Single Top Quark Production with and without a Higgs Boson

Qing-Hong Cao,1, 2, 3, ∗ Hao-ran Jiang,1, † and Guojin Zeng1, ‡

1School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, China2Collaborative Innovation Center of Quantum Matter, Beijing 100871, China3Center for High Energy Physics, Peking University, Beijing 100871, China

One way to probe new physics beyond standard model is to check the correlation among higherdimension operators in effective field theory. We examine the strong correlation between theprocesses of pp→ tHq and pp→ tq which both depend on the same three operators. The correlationindicates that, according to the data of pp → tq, σtHq =

[106.8 ± 64.8

]fb which is far below the

current upper limit σtHq ≤ 900 fb.

1. Introduction.One way to probe new physics (NP) beyond the

Standard Model (SM) is from the so-called bottom-upapproach, i.e. one describes the unknown NP effectsthrough high-dimensional operators constructed with theSM fields at the NP scale Λ, obeying the well-establishedgauge structure of the SM, i.e. SU(2)W ⊗ U(1)Y . TheLagrangian of effective field theory (EFT) is

LEFT = LSM +1

Λ2

∑i

CiO(6)i +O

(1

Λ4

), (1)

where Ci’s are Wilson coefficients. The aim of EFTanalysis is to find nontrivial correlations among Ci’sas those relations would shed lights on the structureof NP models. Various basis of dimension-6 operatorshave been introduced in the literature, e.g. Warsawbasis [1], HISZ basis [2, 3], and SILH basis [4, 5].Unfortunately, each basis consists of 59 independentoperators, and it is difficult to explore the correlationsamong 59 operators in practice. One way out of thepredicament is to find independent observables that aresensitive to a small set of operators and then examinecorrelations among those operators. That has beenstudied in single top productions [6–12]. In this work weexplore the strong correlation among the t-channel singletop (single-t) production and the associated productionof single top quark with a Higgs boson (named as the tHqproduction); see Fig. 1. The two channels mainly involvethree operators which can be measured in the single-tproduction. That yields a strong constraint on the crosssection of the tHq production.

As some high-dimensional operators can be absorbedby field redefinition, various schemes of input parametershas been widely discussed in the literature [13, 14].We choose the Warsaw basis [1] and follow the schemeof input parameters proposed in [15] to normalize theSM fields and masses. Several approximations madein our analysis are listed as follows: i) only diagonalelements of CKM matrix have been considered for they

∗Electronic address: qinghongcao@pku.edu.cn†Electronic address: H.R.Jiang@pku.edu.cn‡Electronic address: guojintseng@pku.edu.cn

FIG. 1: Feynman diagrams of the single-t channel (a) and thetHq production (b, c).

are much larger than off-diagonal elements; ii) we ignorethe masses of light quarks and keep only the top quarkmass hereafter; iii) gauge couplings of light quarks are setto be the same as the SM as these couplings are strictlyconstrained by hadron experiments, iv) the influence ofbranch ratio from top quark decay is neglected since theeffects on partial width and total width will cancel witheach other, leaving a next leading order contribution; v)CP conservation is supposed to avoid complex Wilsoncoefficients.

The SMEFT analysis in the single-t channel has beencarried out in the literature [6–9]. Only 4 independentdimension-6 operators are involved in the t-channelproduction:

OuW = (qpσµνur)τ

I φW Iµν + h.c.,

Oφq3 = (φ†i↔DIµ φ)(qpτ

Iγµqr) + h.c.,

O(1)qq = (qpγµqr)(qsγ

µqt) + h.c.,

O(3)qq = (qpγµτ

Iqr)(qpγµτ Iqr) + h.c. . (2)

where φ denotes the SM Higgs boson doublet, Dµ

the covariant derivative, qi the left-handed SU(2)doublet of the i-th generation, and ur the right-handedisosinglets [16]; τ I denote the usual Pauli matrices inthe weak isospin space. Considering the flavor structureof four-fermion operators, there can be flavor changingcurrents from both the first and the second generationsto the third generation. However, we would ignore thecontribution from the second generation, which suffers asuppression from the PDF. With Fierz Identity, it can beproved that only two independent flavored four-fermion

arX

iv:2

105.

0446

4v2

[he

p-ph

] 2

0 M

ay 2

021

2

operators have contribution to the t-channel production.They are

O(1)3113 = (q3γµq1)(q1γ

µq3) + h.c.,

O(3)3311 = (q3γµτ

Iq3)(q1γµτ Iq1) + h.c. . (3)

For convenience, we will still use O(1)qq and O

(3)qq to denotes

these two flavored operators respectively. Indeed, the

interference between O(3)qq and the SM is only three times

as large as that of O(1)qq due to a color factor. Although

their contributions can be separated by quadratic terms,they can be considered as a same degree of freedom in theleading order analysis, i.e. keeping only the interference

between the SM and NP operators. We thus use O(3)qq to

denote the degree of freedom hereafter.The cross sections of the t-channel single top-quark

productions at the 13 TeV LHC are

σt =[214− 13C(3)

qq + 16CuW + 13Cφq3

]pb,

σt =[81− 4C(3)

qq + 5CuW + 5Cφq3

]pb, (4)

where

C(3)qq ≡ C(3)

qq

(TeVΛ2

)2,

CuW ≡ CuW(

TeVΛ2

)2,

Cφq3 ≡ Cφq3(

TeVΛ2

)2,

and the first constant term denotes the SM contributionwith NNLO QCD corrections [17]. Both FeynRules andMadGraph packages are used in our calculation [18, 19].

The tHq production involves more operators than thesingle-t channel. Figure 2 shows possible modificationsto the tHq production from various operators; see theblack thick dots. In addition to the operators affectingthe single-t production, the tHq production involvesoperators as follows [1]:

OφD = (φ†Dµφ)∗(φ†Dµφ),

Ouφ = (φ†φ)(qpurφ) + h.c. ,

Oφ2 = (φ†φ)2(φ†φ),

OφW = φ†φW IµνW

Iµν . (5)

The operator Ouφ is tightly constrained from gluon-gluon fusion measurements [20], and the operator OφW isseverely bounded by the combined analysis of the Hγγ,HZZ and HZγ couplings [21]. The operators OφD andOφ2 are constrained by the oblique parameters [22, 23]and also are mildly sensitive to the tHq production [10].Therefore, when keeping only the interference effectbetween the SM and NP channels, we end up with

only three independent operators, O(3)qq , Oφq3 and OuW ,

which affect both the single-t production and the tHqproduction.

The tHq production is not found yet owing to its smallproduction rate. We sum up both the tHq and tHq′

FIG. 2: The tHq production induced by effective operators

productions hereafter and denote the production crosssection of both the tHq and tHq′ productions as σtHq,which is

σtHq =[74.3− 11.3C(3)

qq + 22.0CuW − 2.6Cφq3

]fb. (6)

2. Collider phenomenology.Thanks to the large production rate of the single-t

production, bot the total cross section and the differentialdistributions are well measured at the LHC and can beused to search for new physics beyond the SM. Theleptonic decays of the top quark provide clean collidersignature at the LHC and are used widely in experimentalsearches. We thus focus on the channel of t→ µ+ νµ + band choose Br(t→ bµνµ) = (13.4± 0.6)% [24].

Three observables are needed to probe the three

operators O(3)qq , Oφq3 and OuW . In our analysis we

consider both the single top-quark production and thesingle antitop-quark production. The first observable inour analysis is the total cross section of the t-channelsingle-top production, i.e.

σt+t = σ(tq) + σ(tq). (7)

The second observable is the cross section ratio of thesingle-t production and single-t production, defined as

Rt ≡σtσt. (8)

The difference in production rates mainly arises from theparton distribution functions (PDFs) of initial state.

The third observable is based on the spin correlationbetween the charged lepton and top-quark. Owing to itslarge production rate the single top-quark processes arewell measured at the LHC such that one can examinedifferential distributions of various observables. Weadopt the so-called “spectator basis” to maximize spincorrelations by taking advantage of the fact that the topquark produced through the single-t processes is almost100% polarized along the direction of the spectatorquark, the light jet produced in association with the top

3

quark [25–29]. The distribution of the cosine of the anglebetween the charged lepton and the spectator jet in therest frame of top quark is

1

σ

dσ

d cos θ=

1

2

(1 +

N− −N+

N− +N+cos θ

), (9)

where N− (N+) is the number of top quark withpolarization against (along) the direction of the spectatorjet, respectively. The third observable AFB is defined as

AFB =σF − σBσF + σB

(10)

where

σF ≡∫ 1

0

dσ

d cos θd cos θ, σB ≡

∫ 0

−1

dσ

d cos θd cos θ. (11)

The distribution of cos θ is modified by the operators asfollowing:

dσ

d cos θ=

1

2

(σ0t +

∑i

Ciσit

)

+

(A0FBσ

0t +

∑i

CiAiFBσ

it

)cos θ,

(12)

which yields

AFB =A0FBσ

0t +

∑3i=1 CiA

iFBσ

it

σ0t +

∑3i=1 Ciσ

it

. (13)

where σit and AiFB represent the cross section andasymmetry generated by the SM (i = 0) and heavyoperators (i = 1, 2, 3). We take both the single-t andsingle-t into account in the σt and AFB analysis. Thenumerical values of the σit’s can be read out from Eq. 4,and the values of AFB ’s are listed as follows:

A0FB = 0.4596, A

Oφq3FB = 0.4596,

AO(3)qq

FB = 0.4721, AOuWFB = 0.3548, (14)

The operator Oφq3 modifies only the magnitude of thegW coupling and do not affect the shape of all thedifferential distributions, therefore, it yields exactly thesame asymmetry AFB as the SM.

In our analysis we normalize the three observables bythe SM predictions as follows:

σt ≡σt+tσ0t+t

, A ≡ AFBA0FB

, R ≡ RtR0t

. (15)

Since the contribution of NP operators are assumed tobe smaller than the SM, keeping only the linear term ofWilson coefficients yields

σt = 1− 0.059× C(3)qq + 0.075× CuW + 0.060× Cφq3,

A = 1− 0.0033× C(3)qq − 0.0349× CuW ,

R = 1− 0.0308× C(3)qq + 0.0328× CuW , (16)

from which we obtain

C(3)qq = 57− 28A− 29R,

CuW = 23− 26A+ 3R,Cφq3 = 10 + 17σt + 6A− 33R. (17)

Note that the three observables exhibit quite distinctsensitivity to the three operators when considering onlythe leading effect of NP operators. First, the operatorOφq3 only modify the magnitude of the W -t-b couplingand does not alter any differential distribution, therefore,it does not contribute to either the asymmetry A orthe charge ratio R. Second, the asymmetry A dependsmainly on OuW owing to the fact that OuW flips thechirality of the top quark and yields an obvious deviation

in the cos θ distribution. Third, both the O(3)qq and OuW

yields comparable interference contribution to the single-t production and thus contribute equally to the R whichis more sensitive to the parton distributions of valenceand sea quarks in the initial state. Finally, the crosssection σt is sensitive to all the three operators.

Combined analysis of the three observables help usprobing NP beyond the SM. For example, the relation

of CuW and C(3)qq can be determined from the A and R

measurements; see Eq. 17. In particular, if R ' 1 and

A 6= 1, then CuW ' C(3)qq 6= 0; if A ' 1 and R 6= 1, then

C(3)qq ∼ −10CuW . If no deviations are found in the A andR measurements, say A ' R ' 1, then the only possibleNP source in the single-t process will be the Oφq3 whichcould be obtained from the σt.

The current bound of the total cross section of thesingle-t production is 176 pb ≤ σt ≤ 238 pb at the 1σconfidence level [30], and the relative error of the crosssection measurement is δσt = 15%. It yields

σt = 0.95± 0.14 . (18)

The charge ratio [30–32] and the cos θ distribution [32–34] in the single top processes have been measured at the13 TeV LHC. Taking advantage of the unfolding resultson the charge ratio and angular distribution providedby the CMS collaboration [32, 33], we compare ourparton level analysis directly to the experimental dataand obtain

R = 1± 0.035, (19)

A = 0.877± 0.154, (20)

where the A is obtained from the linear fitting of the Aµdistribution in Ref. [32]. That gives rise to the constraintson the three operators as follows:

− 1.9 ≤ C(3)qq ≤ 8.8 ,

− 0.9 ≤ CuW ≤ 7.3 ,

− 5.2 ≤ Cφq3 ≤ 3.9 . (21)

Even though the current measurements impose onlyrough bounds on operators, more accurate measurements

4

0.6 0.8 1 1.2 1.40

0.04

0.08

0.12

0.16

σ- t

σtHq

(a) (b)

0.6 0.8 1 1.20

0.05

0.1

0.15

0.2

σtHq

(c)

0.9 0.95 1 1.05 1.10

0.04

0.08

0.12

0.16

ℛ

σtHq

FIG. 3: Correlations among σtHq and the cross section σt (a) , the asymmetry A (b), and the charge ratio R (c) at the 1σconfidence level at the current 13 TeV LHC (orange) and the HL-LHC (black). The central values of all the three observablesare chosen as 1 at the HL-LHC.

are anticipated at the high luminosity LHC with anintegrated luminosity of 3000 fb−1 (HL-LHC). Theuncertainties are mainly due to the parton distributionfunctions and the systematic errors. The majoruncertainties of the cross section and forward-backwardasymmetry measurements are due to the systematicerror which are roughly about 15% [30]. On the otherhand, the statistical uncertainty is about 1% which canbe ignored in our analysis. Therefore, we expect theuncertainties of the cross section and AFB measurementsto be reduced to 10% in our projection of the HL-LHCpotential of measuring the single-t channel.

In the charge ratio measurement the large systematicerrors tend to cancel out and give rise to a net valuecomparable to the PDF uncertainty, e.g. both thePDF and systematic errors are ∼ 3.0% in the recentCMS data [30]. In accord to the NNPDF study thePDF error is expected to less than 3% when moreprecision measurements are available [35]. Therefore,we optimistically expect the charge ratio to be measuredwith an accuracy of 3% in the forthcoming HL-LHC.

Assuming the central values of the three operators arethe same as the SM predictions, i.e. centralizing aroundzero, we obtain the projected constraints on the operatorsat the HL-LHC as follows:

− 3.7 ≤ C(3)qq ≤ 3.7 ,

− 2.7 ≤ CuW ≤ 2.7 ,

− 3.2 ≤ Cφq3 ≤ 3.2 , (22)

where both the cross section and AFB measurement areof 10% uncertainties, and more optimistically,

− 2.3 ≤ C(3)qq ≤ 2.3 ,

− 1.4 ≤ CuW ≤ 1.4 ,

− 2.1 ≤ Cφq3 ≤ 2.1 , (23)

when the systematic uncertainties are improved to be 5%.4. Single-t channel versus tHq channel

We now examine the correlation between the single-tproduction and the tHq production as both the single-t

channel and the tHq channel mainly depend on the threeoperators. A simple algebra gives rise to

σtHq =[− 95.1− 44.0σt − 266.0A+ 479.4R

]fb, (24)

which serves well for checking the consistence of theexperimental measurement and operator analysis. Ittells us that the information of the tHq production ratecan be inferred from the single-t production as long asone keeps the interference effect and neglects those sub-leading operators explained above. For example, basedon the current measurement of the single-t production,our operator analysis shows that

σtHq =[106.8± 64.8

]fb (25)

at the 1σ confidence level, which is far below the currentupper limit σtHq ≤ 900 fb [36, 37].

Figure 3 displays the correlations among σtHq andthe cross section σt (a) , the asymmetry A (b), andthe charge ratio R (c) at the 1σ confidence level atthe current 13 TeV LHC (orange) and the HL-LHC(black). In the study of the HL-LHC, the centralvalues of all the three observables are chosen as 1, theuncertainties of the σt and AFB are set to be 10%,and the uncertainty of the charge ratio R is 3%. Thecapability of the operator analysis is highly limited bythe large systematic uncertainties in the cross section σtand the asymmetry A. The strongest bound on σtHq isderived from the charge ratio R measurement owing tothe small systematic error.

Assuming the central value of σtHq is still the same asthe SM prediction at the 13 TeV HL-LHC, we obtain theprojected σtHq as following:

σtHq =[74.3± 45.4 (29.9)

]fb, (26)

where the systematic errors of both the cross section andAFB measurements are 10% (5%), respectively, and theerror of charge ratio measurement is 3%.

5

5. Conclusion.Effective field theory is a very powerful tool to probe

new physics beyond the Standard Model in a model-independent approach. More accurate informationof higher dimension operators are anticipated at thelarge hadron collider in the operation phase of highluminosity. When new physics resonances decouplefrom the electroweak scale, they leave their footprintsin various relations among higher-dimension operators,i.e., strong correlations among the Wilson coefficients.We pointed out that the single-t production and the tHqchannel are highly correlated as both processes dependmainly on three dimension-6 operators. At the leadingorder of operator expansion, we obtain a relation asfollowing:

σtHq =

[− 95.1− 44.0× σt+t

σSMt+t

− 266.0× AFB

ASMFB

+ 479.4× RtRSMt

]fb,

where σt+t, AFB and Rt = σt/σt denotes the total crosssection, the asymmetry and the charge ratio measured

in the single-t processes, respectively. The relationpredicts that, according to the current data of single-tchannel, the yet-to-be measured cross section of the tHqproduction at the 13 TeV LHC is

σtHq =[106.8± 64.8

]fb (27)

at the 1σ confidence level, which is far below the currentupper limit σtHq ≤ 900 fb. Assuming the central value ofσtHq is still the same as the SM prediction at the 13 TeVHL-LHC, we obtain the projected σtHq as following:

σtHq =[74.3± 45.4

]fb, (28)

where the systematic errors of both the cross section andAFB measurements are 10% and the error of charge ratiomeasurement is 3%.

Acknowledgments. We thank Yandong Liu and RuiZhang for enlightening discussions and comments. Thework is supported in part by the National ScienceFoundation of China under Grant Nos. 11725520,11675002, and 11635001.

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