Threshold resummation for the production of a color sextet (antitriplet) scalar at the LHC

We investigate threshold resummation effects in the production of a color sextet (antitriplet) scalar at next-to-next-to-leading logarithmic (NNLL) order at the LHC in the frame of soft-collinear effective theory. We show the total cross section and the rapidity distribution with NLO+NNLL accuracy, and we compare them with the NLO results. Besides, we use recent dijet data at the LHC to give the constraints on the couplings between the colored scalars and quarks.


I. INTRODUCTION
There is no clear evidence of new physics beyond the Standard Model found at the LHC so far [1][2][3], and the most favored Supersymmetry, Extra-Dimensions, and many others all receive somewhat strong constraint [1,4,5].Then it is a preferable way to concern more about the model independent theory rather than considering some specific models.Here we study the model independent color sextet (antitriplet) scalars, which have many significant effects in the phenomenology.Actually, color sextet scalars have been included in many new physics models, such as unification theories [6][7][8], Supersymmetry with R-parity violation [9], diquark Higgs [10].And their masses can be as low as TeV scale or less [11], which leads to many physics impacts.For example, in the supersymmetric Pati-Salam SU (2) R × SU (2) L × SU (4) C model, light color sextet scalars can be realized around the weak scale even though the scale of SU (2) R × SU (4) C symmetry breaking is around 10 10 GeV [10,11].Observation of the color sextet scalars will be a direct signal of new physics beyond the Standard Model.
Considering the interaction of the color sextet (antitriplet) scalars with quarks, which is parameterized, the relevant Lagrangian can be written as [12] where K ab i is the Clebsch-Gordan coefficient of the sextet (antitriplet), λ L/R is the Yukawa-like coupling, and a, b are the color indices.The quantum numbers of the colored scalars are listed in Table I, and more information can be find in [12,13].For the requirement of symmetry, the colored scalars couple to same sign quarks, and then have fractional electronic charges, and in the cases of antitriplet the couplings should be antisymmetric in flavor.For convenience, we label the colored scalars as sextet I , sextet II and sextet III with electronic charge 1/3, −2/3 and 4/3, respectively.
For antitriplet, the labels are antitriplet I , antitriplet II and antitriplet III in the same way.
It has been shown [10,14,15] that the measurements of D 0 − D0 mixing and the rate of D → π + π 0 (π + φ) decay can constrain the couplings of the colored scalars to two up-type quarks: Besides, the left-handed coupling λ L also receives tight constraint due to minimal flavor violation.Since we use the model independent coupling λ 2 = λ 2 L + λ 2 R , above constraints can be relaxed in the scenario considered below.Production and decay of the colored scalars at hadron colliders have been extensively discussed in [12,[15][16][17][18][19][20].The NLO QCD corrections to the single production of the color sextet (antitriplet) scalars at hadron colliders also have been studied in [12], which includes the transverse momentum resummation at the leading logarithmic level.Although the NLO QCD corrections to the cross section were studied, the threshold resummation has not been available in the literature so far.
In this work we investigate the threshold resummation effects in the single production of the color sextet (antitriplet) scalars, and also discuss the rapidity distribution of the colored scalars at NLO+NNLL accuracy at the LHC with soft-collinear effective theory (SCET) [21][22][23][24][25].As a cross check, we also calculate the NLO corrections using the analytical phase space integral method, and present their analytical expressions.Actually, when the masses of the colored scalars approach the threshold limit, there are large logarithms left after cancelling the divergences, because the scale of the soft radiations is rather small compared to the scalar mass.These threshold logarithms should be resummed to avoid spoiling the reliability of the perturbative expansion.The PDF is strongly suppressed near the endpoint region x → 1, so the cross sections fall precipitously as the scalar mass approaches the kinematical endpoint.The resummation effects are therefore dynamically enhanced.This paper is organized as follows: In Sec.II, we present the fixed-order calculations.In Sec.III, we calculate the soft function and present solutions of the renormalization group equations obeyed by hard and soft functions.In Sec.IV, we present detailed numerical analyses and compare the NLO+NNLL rapidity distributions with the NLO results.We also use recent dijet data at the LHC to give constraint on the couplings between the colored scalars and quarks.We conclude in Sec.V.

II. FIXED-ORDER CALCULATIONS
We consider the process h 1 + h 2 → φ + X, where h 1 and h 2 are the incoming hadrons with momenta P 1 and P 2 , and define the rapidity of the colored scalar φ as Y = 1 2 ln E+pz E−pz , where E and p z represent the energy and longitudinal momentum of the scalar in the center-of-mass frame of the colliding hadrons.We write the cross section as [26,27]: with where PS f is the final state phase space, and µ f is the factorization scale.For one-particle final state, there is no y dependence, and then the delta function can be reduced to (δ(y) + δ(1 − y))/2.
According to our calculation, the LO and NLO results of C ij (y, z, m φ , µ f ) are as follows: where Following the method in [26], we rearrange the results as and define C(z, m φ , µ f ) as the leading singular terms (threshold terms).
The comparison between the leading singular results and the complete fixed-order results is shown in Fig. 1.Here we use the PDFsets MSTW2008 [28][29][30], and will continue using them throughout this work.We find that the leading singular terms give the dominant contribution.And the leading singular contribution of sextet is smaller than the one of antitriplet.The reason is that the terms associated with C D give negative contribution, and then larger C D of sextet leads to smaller leading singular results.

III. RESUMMATION
The coefficient C(z, m φ , µ f ) in Eq.( 6) can be factorized as [26,31] The coupling λ satisfies the renormalization group equation where the one-loop level γ λ is given by The hard function encodes short distance information, which is evaluated with the current We read off the results from the virtual correction: C H satisfies the RGE [26] with γ q is the anomalous dimension of massless quark [32], and γ D is the one of the final state colored scalar, which is given by [33] The solution of Eq.( 13) is [26] C with where µ h is the hard matching scale, and a γ H has similar expression.We write the soft function as After calculating the diagrams shown in Fig. 2 in eikonal approximation, we obtain the results

S( √
It satisfies the RGE [26] d W(ω, µ) with where γ φ is the anomalous dimension of PDF [34].And its solution is [26] with where ∂ η is a derivative with respect to η, and s is obtained by Laplace transformation with Combining the above formulae, the RG-improved integral kernel is given by with For convenience, we list the counting scheme in Table II, which shows corresponding requirements of different levels of accuracy [26].Currently the two-loop γ λ is not available in the literature, so we just use the one-loop γ λ .The contribution of γ λ in the evolution function U (m, µ λ , µ h , µ s , µ f ) cancels out when µ λ ∼ µ h , so γ λ only affects the running of λ(µ λ ), which gives a subordinate contribution.We then call our resummation as approximate next-to-next-to-leading logarithmic (NNLL approx ), which is combined with the NLO results as follows: IV. NUMERICAL DISCUSSION In this section, we discuss the numerical results for threshold resummation effects in the single production of the color sextet (antitriplet) scalars at the LHC.Throughout our work MSTW2008lo and MSTW2008nlo are used for LO, NLL and NLO, NNLL approx , respectively.If not explained specially, we will assume the coupling λ 2 (M Z ) = 0.01α s (M Z ), and choose the initial state quarks uu for sextet and ud for antitriplet.Taking the perturbative convergence of C H and s as the guiding principle, we can obtain the matching scales µ h and µ s .In Fig. 3 we show the µ h dependence of the expansion coefficient c 1 defined in Eq. (11).We choose the hard scale µ 0 h = 0.535m φ for sextet and µ 0 h = 1.63m φ for antitriplet, respectively.The µ s dependence of the soft function is shown in Fig. 4. We fit the results and obtain the empirical functions: It is required that µ λ reflects the intensity of the interaction between the colored scalars and quarks, and In Table III, we list the typical results of total cross sections, which compare NLO+NNLL approx with LO and NLO results.From Table III, we can see that the resummation effects increase the NLO total cross section by about 2% and 0.2% for 1 TeV antitriplet and sextet, respectively, and 5% and 3% for 2 TeV antitriplet and sextet, respectively, at the 8 TeV LHC.And the resummation effects at the 14 TeV LHC are smaller than the ones at the 8 TeV LHC.
In Fig. 5, we show the dependence of the total cross section on the scalar masses including perturbative uncertainty bands due to variation of scale µ f at the 8 TeV LHC.We find that the threshold resummation reduces the scale dependence of the total cross section.The scenario at the 14 TeV LHC is very similar, so we do not present it in the figures.
Fig. 6 show the dependence of the resummed total cross section on µ h and µ s .The scales are varied over the ranges µ 0 h /2 < µ h < 2µ 0 h and µ 0 s /2 < µ s < 2µ 0 s , respectively.From Fig. 6, we can see that the µ h dependence of sextet is more sensitive than antitriplet.
In Fig. 7, we present the rapidity distributions, which compare the resummation results combined in Eq.( 29) with the fixed-order results.The scale µ f is varied over the range m φ /2 < µ f < 2m φ .We find that the shapes of the rapidity distribution of the resummation change slightly over the fixed-order results, and resummation reduces the scale dependence, except the NNLL approx results of the sextet cases.This is caused by the large color factor for sextet (C D = 10/3 for sextet, while C D = 4/3 for antitriplet).And the terms containing large color factor C D , which is associated with the scale dependence of λ and α s , will enlarge the scale dependence of NNLL approx results of sextet.
Finally, we use recent dijet data at the LHC to give constraint on the couplings λ.The CMS collaboration published the results of dijet production based on 5 fb −1 of 7 TeV data and 4 fb −1 of 8 TeV data [35][36][37], and the ATLAS collaboration based on 4.8 fb −1 of 7 TeV data and 13 fb −1 of 8 TeV data [38,39].Using the narrow-width-approximation [40], the total cross section can be written as After fitting the dijet data, we can give the constraints on the couplings.Since there is no direct theoretical requirement on the couplings between the colored scalars and different quarks, we use a common value of the coupling λ here.The colored scalars with different electronic charges couple to different quarks, and then they receive different constraints.In Fig. 8, we show the results of the constraints on the couplings.The most stringent constraint on sextet I is λ 2 (M Z ) ≥ 0.006α s (M Z ), and similarly the other constraints are 0.024α s (M Z ), 0.006α s (M Z ), 0.011α s (M Z ), 0.16α s (M Z ) and 0.16α s (M Z ) for sextet II , sextet III , antitriplet I , antitriplet II and antitriplet III , respectively.
The color factors are N D = 6, C D = 10/3 for the sextet and N D = 3, C D = 4/3 for the antitriplet.In the above results, we have set the renormalization scale µ r = µ f .

FIG. 1 :
FIG.1: Comparison of the complete fixed-order results and the leading singular results.The long-dashed, dashed and solid lines correspond to LO, leading singular NLO and complete NLO results, respectively.The mass of the colored scalars is set to be 1 TeV in the rapidity distributions.PDF sets MSTW2008lo and MSTW2008nlo are used for LO and NLO results, respectively, and the initial state quarks are chosen as uu for sextet and ud for antitriplet.The center-of-mass energy of the colliding hadrons is set to be 14 TeV.

1 FIG. 3 :
FIG. 3:The µ h dependence of the expansion coefficients c 1 in the hard function.

FIG. 4 :
FIG. 4:The µ s dependence of the soft function with different masses of the colored scalars.

spectively, at the 8 FIG. 5 :
FIG. 5:The fixed-order and RG-improved cross section predictions including perturbative uncertainty bands due to variations of scale µ f .

FIG. 6 :
FIG. 6:The µ h and µ s dependence of the resummed total cross sections.The solid and dashed lines represent µ h and µ s dependence, respectively.We set the scalar mass to be 1 TeV.

TABLE II :
Schemes for resummation with different levels of accuracy.

TABLE III :
Numerical results of the total cross section (unit: pb).