Supplementation in Theory Parts - ATLAS実験における重心系衝突エネルギー8 TeVでの陽子-陽子衝突のデータを用いたトップクォークとボトムクォークに崩壊

A.1 Crystal-Ball Function

This function is a probability density function, which consists of a Gaussian with a low tail separated by a certain threshold. We often use the function for describing the effect of radiative energy loss in an invariant mass. The Crystal-Ball function is expressed by

f(x;x, σ, α, n) =





A·exp£

−¹₂¡_x₋_x

¢₂¤

(^x⁻_σ^x >−α), A·(ⁿα)ⁿ^exp[−¹2α²]

(^x−x_σ ⁺ⁿ_α−α)ⁿ (^x⁻_σ^x <−α), (A.1) wherex is a mean value, σ is a deviation,α andnare fitting parameters. A in the equation is expressed by

A⁻¹ =σ·

·n α · 1

n−1exp

−1 2α²

¸ +

rπ 2erf

µ α

√2

¶¸

, (A.2)

whereerf indicates a error function.

A.2 MSSM Benchmark Scenarios after the Discovery of the Higgs Boson

There are MSSM scenarios with different mass ranges of the SM Higgs boson. In order to keep consistency in the scenarios after the discovery of the Higgs boson at the LHC, we have to modify the scenarios by adjusting to the SM Higgs mass of 125.5 GeV [24].

The SUSY parameters are important to decide the mass. The mass matrices in the eigenstates of chirality for the stop and sbottom sectors of the MSSM is expressed by

M²˜t = Ã M_˜_t²

L+m²_t + cos 2β(¹₂−²₃s²_W)M_Z² m_tX_t^∗ m_tX_t M_˜_t²

R+m²_t +²₃cos 2βs²_WM_Z²

, (A.3) M²˜b =

Ã M_˜²

bL+m²_b + cos 2β(−¹₂ +¹₃s²_W)M_Z² m_bX_b^∗

m_bX_b M_˜²

bR +m²_b−¹₃cos 2βs²_WM_Z²

, (A.4) where

mtXt=mt(At−µ^∗cotβ), m_bX_b =m_b(A_b−µ^∗tanβ). (A.5) A_t denotes the trilinear Higgs-stop coupling, A_b denotes the Higgs-sbottom coupling, µ is the higgsino mass parameter, ands_W is expressed by

1−c²_W withc_W =M_W/M_Z. We concentrate on the case

M_˜_t_L=M_˜_b

L=M_˜_t_R =M_˜_b

R =:M_SUSY. (A.6)

There are the corresponding soft SUSY-breaking parameters in the scalar tau/neutrino sector, which are denoted asA_τ andM_l_˜

3. We assume that the diagonal soft SUSY-breaking entries in the stau/sneutrino mass matrices to be equal to each other. We also assume that the mass matrices for the squarks and sleptons of the first and second generations have the equality of the diagonal soft SUSY-breaking parameters, which are denoted as M_q_˜_1,2 for the squarks and M_˜_l

1,2 for the sleptons. In the off-diagonal components of the squarks and sleptons, theA-terms multiplied by with the corresponding fermion mass always appear. In the benchmark scenarios, we can neglect theA-terms associated with the first and second sfermion generations.

The gaugino masses affect the Higgs sector. We denote theSU(2)andU(1)gaugino mass parameters asM₂ andM₁respectively. The parameters are usually assumed to be related via the GUT relation,

M1 = 5 3

s²_W

c²_WM2. (A.7)

There are several approaches to evaluate the loop corrections to the MSSM Higgs boson sector.

The program FEYNHIGGS is based on results obtained in the Feynman-diagrammatic (FD) ap-proach. The FD results have been obtained in the on-shell (OS) renormalization scheme. The program CPSUPERH is based on results obtained in the renormalization group (RG) improved effective potential approach. The RG results have been calculated by using theMSscheme. The two approaches provide different values of the parametersX_tandM_SUSYbecause the parameters are scheme-dependent. The change of scheme induces in general only a minor shift, of the order of 4%, in the parameterM_SUSY, but sizable differences can occur between the numerical values ofX_tin the two schemes.

The parameters Mq˜1,2, M˜l1,2, and Af with f = c, s, u, d, µ, ehave a minor impact on the MSSM Higgs sector prediction, therefore, we fix them to the following values,

M_q_˜_1,2 = 1500GeV, (A.8)

M_˜_l

1,2 = 500GeV, (A.9)

A_f = 0. (A.10)

We set the top quark mass to 173.2 GeV. For each MSSM scenario, the parameters are set as follows.

• The m^max_h scenario (The value of Xt is chosen to maximize the lightest CP-even Higgs mass,m_h⁰. This is an old-fashioned benchmark scenario.): M_SUSY is 1000 GeV,µis 200 GeV,M₂ is 200 GeV,X_t^OSis 2M_SUSY (FD calculation),X_t^MSis√

6M_SUSY (RG calcula-tion),A_b =A_τ =A_t,M_˜_gis 1500 GeV,M_˜_l

3 is 1000 GeV.

• Them^mod+_h scenario (them_h⁰ is close to the discovered Higgs boson mass by reducing the amount of mixing in the stop sector and the sign ofX_tis plus.): M_SUSY is 1000 GeV,µ is 200 GeV,M₂ is 200 GeV,X_t^OSis 1.5M_SUSY (FD calculation), X_t^MS is 1.6M_SUSY (RG calculation),A_b=A_τ =A_t,M_g_˜is 1500 GeV,M_˜_l

3 is 1000 GeV.

• Them^mod_h ⁻scenario (them_h⁰ is close to the discovered Higgs boson mass by reducing the amount of mixing in the stop sector and the sign ofX_tis minus.):M_SUSYis 1000 GeV,µis 200 GeV,M2is 200 GeV,X_t^OSis−1.9MSUSY(FD calculation),X_t^MSis−2.2MSUSY (RG calculation),A_b=A_τ =A_t,M_g_˜is 1500 GeV,M_˜_l

3 is 1000 GeV.

• The light stop scenario (the m_h⁰ is close to the discovered Higgs boson mass by a large value of|X_t|and a relatively low value ofM_SUSY, and a light stop may lead to a relevant modification of the gluon fusion rate on the lightest Higgs boson production.): M_SUSY is 500 GeV,µ is 350 GeV,M₂ is 350 GeV,X_t^OS is 2.0M_SUSY (FD calculation), X_t^MS is 2.2M_SUSY (RG calculation),A_b=A_τ =A_t,M_˜_gis 1500 GeV,M_˜_l

3 is 1000 GeV.

• The light stau scenario with∆τ calculation (them_h⁰ is close to the discovered Higgs boson mass and a light stau may lead to important modification of the diphoton decay width of the lightest Higgs boson by the large mixing in the stau sector. The∆τ calculation indicates that the∆τ corrections are not neglected in the stau mass matrix.):M_SUSY is 1000 GeV,µ is 450 GeV,M₂ is 400 GeV,X_t^OSis 1.6M_SUSY (FD calculation), X_t^MS is 1.7M_SUSY (RG calculation),A_b=A_t,A_τ = 0,M_˜_gis 1500 GeV,M_˜_l

3 is 250 GeV.

A.3 Sudakov Form Factor

This factor represents the probability of evolving into softer scale without gluon emissions from harder scale. For example, the Sudakov form factor of an initial state parton is given by

∆(t, t0) = exp

− Z t

dt^′ t^′

Z dz z

α_s

2πP(z)f(x/z, t) f(x, t)

, (A.11)

wheretis the hard scale, t₀ is the cutoff scale, zis the momentum fraction, f(x, t)is the PDF with the scale t and the fraction x, and P(z) is the splitting function for the branching under consideration. In this case, the factor describes the probability that no emission has been occurred betweentandt₀.

In case of the clustering procedure in the CKKW technique, the NLL Sudakov form factors are used:

∆_q(Q, Q₀) = exp

− Z _Q

dqΓ_q(Q, q)

, (A.12)

∆g(Q, Q0) = exp

− Z Q

dq(Γg(Q, q) + Γ_f(q))

, (A.13)

whereQis the hard scale,Q₀is the cutoff scale,qis the resolution scale,Γ_q,g,fare the integrated splitting functions forq→qg,g→ggandg→qqprocesses. TheΓare expressed by

Γ_q(Q, q) = 2C_F π

α_s(q) q

µ lnQ

q −3 4

, (A.14)

Γ_g(Q, q) = 2C_A π

α_s(q) q

µ lnQ

q −11 12

, (A.15)

Γ_f(q) = N_f 3π

α_s(q)

q , (A.16)

whereN_f is the number of quark flavours,C_F is the color factor forq →qgandC_Ais the color factor forg→gg. The color factors are defined by

t^a_ikt^a_kj =X

(t^at^a)≡C_Fδij with t^a≡ λ^a 2 and

·λi

2,λj

=iX

f_ijkλ_k

2 , (A.17) X

a,b

f_abcf_abc=CAδ_cd, (A.18) wheret^a_ik,kj are theSU(3)generators,f_ijk,abc are theSU(3)structure constants. In QCD,C_F is 4/3 andCAis 3. We employ the Sudakov form factors for the reweighting related to the appearance of external parton lines. For the reweighting related to internal parton lines, we use a ratio of two Sudakov form factors ∆(Q, Q₀)/∆(q, Q₀), which describes the probability of no emission resolvable atQ0betweenQandq.

A.4 Treatment of W Boson with Two Jets Process

For example, we discuss the W boson production in association with two gluons from qq annihilation. Figure A.1 shows the Feynman diagrams of the process with a soft gluon. In the figure, gluon 1 represents the soft gluon. The upper two diagrams have a gluon emission from the internal line of an adjacent propagator and the lower two diagrams have a gluon emission from the external line of a gluon. In case of the limit that one of the gluons is soft, the singularities in the matrix elements occur in the upper diagrams.

A.4.1 Color Flow

We denote the momenta of the gluon 1 and the gluon 2 asp₁ andp₂ respectively. We also denote the color labels of p₁ and p₂ by t^A and t^B respectively, where t^A,B indicate the color matrix of SU(3). The diagram D1 in Figure A.1 is proportional tot^Bt^A. The diagram D2 in the figure is proportional tot^At^B. Both diagram D3a and D3b are proportional tof^ABCt^C, which can be expressed by(t^At^B−t^Bt^A). We can write the amplitude in the limit of the softp₁ as

M^qq^→^{W gg} =t^At^B(D₂+D₃) +t^Bt^A(D₁−D₃), (A.19) whereD1andD2are the amplitude from the Feynman diagram D1 and D2 respectively,D3is the sum of diagrams D3a and D3b. By squaring the amplitude in Equation (A.19), we obtain

|M^qq^→^{W gg}|² = N C_F² £

|D₂+D₃|²+|D₁−D₃|²]−C_FRe[(D₂+D₃)(D₁−D₃)^∗¤ ,

= N²C_F 2

|D₂+D₃|²+|D₁−D₃|²− 1

N²|D₁+D₂|²

, (A.20)

ドキュメント内 ATLAS実験における重心系衝突エネルギー8 TeVでの陽子-陽子衝突のデータを用いたトップクォークとボトムクォークに崩壊する荷電ヒッグス粒子の探索 (ページ 196-200)