 2 while theoretical uncertainties are discussed in section 2.

The likelihood function L H contains information from single Higgs boson production and decay measurements, the quantity -2 lnL H is a χ 2 function defined as:

-2 lnL H = χ 2 H = µ exp -µ theo (κ λ ) T C(κ λ ) -1 µ exp -µ theo (κ λ )
where µ exp is a fifteen dimensional vector containing the measurements µ i × µ f listed in Table 3 and their theoretical expectation as a function of κ λ described in section 2. The matrix C(κ λ ) -1 is the inverse of the covariance matrix C(κ λ ) = C theo (κ λ ) + C exp where C exp is built from the uncertainties shown in Table 3 and the correlation matrix ρ shown in Table 4, while C theo (κ λ ) is a diagonal matrix containing the square of the theoretical uncertainties on µ i × µ f due to missing higher order terms as discussed in section 2.

Results

The value of -2 lnΛ as a function of the κ λ parameter is shown in the left panel of Figure 2. For positive κ λ values, i.e. when the interference between the box and the self-coupling diagram in the pp → HH process is destructive and brings to a sensitivity loss of the double-Higgs channel, all the three measurements (HH, single-H and EWPO) show a comparable constraining power, with a stronger impact of the EWPO for low values of κ λ . On the other hand, for negative κ λ values, the higher statistics of the single-Higgs analyses allows to reach a better constraint on κ λ while the EWPO have a smaller impact on the result. The fit results are summarised in Table 5. All the 95% CL intervals include the value κ λ = 0 that in our analysis does not represent a special value. Indeed, all the processes we are considering contain contributions that are independent from κ λ and all observables have a continuous behavior as a function of κ λ . In order to compare the impact on the fit of the two EWPO we disentangle the likelihood functions of m W and sin 2 θ lep eff from the m W +sin 2 θ lep eff combination and from the combination of single-Higgs plus double-Higgs results, as shown on the right panel of Figure 2. The sensitivity of the EWPO is dominated by the m W measurement that represents an important addition to the single-Higgs and double-Higgs combination. In this analysis the sensitivity of the sin 2 θ lep eff measurement is expected to be reduced by the use of the scale factor described in section 3. In order to check the impact of the rescaled error on the fit results, the analysis has been performed again without applying the scale factor. The 95% CL interval from sin measurement is combined with the m W one the improvement is much smaller, a reduction of 0.2 at both edges of the interval. Furthermore the total combination with single and double Higgs boson measurements shows just a reduction of 0.1 at the upper edge of the interval, while no appreciable variation is observed on the lower edge.

In order to investigate if the impact of the EWPO measurements on the result is due to its intrinsic sensitivity, we have performed the likelihood scan setting all the fit parameters to their SM expectations. For the HH analyses the expected likelihood function has been taken directly from ref. [52], while for the single-Higgs analyses and the EWPO, it has been assumed that the correlation matrix and the fractional error on the fitted parameters don't change when the parameters move from their observed values to their expected ones. The resulting -2 lnΛ functions are shown in Figure 3.

The functional shapes of -2 lnΛ show that the constraining power of the EWPO is expected to be lower than what observed in data, in fact the full combined -2 lnΛ doesn't show large differences with respect to the combination of only the single-Higgs and double Higgs -2 lnΛ. From Figure 2 is possible to see that the combined H and HH -2 lnΛ has a minimum far from its SM expectation of κ λ = 1, while the minimum of the EWPO -2 lnΛ is closer to its SM expectation. Therefore the EWPO have an higher impact on the final observed result, in particular at the upper edge of the confidence interval.

Conclusion

In this paper we combine the ATLAS data analyses of the single-Higgs and double-Higgs processes with the information coming from the EWPO in order to constraint the Higgs boson trilinear self-coupling modifier κ λ = λ 3 /λ SM 3 . Under the assumption that NP affects only the Higgs potential we find as the best fit value of the trilinear self-coupling modifier κ λ = 4.0 +2.9 -3.3 excluding values outside the interval -1.8 < κ λ < 9.2 at 95% CL. With respect to analyses where single-Higgs data [22] or double-Higgs data [27] or a combination of both [22] are taken into account, our study shows that the inclusion in the fit of the information coming from the EWPO m W and sin 2 θ lep eff gives rise to a stronger constraint on κ λ , in particular on the positive side of the CL interval.

At the moment the information coming from EWPO gives an indication for λ 3 values closer to λ SM 3 than the single and double-Higgs analyses. It is interesting to see if, in the future, with the LHC collaborations analysing larger set of single and double-Higgs data and with possible improvements on the measurement of the m W from LHC, this different indication will remain in the data.

  × 10 -6 -1.54 × 10 -6 sin 2 θ lep eff -1.56 × 10 -5 4.55 × 10 -6

Figure 1 :

 1 Figure 1: Likelihood function used in the fit procedure from the combination of the HH → b bb b, the HH → b bγγ and the HH → b bτ + τ -channels.

Figure 2 :

 2 Figure 2: Values of -2 lnΛ as a function of κ λ . Left panel: the mW + sin 2 θ lep eff combination, the single-Higgs analyses, the double-Higgs analyses and their combination. Right panel: The mW , the sin 2 θ lep eff , their combination and the combination of the single-Higgs and double-Higgs analyses.

Figure 3 :

 3 Figure 3: Value of -2 lnΛ as a function of κ λ obtained setting all observables to their SM expectation. The function is shown for the EWPO, the single-Higgs, the double-Higgs observables, the single-Higgs plus double-Higgs combination and the full combination.

Table 1 :

 1 Values of the coefficients C 1 and C 2 for the EWPO.

Table 2

 2

		summarises all the input measurements used in
	this work.
	Double Higgs-boson production (ATLAS data)
	Channel		L [fb -1 ]
	pp → HH → b bγγ	36.1
	pp → HH → b bb b		27.5
	pp → HH → b bτ + τ -	36.1
	Single Higgs-boson production (ATLAS data)
	Decay Channel	Production Mode	L [fb -1 ]
	H → γγ	ggF, VBF, W H, ZH	139
	H → ZZ *	ggF, VBF, W H, ZH, t tH	36.1 -139
	H → W + W -	ggF, VBF, t tH	36.1
	H → τ + τ -	ggF, VBF, t tH	36.1
	H → b b	VBF, W H, ZH, t tH	24.5 -139
		Precision electroweak observables
	Observable	Value	Reference
	m W	80.379 ± 0.012 GeV	PDG World Average
	sin 2 θ lep eff	0.23151 ± 0.00021	LEP/SLD/Tevatron/LHC

Table 2 :

 2 Input measurements used in the present work. For the ATLAS measurements the analysed dataset has been specified being the experiment still on-going and analyses updates on larger datasets are expected in the future.

Table 3 :

 3 Values of the product µ i × µ f used in the fit procedure. The values corresponds to the product σ × Br normalised to its Standard Model expectation. Rows span over the decay mode f while colums over the production mode i.

		ggF	VBF	V H	t tH
	ρ ZZ ggF τ + τ -γγ
	γγ	0.06	-0.11
	ZZ *		-0.21	-0.28
	W + W -		-0.08
	τ + τ -		-0.45
	VBF
	γγ		0.07
	V H
	ZZ *				-0.07
	t tH
	W + W -				-0.42

* W + W -τ + τ -ZZ * W + W -τ + τ -

Table 4 :

 4 Correlation matrix of the signal strength measurements used in the fit. Only elements larger or equal to 0.05 are used in the present fit and are shown in the table. The correlation matrix is symmetric therefore only its upper triangle section is reported.

Table 5 :

 5 Best fit results, 68% and 95% CL intervals for all measurements used in this work and their combination.

Acknowledgements

The work of G.D. was partially supported by the Italian Ministry of Research (MUR) under grant PRIN 20172LNEEZ. The work of PPG has received financial support from Xunta de Galicia (Centro singular de investigación de Galicia accreditation 2019-2022), by European Union ERDF, and by "María de Maeztu" Units of Excellence program MDM-2016-0692 and the Spanish Research State Agency.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: