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Probing the CP violating Hγγ coupling using interferometry

  • The diphoton invariant mass distribution from the interference between ggHγγ and ggγγ is almost antisymmetric around the Higgs mass MH. We propose a new observable Aint, the ratio of the sign-reversed integral around MH (e.g. MHMH5 GeVMH+5 GeVMH) and the cross-section of the Higgs signal, to quantify this effect. We study Aint both in the Standard Model (SM) and new physics with various CP violating Hγγ couplings. Aint in SM could reach a value of 10%, while for CP violating Hγγ coupling Aint could range from 10% to −10%, which could probably be detected in the HL-LHC experiments. Aint with both CP violating Hγγ and Hgg couplings is also studied, and its range of values is found to be slightly larger.
  • The study on exotic hadron states elucidates the property of QCD and also explores a new form of matter. However, the existence of exotic states remains an open question. Most of the exotic states can be classified in the conventional quark model, and the properties of the exotic states can be explained in the framework of the quark model with a few improvements. For example, the well-known exotic state [1-4], X(3872), can be explained as the traditional cˉc state with a large component D¯D+DˉD in our unquenched quark model [5].

    In fact, several experimental collaborations have been searching for exotic states for the past two decades. In 2016, the D0 Collaboration observed a narrow structure, which is denoted as X(5568), in the B0sπ± invariant mass spectrum with a 5.1σ significance [6]. Owing to the B0sπ± decay mode, X(5568) was interpreted as the sˉduˉb(sˉudˉb) tetraquark state. However, it is difficult to determine the candidate for X(5568) in various approaches, if the requirements for ordinary hadrons can be described well in the approaches [7]. In our chiral quark model calculation, all the possible candidates for X(5568) are scattering states [8], while we predicted one shallow bound state [9, 10], BˉK with 6.2 GeV, in the IJP=00+bsˉqˉq system. Indeed, other experimental collaborations did not find the existing evidence of X(5568) [11]. Recently, the LHCb Collaboration coincidentally reported their observation of the first two fully open-flavor tetraquark states named X0(2900) and X1(2900) in the csˉqˉq system, whose statistical significance is more than 5σ [12]. If these two states are confirmed by other collaborations in the future, the X(2900) could be the first exotic state with four different flavors that cannot be quark-antiquark systems.

    \begin{array}{ll}  M_{X_0(2900)} \hspace*{-3mm} &=  2866\pm 7 \;{\rm MeV}, \nonumber \\ \Gamma_{X_0(2900)} \hspace*{-3mm} &=  57\pm 3\; {\rm MeV}, \nonumber \\ M_{X_1(2900)} \hspace*{-3mm} &=  2904\pm 5\; {\rm MeV}, \nonumber \\ \Gamma_{X_1(2900)} \hspace*{-3mm} &=  110\pm 12 \;{\rm MeV}.  \end{array}

    Owing to the report on X(2900), several possible candidates have emerged to elucidate X(2900) in different frameworks [13-30], and most of them can be divided into two categories: dimeson and diquark structures. Xue et al. obtained a 0+ D¯K resonance that can elucidate the X0(2900) in the quark delocalization color screening model [13], and via the qBSE approach, He et al. also arrived at the same conclusion [14]. Karliner et al. approximately estimated the diquark structure of csˉqˉq, and obtained a resonance that can be assigned as the candidate for X0(2900) [22]. In addition, a resonance with JP=0+ of bsˉqˉq system with a mass of 6.2 GeV was also proposed. In the framework of the QCD sum rule, Chen et al. assigned X0(2900) as a 0+ D¯K molecular state, while X1(2900) was assigned as a 1 csˉqˉq diquark state [23]. However, using a similar method, Zhang regarded both X0(2900) and X1(2900) as diquark states [24]. In addition, before the report on X(2900), Agaev et al. [25] obtained a resonance with 2878±128 MeV in the 0+ csˉqˉq system. A few studies have also disfavored these findings. Liu et al. hypothesized that the two rescattering peaks may simulate the X(2900) without introducing genuine exotic states [28]. Burns et al. interpreted the X(2900) as a triangle cusp effect originating from D¯K and D1ˉK interactions [29]. Based on an extended relativized quark model, the study reported in [30] determined four resonances, 2765, 3055, 3152, and 3396MeV, and none of them could be the candidate for X0(2900) in the 0+ csˉqˉq system.

    In fact, both molecular D¯K and diquark csˉqˉq configurations have energies approximate to the mass of X(2900). System dynamics should determine the preferred structure. Hence, the structure mixing calculation is required. Owing to the high energy of X(2900), the combinations of the excited states of cˉq and sˉq are possible. More importantly, these states will couple with the decay channels, DˉK, D¯K, and DˉK. Do these states survive after the coupling? Owing to the finite space used in the calculation, a stability method has to be employed to identify the genuine resonance. In this study, a structure mixing calculation of meson-meson and diquark-antidiquark structures is performed in the framework of the chiral quark model via the Gaussian expansion method (GEM), and the excited states of subclusters are included. Therefore, four kinds of states with quantum numbers, IJP=00± and 01±, are investigated. Owing to the lack of orbital-spin interactions in our calculation, we adopt the symbol 2S+1LJ to denote P-wave excited states. Accordingly, 0 and 1 may be expressed as 1P1, 3PJ, and 5PJ. To determine the genuine resonance, the real-scaling method [31] is adopted.

    This paper is organized as follows. In Sec. II, the chiral quark model, real-scaling method, and the wave-function of csˉqˉq systems are presented. The numerical results are provided in Sec. III, and the last section summarizes the study.

    The constituent chiral quark model (ChQM) has been successful both in describing the hadron spectra and hadron-hadron interactions. Details on the model can be found in Refs. [32, 33]. The Hamiltonian of ChQM for the four-quark system is written as

    H=4i=1mi+p2122μ12+p2342μ34+p212342μ1234+4i<j=1(VGij+VCij+χ=π,K,ηVχij+Vσij),

    (1)

    where mi is the constituent mass of the i-th quark (antiquark), and μ is the reduced mass of two interacting quarks or quark-clusters.

    μij=mimjmi+mj,μ1234=(m1+m2)(m3+m4)m1+m2+m3+m4,pij=mjpimipjmi+mj,p1234=(m3+m4)p12(m1+m2)p34m1+m2+m3+m4.

    (2)

    The quadratic form of color confinement is used here:

    VCij=(acr2ijΔ)λciλcj.

    (3)

    The effective smeared one-gluon exchange interaction takes the form

    VGij=αs4λciλcj[1rij2π3mimjσiσjδ(rij)],δ(rij)=erij/r0(μij)4πrijr20(μij),r0(μij)=r0μij.

    (4)

    The last piece of the potential is the Goldstone boson exchange, which originates from the effects of the chiral symmetry spontaneous breaking of QCD in the low-energy region,

    Vπij=g2ch4πm2π12mimjΛ2πΛ2πm2πmπvπij3a=1λaiλaj,VKij=g2ch4πm2K12mimjΛ2KΛ2Km2KmKvKij7a=4λaiλaj,Vηij=g2ch4πm2η12mimjΛ2ηΛ2ηm2ηmηvηij×[λ8iλ8jcosθPλ0iλ0jsinθP],Vσij=g2ch4πΛ2σΛ2σm2σmσ[Y(mσrij)ΛσmσY(Λσrij)],vχij=[Y(mχrij)Λ3χm3χY(Λχrij)]σiσj,Y(x)=ex/x.

    (5)

    In the above expressions, σ indicates the SU(2) Pauli matrices; λ and λc are the SU(3) flavor and color Gell-Mann matrices, respectively; andαs is an effective scale-dependent running coupling,

    αs(μij)=α0ln[(μ2ij+μ20)/Λ20].

    (6)

    The model parameters are determined by the requirement that the model can accommodate all the ordinary mesons, from light to heavy, considering only a quark-antiquark component. Details on the meson spectrum fitting process can be found in the work by Vijande et al. [32]. Here, we provide a brief introduction of the process. First, the mass parameters, ms,s=σ,η,κ,π take their experimental values, while the cut-off parameters, Λs,s=σ,η,κ,π, are fixed at typically used values [32]. Second, the chiral coupling constant gch can be obtained from the experimental value of the πNN coupling constant

    g2ch4π=925g2πNN4πm2udm2N.

    Finally, our confinement potential takes the ordinary quadratic form, which differs from the expression used in Ref. [32], where the effect of sea quark excitation is considered. Here, we leave the effect of sea quark excitation to the unquenched quark model. All of the parameters are presented in Table 1, and the masses of mesons obtained are presented in Table 2.

    Table 1

    Table 1.  Quark Model Parameters (mπ=0.7 fm−1, mσ=3.42 fm−1, mη=2.77 fm−1, and mK=2.51 fm−1).
    Quark masses mu=md/MeV 313
    ms/MeV 536
    mc/MeV 1728
    mb/MeV 5112
    Goldstone bosons Λπ=Λσ/fm1 4.2
    Λη=ΛK/fm1 5.2
    g2ch/(4π) 0.54
    θp/() −15
    Confinement ac/MeV 101
    Δ/MeV −78.3
    μc/MeV 0.7
    OGE α0 3.67
    Λ0/fm1 0.033
    μ0/MeV 36.976
    ˆr0/MeV 28.17
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    Table 2

    Table 2.  Meson spectrum (unit: MeV).
    D D DJ D1
    QM 1862.6 1980.5 2454.7 2448.1
    exp 1867.7 2008.9 2420.0 2420.0
    K K KJ K1
    QM 493.9 913.6 1423.0 1400.0
    exp 495.0 892.0 1430.0 1427.0
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    The csˉqˉq system has two structures, meson-meson and diquark-antidiquark, and the wave function of each structure comprises four parts: orbital, spin, flavor, and color wave functions. In addition, the wave function of each part is constructed by coupling two sub-clusters wave functions. Therefore, the wave function for each channel will be the tensor product of the orbital (|Ri), spin (|Sj), color (|Ck) and flavor (|Fl) components,

    |ijkl=A|Ri|Sj|Ck|Fl,

    (7)

    A is the antisymmetrization operator.

    1   orbital wave function

    The orbital wave function comprises two sub-clusters orbital wave functions and the relative motion wave function between two sub-clusters,

    Φ(r)=CLl1,l2,l3Ψl1(r12)Ψl2(r34)Ψl3(r1234).

    (8)

    The negative parity requires the P-wave angular momentum, and only one orbital angular momentum is set to 1. Accordingly, the following combinations are obtained: l1=1, l2=0,l3=0 as "|R1," l1=0,l2=1,l3=0 as "|R2," and l1=0,l2=0,l3=1 as "|R3." However, for the positive parity state, we set all orbital angular momentum to 0, l1=0,l2=0,l3=0 as "|R0."

    In the GEM, the radial part of the spatial wave function is expanded by Gaussians:

    R(r)=nmaxn=1cnψGnlm(r),

    (9)

    ψGnlm(r)=Nnlrleνnr2Ylm(ˆr),

    (10)

    where Nnl, which is the normalization constant, is expressed as

    Nnl=[2l+2(2νn)l+32π(2l+1)]12.

    (11)

    cn are the variational parameters, which are determined dynamically. The Gaussian size parameters are selected according to the following geometric progression

    νn=1r2n,rn=r1an1,a=(rnmaxr1)1nmax1.

    (12)

    The advantage of the geometric progression is that it enables the optimization of the ranges using just a small number of Gaussians. The GEM has been successfully used in the calculation of few-body systems [34].

    2   spin wave function

    Because there is no difference between the spin of quark and antiquark, the wave functions of the meson-meson structure has the same form as that of the diquark-antidiquark structure.

    |S1=χσ10=χσ00χσ00,|S2=χσ20=13(χσ11χσ11χσ10χσ10+χσ11χσ11),|S3=χσ11=χσ00χσ11,|S4=χσ21=χσ11χσ00,|S5=χσ31=12(χσ11χσ10χσ10χσ11),|S6=χσ12=χσ11χσ11.

    (13)

    Where the subscript of "χσiS" denotes the total spin of the tretraquark, and the superscript is the index of the spin function with a fixed S.

    3   flavor wave function

    The total flavor wave functions can be written as,

    |F1=12(cˉusˉdcˉdsˉu),|F2=12(csˉuˉdcsˉdˉu).

    (14)

    Here, |F1 (|F2) denotes the flavor wave function for the molecular (diquark-antidiquark) structure.

    4   color wave function

    The colorless tetraquark system has four color structures, including 11, 88, 3ˉ3, and 6ˉ6,

    |C1=χm111=19(ˉrrˉrr+ˉrrˉgg+ˉrrˉbb+ˉggˉrr+ˉggˉgg+ˉggˉbb+ˉbbˉrr+ˉbbˉgg+ˉbbˉbb),

    |C2=χm288=212(3ˉbrˉrb+3ˉgrˉrg+3ˉbgˉgb+3ˉgbˉbg+3ˉrgˉgr+3ˉrbˉbr+2ˉrrˉrr+2ˉggˉgg+2ˉbbˉbbˉrrˉggˉggˉrrˉbbˉggˉbbˉrrˉggˉbbˉrrˉbb).|C3=χd1ˉ33=36(rgˉrˉgrgˉgˉr+grˉgˉrgrˉrˉg+rbˉrˉb,rbˉbˉr+brˉbˉrbrˉrˉb+gbˉgˉbgbˉbˉg+bgˉbˉgbgˉgˉb),|C4=χd26ˉ6=612(2rrˉrˉr+2ggˉgˉg+2bbˉbˉb+rgˉrˉg+rgˉgˉr+grˉgˉr+grˉrˉg+rbˉrˉb+rbˉbˉr+brˉbˉr+brˉrˉb+gbˉgˉb+gbˉbˉg+bgˉbˉg+bgˉgˉb).

    (15)

    To write down the wave functions easily for each structure, the different orders of particles are adopted. However, when coupling the different structure, the same order of particles should be used.

    5   total wave function

    In this study, we investigated all possible candidates for X(2900) in the csˉqˉq system. The antisymmetrization operators are different for different structures. For the csˉqˉq system, the antisymmetrization operator becomes

    A=1(34)

    (16)

    for diquark-antidiquark, and

    A=1(24)

    (17)

    for the meson-meson structure. After applying the antisymmetrization operator, some wave function will vanish, which means that the states are forbidden. All of the allowed channels are presented in Table 3. The subscript "8" denotes the color octet subcluster, the superscript of the diquark/antidiquark is the spin of the subcluster, and the subscript is the color representation of subcluster, 3, ˉ3, 6 and ˉ6 , which denote the color triplet, anti-triplet, sextet, and anti-sextet, respectively.

    Table 3

    Table 3.  All of the allowed channels. We adopt |ijkl to donate different states. "i,j,k,l" are indices that denote the orbital, spin, flavor, and color wave functions, respectively.
    csˉqˉq
    |ijkl 3PJ |ijkl 1P1 |ijkl 5PJ
    |1311 D1¯K |1111 D1ˉK |1611 DJ¯K
    |1312 [D1]8[¯K]8 |1112 [D1]8[ˉK]8 |1612 [DJ]8[¯K]8
    |1411 DJˉK |1211 DJ¯K |2611 D¯KJ
    |1412 [DJ]8[ˉK]8 |1212 [DJ]8[¯K]8 |2612 [D]8[¯KJ]8
    |1511 DJ¯K |2111 D¯K1 |3611 (D¯K)P
    |1512 [DJ]8[¯K]8 |2112 [D]8[¯K1]8 |3612 ([D]8[¯K]8)P
    |2311 D¯KJ |2211 D¯KJ |1624 [cs]1,P6[ˉqˉq]1ˉ6
    |2312 [D]8[¯KJ]8 |2212 [D]8[¯KJ]8 |2623 [cs]13[ˉqˉq]1,Pˉ3
    |2411 D¯K1 |3111 (DˉK)P |3624 ([cs]16[ˉqˉq]1ˉ6)P
    |2412 [D]8[¯K1]8 |3112 ([D]8[ˉK]8)P |ijkl 1+
    |2511 D¯KJ |3211 (DˉK)P |0311 D¯K
    |2512 [D]8[¯KJ]8 |3212 ([D]8[ˉK]8)P |0312 [D]8[¯K]8
    |3311 (DˉK)P |1123 [cs]0,P3[ˉqˉq]0ˉ3 |0411 DˉK
    |3312 ([D]8[ˉK]8)P |1224 [cs]1,P6[ˉqˉq]1ˉ6 |0412 [D]8[ˉK]8
    |3411 (DˉK)P |2124 [cs]06[ˉqˉq]1,Pˉ6 |0511 D¯K
    |3412 ([D]8[ˉK]8)P |2223 [cs]13[ˉqˉq]1,Pˉ3 |0512 [D]8[¯K]8
    |3511 (DˉK)P |3123 ([cs]03[ˉqˉq]0ˉ3)P |0324 [cs]06[ˉqˉq]1ˉ6
    |3512 ([D]8[ˉK]8)P |1324 [cs]1,P6[ˉqˉq]1ˉ6 |0423 [cs]13[ˉqˉq]0ˉ3
    |1324 [cs]0,P6[ˉqˉq]1ˉ6 |ijkl 0+ |0524 [cs]16[ˉqˉq]1ˉ6
    |1423 [cs]1,P3[ˉqˉq]0ˉ3 |0111 DˉK
    |1524 [cs]1,P6[ˉqˉq]1ˉ6 |0112 [D]8[ˉK]8
    |2323 [cs]03[ˉqˉq]1,Pˉ3 |0211 D¯K
    |2424 [cs]16[ˉqˉq]0,Pˉ6 |0212 [D]8[¯K]8
    |2523 [cs]13[ˉqˉq]1,Pˉ3 |0123 [cs]03[ˉqˉq]0ˉ3
    |3324 ([cs]06[ˉqˉq]1ˉ6)P |0224 [cs]16[ˉqˉq]1ˉ6
    |3423 ([cs]13[ˉqˉq]0ˉ3)P
    |3524 ([cs]16[ˉqˉq]1ˉ6)P
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    In this section, we present our numerical results. In the calculation of the csˉqˉq system, two structures, the meson-meson and diquark-antidiquark structures, and their coupling are considered. Because the mass of X(2900) is larger than the threshold of the csˉqˉq system, the possible candidates must be resonance states rather than bound states. To verify whether the states survive the coupling to the open channels, DˉK, D¯K, and DˉK, the real-scaling method (RSM) is employed to test stability of these candidates.

    In the JP=0+csˉqˉq system, there are four channels in the meson-meson structure and two channels in the diquark-antidiquark structure (see Table 3). The lowest eigen-energy of each channel is provided in the second column of Table 4. The eigen-energies of the entire channel coupling are presented in the rows that are marked "c.c," and the percentages in the table represent the percent of each channel in the eigen-states with corresponding energies (in the last row of the table). The two lowest eigen-energies and the eigen-energies of approximately 2900 MeV are given. In the channel coupling calculation, we obtain four energy levels, E1(2836), E2(2896), E3(2906) and E4(2936), which could be the candidates for X0(2900). However, the eigen-state with E2(2896) has ~89% of D¯K, and the energy is higher than its threshold, 2894 MeV, and the single-channel calculation of D¯K indicates that the state is unbound, such that it should be in a D¯K scattering state rather than a resonance; hence, E9(2906) does satisfy this. However, both E1(2837) and E4(2936) have more than 30% of the diquark structure, which indicates that the two states may be in resonance states. The stability of these states have to be verified when assigning these resonances to be the candidate for X0(2900).

    Table 4

    Table 4.  Results for IJP=0+,1+ states ("c.c." means channel coupling).
    0+csˉqˉq
    s.c. 1st 2nd 7th 8th 9th 10th
    DˉK 2357.0 90.1% 99.4% 61.9% 6.1% 24.0% 26.2%
    [D]8[ˉK]8 3098.2 0.3% 0.0% 1.5% 0.4% 0.9% 4.2%
    D¯K 2895.8 0.5% 0.0% 0.3% 88.8% 58.8% 28.9%
    [D]8[¯K]8 2863.7 1.5% 0.1% 2.6% 0.1% 1.5% 0.4%
    [cs]03[ˉqˉq]0ˉ3 2656.5 6.9% 0.1% 7.3% 0.3% 10.1% 0.9%
    [cs]16[ˉqˉq]1ˉ6 2965.7 0.7% 0.4% 26.3% 4.4% 4.9% 29.8%
    c.c. 2340.1 2358.9 2836.3 2896.7 2906.9 2935.8
    1+csˉqˉq
    s.c. 1st 10th 11th 12th 13th 14th
    D¯K 2777.6 0.3% 25.4% 0.5% 0.4% 0.2% 9.3%
    [D]8[¯K]8 3111.8 0.4% 3.4% 1.1% 1.2% 1.7% 0.4%
    DˉK 2475.3 87.4% 34.2% 30.1% 55.2% 63.7% 54.8%
    [D]8[ˉK]8 3110.7 0.4% 1.6% 0.6% 1.8% 1.2% 0.9%
    D¯K 2895.9 0.3% 0.6% 52.5% 11.1% 3.7% 0.9%
    [D]8[¯K]8 3005.0 1.1% 0.9% 1.5% 2.8% 3.4% 4.6%
    [cs]06[ˉqˉq]1ˉ6 3112.3 0.1% 0.4% 1.2% 7.3% 4.8% 19.7%
    [cs]13[ˉqˉq]0ˉ3 2690.6 9.8% 0.5% 1.2% 0.8% 4.5% 5.5%
    [cs]16[ˉqˉq]1ˉ6 3040.4 0.1% 16.1% 11.4% 19.4% 17.0% 3.9%
    c.c. 2464.3 2857.1 2896.3 2904.2 2920.3 2941.7
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    Due to three combinations of spin in the JP=1+csˉqˉq system, more channels are presented in Table 3. Consequently, five energy levels near X(2900), which include E5(2857), E6(2896), E7(2904), E8(2920), and E9(2941) emerge in the channel coupling calculation. Similar to the JP=0+ case, the E6(2896) and E7(2904) are dominated by the meson-meson scattering states. However, both E8(2920) and E9(2941) have almost 28% of the diquark structure, which is beneficial to the existence of resonance.

    For the P-wave excited csˉqˉq system, the states are denoted as 1P1, 3PJ(J=0,1,2), and 5PJ(J=1,2,3). The numerical results are presented in Table 5. The channels with negligible percentages are omitted. Because the present calculation only includes the central forces, the states with the same spin are degenerate. Consequently, the P-wave csˉqˉq threshold D1ˉK and (DˉK)P is close to X(2900), and X(2900) may be molecular states. Because every sub-cluster could be the P-wave excited state, the number of channels in this case is large. Therefore, the effects of the channels may play an important role in the formation of X(2900).

    Table 5

    Table 5.  Results for JP=0,1 ("c.c." represents channel coupling).
    1P1csˉqˉq
    s.c. 1st 6th 7th 8th 9th 10th
    D1ˉK 2943.3 0.0% 1.6% 0.0% 0.1% 2.1% 83.9%
    [D1]8[ˉK]8 3554.8 0.0% 0.6% 0.0% 0.0% 0.4% 0.1%
    DJˉK 3369.5 0.0% 0.1% 0.0% 0.1% 0.4% 0.1%
    [DJ]8[ˉK]8 3340.5 0.0% 1.7% 0.0% 0.0% 2.5% 0.3%
    DˉK1 3264.4 0.0% 6.2% 0.0% 1.5% 2.0% 1.3%
    [D]8[ˉK1]8 3544.4 0.0% 0.5% 0.0% 0.0% 1.1% 0.0%
    DˉKJ 3404.8 0.0% 0.1% 0.0% 0.0% 0.8% 0.0%
    [D]8[ˉKJ]8 3334.2 0.0% 0.9% 0.0% 0.0% 0.5% 0.1%
    (DˉK)P 2359.8 100.0% 48.0% 2.0% 10.0% 24.5% 6.9%
    (DˉK)P 2897.5 0.0% 0.1% 98.0% 74.0% 43.8% 0.9%
    [cs]03[ˉqˉq]0ˉ3 3030.1 0.0% 3.3% 0.0% 0.0% 8.6% 1.2%
    [cs]16[ˉqˉq]1ˉ6 3279.4 0.0% 29.9% 0.0% 9.8% 6.6% 4.3%
    [cs]06[ˉqˉq]1ˉ6 3483.4 0.0% 2.1% 0.0% 0.0% 5.2% 0.4%
    [cs]13[ˉqˉq]1ˉ3 3621.6 0.0% 4.5% 0.0% 1.5% 1.5% 0.3%
    c.c. 2359.8 2873 2897 2908 2932 2943.1
    3PJcsˉqˉq
    s.c. 1st 9th 10th 11th 12th 13th
    D1ˉK 3363.6 0.0% 0.9% 0.0% 0.0% 0.1% 0.2%
    [D1]8[ˉK]8 3551.3 0.0% 1.2% 0.0% 0.9% 0.5% 2.5%
    DJˉK 2950.0 0.0% 0.2% 0.0% 0.1% 0.9% 3.1%
    [DJ]8[ˉK]8 3556.2 0.0% 0.8% 0.0% 0.0% 2.1% 0.5%
    DJˉK 3370.2 0.0% 0.1% 0.0% 0.0% 0.1% 0.0%
    [DJ]8[ˉK]8 3448.5 0.0% 4.4% 0.0% 0.0% 0.1% 0.3%
    DˉKJ 3287.4 0.0% 5.9% 0.0% 0.1% 0.2% 0.4%
    [D]8[ˉKJ]8 3543.9 0.0% 0.1% 0.0% 0.1% 0.3% 0.5%
    DˉK1 3382.8 0.0% 0.8% 0.0% 0.4% 5.8% 5.8%
    [D]8[ˉK1]8 3539.9 0.0% 0.2% 0.0% 0.0% 0.3% 0.3%
    DˉKJ 3405.4 0.0% 0.1% 0.0% 0.5% 0.1% 0.9%
    [D]8[ˉKJ]8 3434.9 0.0% 0.8% 0.0% 0.0% 0.4% 0.5%
    (DˉK)P 2779.9 0.0% 49.8% 0.1% 0.4% 2.6% 2.9%
    (DˉK)P 2477.8 100.0% 6.7% 0.1% 3.1% 32.7% 39.3%
    (DˉK)P 2897.9 0.0% 0.3% 99.5% 89.0% 15.3% 3.6%
    [cs]0,P6[ˉqˉq]1ˉ6 3372.1 0.0% 12.7% 0.3% 0.2% 7.0% 0.9%
    [cs]1,P3[ˉqˉq]0ˉ3 3037.3 0.0% 5.5% 0.0% 0.6% 0.6% 2.0%
    [cs]1,P6[ˉqˉq]1ˉ6 3327.4 0.0% 3.1% 0.0% 2.7% 25.7% 30.6%
    [cs]03[ˉqˉq]1,Pˉ3 3625.5 0.0% 2.0% 0.0% 0.1% 0.8% 0.2%
    [cs]16[ˉqˉq]0,Pˉ6 3477.1 0.0% 3.3% 0.0% 0.4% 0.4% 1.2%
    [cs]13[ˉqˉq]1,Pˉ3 3640.1 0.0% 0.8% 0.0% 0.3% 3.4% 4.2%
    c.c. 2477.8 2867 2897 2908 2928 2944
    5PJcsˉqˉq
    s.c. 1st 2nd 3rd
    DJˉK 3370.2 0.0% 0.1% 0.1%
    [DJ]8[ˉK]8 3653.3 0.0% 0.7% 3.0%
    DˉKJ 3405.4 0.0% 0.6% 3.8%
    [D]8[ˉKJ]8 3649.8 0.0% 0.1% 0.5%
    (DˉK)P 2897.8 100.0% 96.5% 71.2%
    [cs]1,P6[ˉqˉq]1ˉ6 3413.2 0.0% 1.7% 17.6%
    [cs]13[ˉqˉq]1,Pˉ3 3675.6 0.0% 0.3% 2.9%
    c.c. 2897.8 2909 2933
    DownLoad: CSV
    Show Table

    For the 1P1 system, there are five energy levels near the X(2900), E10(2873), E11(2897), E12(2908), E13(2932), and E14(2943) that emerged in the channel coupling calculation. Obviously, owing to both D1ˉK and (D¯K)P being scattering states, the E11(2897) with 98% (D¯K)P may be a scattering state. Furthermore, E12(2908) and E14(2943) also have large scattering state percentages; hence, they are not possible candidate for X(2900). In contrast, E10(2873) with 40% diquark structure may be good candidate for X(2900). Regarding the E13(2932) state with only 20% diquark structure, it seems impossible for it to be the resonance; hence, further calculations are required.

    Similar to the 1P1 case, there are five energy levels, E15(2867), E16(2897), E17(2908), E18(2928) and E19(2944) in the 3PJcsˉqˉq system. E16(2897) and E17(2908) may be scattering states that would decay to (D¯K)P threshold, while E18(2928) and E19(2944) may be possible resonances of X(2900). Regarding the 5PJcsˉqˉq system, the lowest single energy level is the P-wave (D¯K)P, and the lowest energy level is the scattering state. In contrast, the second energy level is DJ¯K with 3370MeV is significantly larger than X(2900). Thus, all of the resonances in the 5PJcsˉqˉq system may be unsuitable for elucidating X(2900).

    In the last subsection, we obtain several possible candidates for X(2900) owing to the structures mixing. However, the LHCb only determined two resonances approximate to 2900 MeV, and the number of candidates may be too rich for X(2900). There are two reasons why our model provides so many candidates. First, we simultaneously consider two different structures in our calculation, which results in molecular and diquark energies filling our energy spectrum. Second, the calculation is performed in a finite space, the behavior of scattering states is similar to that of bound states. Calculations in finite spaces always offer discrete energy levels. Consequently, to check if these states are genuine resonances, the real-scaling method [31] is employed. In this method, the Gaussian size parameters rn for the basis functions between two sub-clusters for the color-singlet channels are scaled by multiplying a factor α, i.e. rnαrn. Then, any continuum state will fall off towards its threshold. A resonant state should not be affected by the variation of α when it stands alone, the coupling to the continuum indicates that the resonance would act as an avoid-crossing structure, as presented in the Fig. 1. The top line represents a scattering state, which would decay to the corresponding threshold. However, the down line, resonance line, would interact with the scattering line, which could result in an avoid-crossing structure. The emergence of the avoid-crossing structure is because the energy of scattering states will get close to the energy of the genuine resonance, with an increase in the scaling factor, and the coupling will become stronger. The avoid-crossing structure is a general property of interacting two-level systems. If the avoid-crossing structure can be repeated with the increase in α, the avoid-crossing structure may be a genuine resonance, and the width can be determined by the following formula [31]:

    Figure 1

    Figure 1.  Resonance shape in the real-scaling method.

    Γ=4V(α)(kr×kc)|krkc|.

    (18)

    V(α) is the minimal energy difference, while kr indicates the slope of the resonance state, and kc represents slope of scattering state.

    The real-scaling results for the positive (Table 4) and negative parity states (Table 5) are illustrated in Figs. 2-6. Here, we only focus on the energy range from 2800 to 3000 MeV because we are interested in the candidates for X(2900); only the D¯K threshold is relevant in our calculation, which is marked with a red line. In Fig. 2, the resonance E1(2836) rapidly falls to the lowest threshold, DˉK with the spaces increases, and both E3(2906) and E4(2936) would decay to the D¯K channel. However, the E4(2936) may combine with a higher energy level into one avoid-crossing structure, and the structure will be repeated at α=2.2, which indicates the existence of a resonance R0(2914), a possible candidate for X0(2900). In the 0+ case, the E4(2936) serves as a resonance level, because it has 30% percent of the diquark structure. According to Eq. (18), we estimate its width to be approximately 42 MeV. In the 1+ case (Fig. 3), there are three possible resonances, E7(2904), E8(2920), and E9(2941), and they all have approximately 30% of the diquark structure. Consequently, the shape of the figure may be very complex. Based on the requirement of the repetitivenes of the avoid-crossing structure, we select one possible resonance, R1(2906). However, the width of R1(2906), which is 29 MeV, may be unsuitable for X1(2900), which has a width of 110 MeV.

    Figure 2

    Figure 2.  (color online) Energy spectrum of 1S0csˉqˉq system.

    Figure 4

    Figure 4.  (color online) Energy spectrum of 1P1csˉqˉq system.

    Figure 5

    Figure 5.  (color online) Energy spectrum of 3PJcsˉqˉq system.

    Figure 6

    Figure 6.  (color online) Energy spectrum of 5PJ.

    Figure 3

    Figure 3.  (color online) Energy spectrum of 3S1csˉqˉq system.

    Now, we consider the negative parity states presented in Table 5. Owing to the threshold of the p-wave D meson and K meson close to X(2900), the resonance states may couple to the scattering states strongly, and the pattern may be more complicate. Similar to the 00+csˉqˉq system, the lines with E12(2908), E13(2932) fall to the threshold of D¯K, and E14(2943) falls within the threshold of D1ˉK (see Fig. 4). However, the state E13(2932) state with 22% of the diquark structure indicates an avoid-crossing structure, which may be a possible resonance, R1(2912) with Γ=10 MeV. Regarding the E10(2873) state, it rapidly decays to the lower threshold and is not a possible candidate for X(2900) for the lower energy. Because the 3PJcsˉqˉq system has 21 channels including two thresholds near X(2900), the pattern of the 3PJcsˉqˉq system is very complex. We determine two possible candidates, RJ(2920) with Γ=9 MeV and RJ(2842) with Γ=24 MeV, for X(2900) (Fig. 5). Finally, the channels in the 5PJcsˉqˉq system have higher energies than X(2900) , and DˉK is a p-wave excited scattering state. From the figure, it can be observed that no resonance survives the coupling to the p-wave scattering state (see Fig. 6). Although the energies of the above resonance states with negative parity are close to the mass of X1(2900), the larger width of X1(2900) prevents the formation of a conclusion.

    In the framework of the chiral constituent quark model, we systematically studied csˉqˉq states to determine the candidates for X(2900), which were reported by the LHCb Collaboration recently. Both the molecular structure, as well as the diquark-antidiquark, with all the possible color, flavor, and spin configurations are considered in the present calculation. The obtained results indicate that there are several states with energies of approximately 2900 MeV in the csˉqˉq system after structure-mixing. These superabundant resonances may be triggered by the structure mixing and finite calculation space. Therefore, the real-scaling method, a stablization method, is adopted to identify the genuine resonances. We obtained five possible resonances, R0(2914) with Γ=42 MeV, R1(2906) with Γ=29 MeV, R1(2912) with Γ=10 MeV, RJ(2920) with Γ=9 MeV, and RJ(2842) with Γ=24 MeV. All the resonances obtained are diquark-antidiquark states. For X0(2900), the resonance, R0(2904) with Γ=42 MeV, may be a good candidate. In this case, X0(2900) would be the positive parity state. However, it is possible to assign candidates for X1(2900) based on energy, but the decay width prevent us from making a definite conclusion. Hence, more information on X(2900) and further studies are required.

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  • [1] A. D. Sakharov, Pisma Zh. Eksp. Teor. Fiz. 5: 32-35 (1967) [Usp. Fiz. Nauk161, 61(1991)]
    [2] Planck Collaboration, P. A. R. Ade et al, Astron. Astrophys., 571: A16 (2014), arXiv:1303.5076[astro-ph.CO doi: 10.1051/0004-6361/201321591
    [3] E. Accomando et al, arXiv: hep-ph/0608079[hep-ph]
    [4] D. E. Morrissey and M. J. Ramsey-Musolf, New J. Phys., 14: 125003 (2012), arXiv:1206.2942[hep-ph doi: 10.1088/1367-2630/14/12/125003
    [5] Y. Gao, A. V. Gritsan, Z. Guo et al, Phys. Rev. D, 81: 075022 (2010), arXiv:1001.3396[hep-ph
    [6] S. Bolognesi, Y. Gao, A. V. Gritsan et al, Phys. Rev. D, 86: 095031 (2012), arXiv:1208.4018[hep-ph
    [7] A. V. Gritsan, R. Rntsch, M. Schulze et al, Phys. Rev. D, 94(5): 055023 (2016), arXiv:1606.03107[hep-ph
    [8] J. Ellis and T. You, JHEP, 06: 103 (2013), arXiv:1303.3879[hep-ph
    [9] M. R. Buckley and D. Goncalves, Phys. Rev. Lett., 116(9): 091801 (2016), arXiv:1507.07926[hep-ph doi: 10.1103/PhysRevLett.116.091801
    [10] G. Li, H.-R. Wang, and S.-h. Zhu, Phys. Rev. D, 93(5): 055038 (2016), arXiv:1506.06453[hep-ph
    [11] A. Hayreter, X.-G. He, and G. Valencia, Phys. Rev. D, 94(7): 075002 (2016), arXiv:1606.00951[hep-ph
    [12] K. Hagiwara, K. Ma, and S. Mori, Phys. Rev. Lett., 118(17): 171802 (2017), arXiv:1609.00943[hep-ph doi: 10.1103/PhysRevLett.118.171802
    [13] M. J. Dolan, P. Harris, M. Jankowiak et al, Phys. Rev. D, 90: 073008 (2014), arXiv:1406.3322[hep-ph
    [14] Y. Chen, A. Falkowski, I. Low et al, Phys. Rev. D, 90(11): 113006 (2014), arXiv:1405.6723[hep-ph
    [15] A. Yu. Korchin and V. A. Kovalchuk, Eur. Phys. J. C, 74(11): 3141 (2014), arXiv:1408.0342[hep-ph
    [16] X. Chen, G. Li, and X. Wan, Phys. Rev. D, 96(5): 055023 (2017), arXiv:1705.01254[hep-ph
    [17] L. Bian, N. Chen, and Y. Zhang, Phys. Rev. D, 96(9): 095008 (2017), arXiv:1706.09425[hep-ph
    [18] J. Brehmer, F. Kling, T. Plehn et al, Phys. Rev. D, 97(9): 095017 (2018), arXiv:1712.02350[hep-ph
    [19] CMS Collaboration, V. Khachatryan et al, Phys. Rev. D, 92(1): 012004 (2015), arXiv:1411.3441[hep-ex
    [20] CMS Collaboration, V. Khachatryan et al, Phys. Lett. B, 759: 672-696 (2016), arXiv:1602.04305[hep-ex
    [21] CMS Collaboration, A. M. Sirunyan et al, Phys. Lett. B, 775: 1-24 (2017), arXiv:1707.00541[hep-ex
    [22] CMS Collaboration, arXiv:1901.00174
    [23] ATLAS Collaboration, G. Aad et al, Eur. Phys. J. C, 76(12): 658 (2016), arXiv:1602.04516[hep-ex
    [24] ATLAS Collaboration, M. Aaboud et al, JHEP, 03: 095 (2018), arXiv:1712.02304[hep-ex
    [25] M. B. Voloshin, Phys. Rev. D, 86: 093016 (2012), arXiv:1208.4303[hep-ph
    [26] F. Bishara, Y. Grossman, R. Harnik et al, JHEP, 04: 084 (2014), arXiv:1312.2955[hep-ph
    [27] Y. Chen, R. Harnik, and R. Vega-Morales, Phys. Rev. Lett., 113(19): 191801 (2014), arXiv:1404.1336[hep-ph doi: 10.1103/PhysRevLett.113.191801
    [28] D. A. Dicus and S. S. D. Willenbrock, Phys. Rev. D, 37: 1801 (1988)
    [29] L. J. Dixon and M. S. Siu, Phys. Rev. Lett., 90: 252001 (2003), arXiv:hep-ph/0302233[hep-ph doi: 10.1103/PhysRevLett.90.252001
    [30] S. P. Martin, Phys. Rev. D, 86: 073016 (2012), arXiv:1208.1533[hep-ph
    [31] D. de Florian, N. Fidanza, R. J. Hernndez-Pinto et al, Eur. Phys. J. C, 73(4): 2387 (2013), arXiv:1303.1397[hep-ph
    [32] S. P. Martin, Phys. Rev. D, 88(1): 013004 (2013), arXiv:1303.3342[hep-ph
    [33] L. J. Dixon and Y. Li, Phys. Rev. Lett., 111: 111802 (2013), arXiv:1305.3854[hep-ph doi: 10.1103/PhysRevLett.111.111802
    [34] J. Campbell, M. Carena, R. Harnik et al, arXiv: 1704.08259[hep-ph]
    [35] A. Djouadi, J. Ellis, and J. Quevillon, JHEP, 07: 105 (2016), arXiv:1605.00542[hep-ph
    [36] B. Lillie, J. Shu, and T. M. P. Tait, Phys. Rev. D, 76: 115016 (2007), arXiv:0706.3960[hep-ph
    [37] L. Bian, D. Liu, J. Shu et al, Int. J. Mod. Phys., 31(14n15): 1650083 (2016), arXiv:1509.02787[hep-ph doi: 10.1142/S0217751X16500834
    [38] Z. Bern, A. De Freitas, and L. J. Dixon, JHEP, 09: 037 (2001), arXiv:hep-ph/0109078[hep-ph
    [39] Z. Bern, L. J. Dixon, and C. Schmidt, Phys. Rev. D, 66: 074018 (2002), arXiv:hep-ph/0206194[hep-ph
    [40] J. M. Campbell, R. K. Ellis, and C. Williams, JHEP, 07: 018 (2011), arXiv:1105.0020[hep-ph
    [41] CMS Collaboration, C. Collaboration, CMS-PAS-HIG-16-040
    [42] ATLAS Collaboration, M. Aaboud et al, arXiv: 1802.04146[hep-ex]
    [43] ATLAS Collaboration, ATLAS, Tech. Rep. ATL-PHYS-PUB-2013-014, CERN, Geneva, Oct, 2013. http://cds.cern.ch/record/1611186
  • 加载中

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13. Huang, H.X., Jin, X., Tan, Y. et al. The exotic hadron states in quenched and unquenched quark models[J]. EPL, 2021, 135(3): 31001. doi: 10.1209/0295-5075/ac25aa

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Xia Wan and You-Kai Wang. Probing the CP violating Hγγ coupling using interferometry[J]. Chinese Physics C, 2019, 43(7): 073101. doi: 10.1088/1674-1137/43/7/073101
Xia Wan and You-Kai Wang. Probing the CP violating Hγγ coupling using interferometry[J]. Chinese Physics C, 2019, 43(7): 073101.  doi: 10.1088/1674-1137/43/7/073101 shu
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Probing the CP violating Hγγ coupling using interferometry

    Corresponding author: Xia Wan, wanxia@snnu.edu.cn
    Corresponding author: You-Kai Wang, wangyk@snnu.edu.cn
  • School of Physics & Information Technology, Shaanxi Normal University, Xi’an 710119, China

Abstract: The diphoton invariant mass distribution from the interference between ggHγγ and ggγγ is almost antisymmetric around the Higgs mass MH. We propose a new observable Aint, the ratio of the sign-reversed integral around MH (e.g. MHMH5 GeVMH+5 GeVMH) and the cross-section of the Higgs signal, to quantify this effect. We study Aint both in the Standard Model (SM) and new physics with various CP violating Hγγ couplings. Aint in SM could reach a value of 10%, while for CP violating Hγγ coupling Aint could range from 10% to −10%, which could probably be detected in the HL-LHC experiments. Aint with both CP violating Hγγ and Hgg couplings is also studied, and its range of values is found to be slightly larger.

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    1.   Introduction
    • CP violation, as one of the three Sakharov conditions [1], is necessary for explaining the matter-antimatter asymmetry in our universe [2]. Its source could have a close relation with Higgs dynamics [3, 4]. Thus the CP properties of the 125 GeV Higgs boson with spin zero are proposed to be probed in various channels at the Large Hadron Collider (LHC) [5-24]. Among them, the golden channel HZZ4 has been studied extensively and it gives relatively stringent experimental constraints [19, 21, 22, 24]. On the contrary, the Hγγ process is another golden channel for discovering the Higgs boson and has a relative clean signature, but it suffers from a lack of CP-odd observable constructed from the self-conjugated diphoton kinematic variables. The CP property of the Hγγ coupling can also be studied in the Hγγ4 process [25-27]. However, it is challenging due to the low conversion rate of the off-shell photon decaying into two leptons. In this paper, we study the CP property of the Hγγ coupling using the interference between ggHγγ and ggγγ.

      This interference has been studied in many papers [28-35]. Compared to the Breit-Wigner line shape of the Higgs boson signal, the line shape of the interference term can be roughly divided into two parts: one is symmetric around MH , and the other is antisymmetric around MH. These two kinds of interference line shapes have different effects. After integrating over a symmetric mass region around MH, the symmetric interference line shape could reduce the signal Breit-Wigner cross-section by ~ 2% [34], while the antisymmetric one has no contribution to the total cross-section, but could distort the signal line shape, and shift the resonance mass peak by 150 MeV [30, 33]. Besides, a variable Ai is proposed [36, 37] to quantify the interference effect in a sophisticated way, which defines a sign-reversed integral around MH (e.g. MHMH5GeVdMMH+5GeVMHdM) in its numerator and a sign-conserved integral around MH (e.g. MHMH5GeVdM+MH+5GeVMHdM) in its denominator, where both integrands have an overall line shape which is a superposition of the signal line shape, the symmetric interference line shape and the antisymmetric interference line shape. In principle, all three effects from the interference, the changing signal cross-section, shifting resonance mass peak and Ai (the ratio of sign-reversed integral and sign-conserved integral), could be used to probe CP violation in Hγγ coupling, but their sensitivities are different. As the symmetric interference line shape derives mainly from the next-to-leading order, while the antisymmetric one comes from the leading order [29, 34], the effect from antisymmetric interference line shape has a better sensitivity, which means that the latter two effects could be more sensitive to CP violation.

      Obtaining Ai experimentally is not trivial, and can be affected greatly by the mass uncertainty of MH [37]. The main reason is that if MH slightly changes, the sign-reversed integral in the numerator gets a large extra value from the signal line shape. To solve this problem, we suggest to first separate the antisymmetric interference line shape from the overall line shape, and then replace the integrand in the numerator with only the antisymmetric interference line shape. Thus the effect of mass uncertainty in the observable is suppressed. The new modified observable is named Aint , and it is used to quantify the interference effect in our analysis.

      In this paper, we study the CP property of the Hγγ coupling using the interference between ggHγγ and ggγγ. The rest of the paper is organized as follows. In Section 2, we introduce the effective model with a CP violating Hγγ coupling, and calculate the interference between ggHγγ and ggγγ. Then, we introduce the observable Aint and study its dependence on CP violation. In Section 3, we simulate the line shapes of the signal and the interference, and get Aint in SM and various CP violation cases. After that, we estimate the feasibility of measuring Aint at the LHC and the High Luminosity Large Hadron Collider (HL-LHC). In Section 4, we build the general framework for the CP violating Hγγ and Hgg couplings, and study Aint using the same procedure as above. In Section 5, we give a conclusion and discussion.

    2.   Theoretical calculation
    • The effective model with a CP violating Hγγ coupling is given as,

      Lh=cγcosξγvhFμνFμν+cγsinξγ2vhFμν˜Fμν+cgvhGaμνGaμν,

      (1)

      where F, Ga denote the γ and gluon field strengths, a = 1, ..., 8 are SU(3)c adjoint representation indices for the gluons, v = 246 GeV is the electroweak vacuum expectation value, the dual field strength is defined as ˜Xμν=ϵμνσρXσρ, cγ and cg are the effective couplings in SM to leading order, and ξγ[0,2π) is a phase that parametrizes CP violation. When ξγ=0, this is the SM case; when ξγ0, there must exist CP violation (except for ξγ=π) and new physics beyond SM. This kind of parametrization makes certain that the total signal strength of the Higgs decay into diphoton is equal to the prediction of SM.

      In SM, to leading order, cγ is brought by the fermion and W loops, and cg is due to the fermion loops only, which can be expressed as

      cg=αs16πf=t,bF1/2(4m2f/ˆs),

      (2)

      cγ=α8π[F1(4m2W/ˆs)+f=t,bNcQ2fF1/2(4m2f/ˆs)],

      (3)

      where αs(α) are the running QCD (QED) couplings, Nc=3, Qf and mf are the electric charge and mass of the fermions, and

      F1/2(τ)=2τ[1+(1τ)f(τ)],

      (4)

      F1(τ)=2+3τ[1+(2τ)f(τ)],

      (5)

      f(τ)={arcsin21/ττ1,14[log1+1τ11τiπ]2   τ<1.

      (6)

      The helicity amplitudes for ggHγγ and ggγγ can be written as [30, 38, 39],

      M=eih3ξγδh1h2δh3h4δabM4γγv24cgcγM2γγM2H+iMHΓH+4ααsδabf=u,d,c,s,bQ2fAh1h2h3h4box,

      (7)

      where a, b are the same as a in Eq. (1), the spinor phases (see their exact formulas in [38, 39] and [16]) are dropped for simplicity, hi are the helicities of outgoing gluons and photons, Qf is the electric charge of the fermion, Ah1h2h3h4box are the reduced 1-loop helicity amplitudes of ggγγ mediated by five flavor quarks, while the contribution from the top quark is considerably suppressed [28] and is neglected in our analysis. Abox for non-zero interference is [30, 38, 39]

      A++++box=Abox=1,A++box=A++box=1+zln(1+z1z)1+z24[ln2(1+z1z)+π2],

      (8)

      where z=cosθ, with θ the scattering angle of γ in the diphoton center-of-mass frame. It may be noted by the careful reader that we use the formulas for A++++/box and A++/++box as in [38, 39], while they are exchanged in [30]. This is because the convention we use here is for outgoing gluons, while the helicities have a reversed sign for incoming gluons. It is also worth noting that Eq. (7) is different from Eq. (2) in Ref. [30] because of the eih3ξγ factor, which ensures that the Higgs signal strength is not affected by the CP violation factor ξγ, while the interference strength has a simple cosξγ dependence (see Eqs. (9) and (10)).

      After considering the interference, the line shape of the smooth background is composed of both the signal and interference line shapes, and can be expressed by

      dσsigdMγγ=G(Mγγ)128πMγγ|cgcγ|2(M2γγM2H)2+M2HΓ2H×dz,

      (9)

      dσintdMγγ=G(Mγγ)128πMγγ(M2γγM2H)Re(cgcγ)+MHΓHIm(cgcγ)(M2γγM2H)2+M2HΓ2H×dz[A++++box+A++box]×cosξγ,

      (10)

      where σsig,σint are the cross-sections of the signal and interference terms, respectively, Mγγ=ˆs, the integral region z depends on the detector angle coverage, and G(Mγγ) is the gluon-gluon luminosity function written as

      G(Mγγ)=1M2γγ/sdxsx[g(x)g(M2γγ/(sx))].

      (11)

      The interference term consists of two parts: the antisymmetric term (the first term in Eq. (10)), and the symmetric term (the second term in Eq. (10)) around the Higgs boson mass. It is worth noting that to leading order Im(cSMgcSMγ) is suppressed by mb/mt compared to Re(cSMgcSMγ), because the imaginary parts of cSMg, cSMγ are mainly from the bottom quark loop, while their real parts are from the top quark or W boson loops. Thus, the symmetric part of the interference term is suppressed to leading order and its contribution to the total cross-section is mainly from the next-to-leading order [28, 34]. In contrast, the antisymmetric term can have a large magnitude around MH.

      The observable Aint extracts the antisymmetric part of the interference by the sign-reversed integral around MH, which is defined as

      Aint(ξγ)=dMγγdσintdMγγΘ(MγγMH)dMγγdσsigdMγγ,

      (12)

      where the region of integration is around the Higgs resonance (e.g. [121,131] GeV for MH=126 GeV), and the Θ-function is

      Θ(x){1,x<01,x>0.

      Therefore, the numerator is the antisymmetric contribution from the interference, and the denomenator is the cross-section from the signal, so that Aint is an observable that roughly gives the ratio of the interference to the signal.

      As ξγ=0 represents the SM case, we can define ASMintAint(ξγ=0) and rewrite Aint(ξγ) simply as

      Aint(ξγ)=ASMint×cosξγ.

      (13)

      The largest deviation Aint(π)=ASMint occurs when ξγ=π, which represents the inverse CP-even Hγγ coupling from new physics without CP violation. It is interesting that this degenerate coupling can only be revealed by the interference effect.

    3.   Numerical results
    • The numerical results are obtained for proton-proton collisions at s=14 TeV by using the MCFM [40] package, in which the subroutines for helicity amplitudes of Eq. (7) are added. The Higgs boson mass and width are set as MH=126 GeV, and ΓH=4.3 MeV. Each photon is required to have pγT>20 GeV and |ηγ|<2.5. Based on the simulation, we study ASMint first, and then Aintfor the CP violation cases. Finally, we estimate the feasibility of obtaining Aint at the LHC.

    • 3.1.   ASMint

    • Fig. 1 shows the theoretical line shapes of the signal (a sharp peak shown in the black histogram) and the interference (a peak and dip shown in the red histogram); Fig. 1(a) is the overall plot, Fig. 1(b) and Fig. 1(c) are close-ups. As shown in Fig. 1(a) and Fig. 1(b), the signal has a mass peak that is about four times larger than the interference. The mass peak of the interference is wider and has a much longer tail. The resonance region [125.9,126.1] GeV is shown in Fig. 1(c) with a bin width reduced from 100 MeV to 2 MeV. The signal exceeds the interference from the energy MγγMH10×ΓH. After integrating, ASMint is 36% , as shown in Table 1, which is quite large. As the smearing from the mass resolution (MR) is not considered yet, we denote this case as σMR=0.

      σMR (GeV)ASMint denominator (fb)ASMint numerator (fb)ASMint (%)
      039.314.336.3
      1.139.34.010.2
      1.339.33.79.4
      1.539.33.48.6
      1.739.33.17.9
      1.939.32.87.2

      Table 1.  ASMint values for different mass resolution widths. σMR=0 represents the theoretical case before Gaussian smearing.

      Figure 1.  (color online) Diphoton invariant mass Mγγ distribution of the signal and the interference given by Eq. (9) and (10). ξγ = 0 represents the SM case, σMR = 0 represents the theoretical distribution before Gaussian smearing. (a) is the overall plot, (b) and (c) are close-ups.

      The invariant mass of the diphoton Mγγ has a mass resolution of about 12 GeV in the CMS experiment [41]. For simplicity we include the mass resolution by convoluting the histograms with a Gaussian function with widths σMR=1.1,1.3,1.5,1.7,1.9 GeV. This convolution procedure is also called Gaussian smearing. Fig. 2 shows the line shapes after Gaussian smearing with σMR=1.5 GeV. The sharp peak of the signal becomes a wide bump (the black histogram), while the peak and dip of the interference are also wider. As they cancel each other near MH, the former peak and dip take a moderately antisymmetric shape around MH (the red histogram). ASMint after Gaussian smearing is thus reduced, and ranges from 10.2% to 7.2% when σMR increases from 1.1 to 1.9 GeV, as shown in Table 1.

      Figure 2.  (color online) Diphoton invariant mass Mγγ distribution after Gaussian smearing with mass resolution width σMR = 1.5 GeV.

    • 3.2.   Aint(ξγ0)

    • Fig. 3 shows the interference line shapes when ξγ=0,π,π/2 and σMR=1.5 GeV. The blue histogram (ξγ=π, sign-reversed CP-even Hγγ coupling) is almost opposite to the red histogram (ξγ=0, SM), and they correspond to the minimum and maximum of Aint . The black dashed histogram (ξγ=π/2, CP-odd Hγγ coupling) looks like a flat line (actually with some tiny fluctuations from the simulation), and corresponds to zero ofAint. Fig. 4 shows Aint and its absolute statistical error δAint. The statistical error is estimated using an integrated luminosity of 30 fb−1, and the efficiency of the detector is assumed to be one. δAint decreases as Aint becomes smaller. However, the relative statistical error δAint/Aint increases quickly and becomes very large as Aint approaches zero. In SM (ξγ=0 in Fig. 4), the relative statistical error δAint/Aint is about 18% with the assumption of zero correlation between symmetric and antisymmetric cross-sections.

      Figure 3.  (color online) Diphoton invariant mass Mγγ distribution of the interference after Gaussian smearing with σMR=1.5 GeV for ξγ=0, π, π/2.

      Figure 4.  (color online) Aint values (red line) and their statistical errors (shade) for different phase ξγ.

    • 3.3.   Aint at the LHC

    • In the CMS or ATLAS experiments, the γγ mass spectrum is fitted by a signal function and a background function. To consider the interference effect, the antisymmetric line shape should also be included. That is, instead of a Gaussian function (or a double-sided Crystal Ball function) as the signal in the LHC experiments [41, 42], a Gaussian function (or a double-sided Crystal Ball function) plus an asymmetric function should be used as the modified signal, while the background should be kept the same.

      To see whether or not the asymmetric line shape could be extracted, we carry out a fit of the modified signal from two background-subtracted data samples. As the background fluctuation would be dealt similarly as in the real experiment, we ignore it here for simplicity. One data sample is from the CMS experiment Ref. [41], from where we take 10 data points with their errors between [121,131] GeV in the background-subtracted γγ mass spectrum for 35.9 fb−1 integrated luminosity with proton-proton collision energy of 13 TeV (see Fig. 13 in Ref. [14]). The fitting function is given as

      f(m)=c1×fsig(mδm)+c2×fint(mδm),

      (14)

      where c1,c2,δm are the fitting parameters, m is the γγ invariant mass, the functions fsig(m),fint(m) are evaluated from the two histograms in Fig. 2 and they describe the signal and interference. Fig. 5 shows the fit result of the CMS data, in which the crosses represent CMS data with their errors, the red solid line is the combined function, and the black dashed line and the blue dotted line are the signal and interference components, respectively. The black dashed line is almost the same as the red solid line, while the blue dotted line is almost flat. The fitting parameter c2 for the interference component has a huge uncertainty that is even larger than the central value of c1, which indicates that it is hard to extract the interference component from 35.9 fb-1 of CMS data. For comparison, we simulate a pseudo-data sample from the combined histogram in Fig. 2, which is normalized to about 80 times the amount of CMS data (corresponding to an integrated luminosity of 3000 fb-1), with a bin width of 0.5 GeV and Poission fluctuation. The result of the fit is shown in Fig. 6, where the red solid line is shifted from the black dashed line, and the blue dotted line can be clearly distinguished. c1 and c2 are fitted as c1=0.999±0.002, c2=0.947±0.028, which are consistent with their SM expected value 1 and deduce to a relative error of Aint 3% according to the error propagation formula. Even though this fitting result looks quite good, it can only reflect that the antisymmetric lineshape could be extracted out when no contamination comes from systematic error. Furthermore, our study shows that the optimal fitting strategy is taking Higgs mass as a free parameter together with c1 and c2. Although MH has been measured in many channels, its fluctuation is usually too large to get a converged fitting if we take it as a known input value.

      Figure 5.  (color online) Fit of the background-substracted CMS data sample. The crosses represent CMS data from Ref. [41]. The red solid line is the combined function, the black dashed line and the blue dotted line represent the signal and interference components, respectively.

      Figure 6.  (color online) Fit of the simulated data sample. The crosses represent simulated data from the combined histogram in Fig. 2 normalized to an integrated luminosity of 3000 fb−1. The red solid line is the combined function, the black dashed line and the blue dotted line represent the signal and interference components, respectively.

      In contrast, a simulation that also studied the interference effect including the systematic errors has been carried out by the ATLAS collaboration for the HL-LHC with an integrated luminosity of 3000 fb−1 [43]. In that simulation, the mass shift of the Higgs boson caused by the interference effect has been studied with different assumptions for the Higgs width. A pseudo-data was produced by smearing a Breit-Wigner distribution with a model of the detector resolution, and the interference effect was described by the shift of the smeared Breit-Wigner distribution. Based on fitting, the mass shift of the Higgs from the interference effect was estimated to be ΔmH=54.4 MeV for the SM case, and the systematic error of the mass difference was about 100 MeV. If this result is used to estimate the mass shift effect for the non-SM ξγ0 cases, ( ξγ=π/2 corresponds to a zero mass shift, and ξγ=πto a reverse mass shift of ΔmH=+54.4 MeV, as shown in Fig. 3), then the largest deviation of the mass shift from the SM case is 2×54.4 MeV (when ξγ=π), which is almost covered by the systematic error of 100 MeV. Therefore, the non-SM ξγ0 cases can not be distinguished using this mass shift effect. Nevertheless, it is worth noting that the antisymmetric line shape of the theoretical interference effect is quite different from the shift of two smeared Breit-Wigner distributions in the ATLAS simulation [43], especially in the region far from the Higgs peak, where the antisymmetric line shape of the interference effect has a long flat tail while the Breit-Wigner distribution falls quickly. The authors of the ATLAS study have also noted this difference and have planned to include it in their new search [43].

    4.   CP violation in Hgg coupling
    • In the above study, Hgg coupling is assumed to be SM-like. Furthermore, the observable Aint could also be used to probe CP violation in Hgg coupling. In this section, we add one more parameter, ξg , to describe CP violation in Hgg coupling, and study Aint following the same procedure as above.

      Based on Eq. (1), the parameter ξg to describe CP violation in Hgg coupling is added, and the effective Lagrangian is modified as

      Lh=cγcosξγvhFμνFμν+cγsinξγ2vhFμν˜Fμν+cgcosξgvhGaμνGaμν+cgsinξg2vhGaμν˜Gaμν.

      (15)

      The helicity amplitude in Eq. (7) and the differential cross-section of the interference in Eq. (10) should be changed correspondingly, and become

      M=eih1ξgeih3ξγδh1h2δh3h4δabM4γγv24cgcγM2γγM2H+iMHΓH+4ααsδabf=u,d,c,s,bQ2fAh1h2h3h4box,

      (16)

      dσintdMγγ(M2γγM2H)Re(cgcγ)+MHΓHIm(cgcγ)(M2γγM2H)2+M2HΓ2H×dz[cos(ξg+ξγ)A++++box+cos(ξgξγ)A++box].

      (17)

      Then, ASMintAint(ξg=0,ξγ=0) and

      Aint(ξg,ξγ)=ASMint×dz[cos(ξg+ξγ)A++++box+cos(ξgξγ)A++box]dz[A++++box+A++box],

      (18)

      where the integral can be calculated numerically once the region of z integration is given. For example, if the pseudorapidity of γ is required to be |ηγ|<2.5, that is, z[0.985,0.985], the integral dzA++box9, and Eq. (18) can be simplified as

      Aint(ξg,ξγ)ASMint×2cos(ξg+ξγ)9cos(ξgξγ)7.

      (19)

      Aint(ξg,ξγ) thus has the maximum and minimum about 1.6 times that of ASMint. If ξg=0, Aint(ξg=0,ξγ) degenerates to Aint(ξγ) in Eq. (13). In contrast, if ξγ=0,

      Aint(ξg)=ASMint×cos(ξg),

      (20)

      which shows the same dependence as Aint(ξγ) on ξγ when ξg=0 as in Eq. (13). Hence, a CP violating Hgg coupling can cause similar deviation of Aint from ASMint as a CP violating Hγγ coupling, and a single observed Aint can not distinguish between them since there are two free parameters for one observable.

      Fig. 7 shows the interference line shapes for different ξg,ξγ . The red histogram (ξg=0,ξγ=0) represents the SM case; the magenta histogram (ξg=π2,ξγ=π2) has the largest Aint; the cyan histogram (ξg=π2,ξγ=3π2) corresponds to the smallest Aint ; and the black histogram is for the case of ξg=0,ξγ=π2 with Aint equal to zero. In the general case where both ξg,ξγ are free parameters, Aint(ξg,ξγ) has a wider range of values than Aint(ξγ), which makes it easier to probe in future experiments.

      Figure 7.  (color online) Diphoton invariant mass Mγγ distribution of the interference after Gaussian smearing in various ξg, ξγ cases.

    5.   Conclusion and discussion
    • The diphoton mass distribution from the interference between ggHγγ and ggγγ to leading order is almost antisymmetric around MH and we propose a sign-reversed integral around MH to get its contribution. After dividing the integral by the cross-section of the Higgs signal, we get the observable Aint. In SM, the theoretical value ofAint , before taking into account the mass resolution, is ~ 39%. After considering the mass resolution of 1.5 GeV, Aint is reduced, but still could be as large as 10%. CP violation in Hγγ could change Aint from 10% to -10% , depending on the CP violation phase ξγ. In the general framework of CP violating Hγγ and Hgg couplings, Aint could have a larger value, ~ ±16%. However, due to the systematic and statistical errors which are both ~ 10% in the current experiments at the LHC, it is difficult to extract the antisymmetric line shape. Even with future high luminosities, the large systematic error is still a serious obstacle.

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