text: 6 gradually with increase of neutron number. In case of Na, TABLE VI: Same as Table IV, for P and S isotopes. N = 3 starts ﬁlling up at 33Na with quadrupole moment osc − Nucl. RMF SHF Exp. deformation parameter β = 0.356 and 0.179 with occu- − 2 − rch β2 BE rch β2 BE rch β2 BE pied orbits [330]1 and [303]7 , respectively. The ﬁlling of 30P 3.138 0.130 246.3 3.189 0.026 249.9 250.6 N = 3 goes on2 increasing fo2r Na with neutron number and 31P 3.158 0.205 258.3 3.201 0.105 261.1 3.189 262.9 osc − − − − it is [330]1 , [310]1 , [321]3 and [312]5 at β = 0.472 32P 3.174 -0.143 267.1 3.216 0.069 270.9 270.9 2 2 2 2 2 for 39Na. Again for the oblate solution the occupation is 33P 3.201 -0.183 277.5 3.246 -0.167 280.5 281.0 − − − − − 34P 3.201 -0.082 285.8 3.248 0.001 289.9 287.2 [301]1 , [301]3 , [303]5 and [303]7 for β = 0.375 for 2 2 2 2 2 35P 3.216 -0.001 295.4 3.265 0.000 299.2 295.6 39Na. In case of Mg isotopes, even for 30,32Mg, the N = 3 osc 36P 3.227 0.120 299.5 3.272 0.007 303.3 299.1 shell has some occupation for the low-lying excited states near 37P 3.246 0.209 305.0 3.290 0.148 307.4 305.9 the Fermi surface. For 30Mg (at β = 0.599 with BE = 237.7 38P 3.260 0.250 310.4 3.313 0.240 311.7 309.6 −2 − MeV) the N = 3 orbit is [330]7 and for 32Mg is [330]1 39P 3.275 0.288 316.1 3.334 0.301 316.1 315.9 osc 2 2 (BE = 248.8 MeV at β = 0.471). With the increase of neu- 40P 3.281 0.274 320.1 3.343 0.290 319.6 319.2 2 tron number in Mg and Si isotopic chains, the oscillator shell 41P 3.288 0.261 324.4 3.355 0.295 322.7 324.2 with N = 3 gets occupied more and more. 42P 3.306 0.301 327.3 3.371 0.320 325.6 326.3 osc − 43P 3.346 -0.323 331.7 3.398 -0.320 329.0 330.7 In Tables (4 6) the results for the ground state solutions 44P 3.346 -0.302 333.3 3.398 -0.293 330.6 are displayed. Thus, the prolate solutions have more bind- 45P 3.315 0.222 335.4 3.397 -0.264 332.4 ing than the oblate one for Ne, Na, Mg and S isotopes. In − 46P 3.342 -0.251 337.5 3.397 -0.237 334.2 some cases, like 24 30Ne the prolate and oblate solutions are 47P 3.341 -0.232 340.0 3.399 -0.218 336.0 in degenerate states. For example, 24Ne has BE = 188.9 and − 48P 3.381 -0.271 341.2 3.379 0.034 337.4 189.1 MeV at β = 0.278 and 0.259 respectively. Contrary 2 49 P 3.328 0.088 343.2 3.387 0.012 339.3 to this, the ground state solutions for Al and Si are mostly 50 P 3.353 0.101 343.7 3.414 -0.061 339.2 oblate. For example, 34Al has BE = 269.9 and 275.1 MeV at 51 P 3.397 -0.166 344.7 3.437 0.068 339.4 β = 0.159 and −0.108 respectively. In such cases, the pro- 52 P 3.403 0.109 345.2 3.462 0.079 339.7 la2te solutions are in low-lying excited intrinsic state. Note that 53 P 3.428 0.109 346.3 3.487 0.089 340.1 in many cases, there exist low laying superdeformed states. 54 P 3.447 0.074 346.6 3.502 0.016 340.5 It is important to list some of the limitations of the results 55 P 3.468 0.037 347.4 3.525 0.001 341.2 33S 3.241 0.197 275.5 3.276 0.119 278.9 280.4 due to the input parameters, mostly comes from Epair and 34S 3.257 -0.168 286.5 3.300 -0.160 289.3 3.285 0.252 291.8 Ecm energies. As one can see from Fig. 3, in many cases there 35S 3.260 -0.078 295.7 3.300 -0.006 299.6 298.8 are solutions of different shapes lying only a few MeV higher, 36S 3.273 0.002 306.2 3.310 0.000 309.6 3.299 0.168 308.7 sometimes even degenerate with the ground states. Such a 37S 3.285 0.152 311.6 3.319 -0.008 315.1 313.0 few MeV difference is within the uncertainty of the predicted 38S 3.300 0.228 318.6 3.340 0.210 320.2 0.246 321.1 binding energies. A slight change in the pairing parameter, 39S 3.312 0.264 325.3 3.354 0.248 326.5 325.4 among others, may alter the prediction for the ground state 40S 3.325 0.299 332.4 3.370 0.300 332.1 0.284 333.2 shape. With few MeV uncertainty in ground state binding en- 41S 3.331 0.287 337.7 3.381 0.294 336.9 337.4 ergies, by reassigning the ground state conﬁgurations, the de- 42S 3.338 0.277 343.2 3.390 0.290 341.0 0.300 344.1 formation may change completely, and make the predictions 43S 3.359 0.318 347.2 3.413 0.326 344.7 346.7 close to each other and agree with the FRDM predictions as 44S 3.381 0.367 351.0 3.440 0.370 348.3 0.254 351.8 well. 45S 3.375 0.312 353.4 3.430 0.311 350.4 354.7 C. Quadrupole deformation 46S 3.371 0.258 356.6 3.420 0.250 352.5 47S 3.385 0.257 358.5 3.428 -0.214 354.8 48S 3.400 0.259 360.8 3.430 -0.200 356.6 The ground and low-lying excited state deformation sys- 49S 3.403 0.227 362.9 3.430 0.127 358.8 tematics for some of the representative nuclei for Ne, Na, Mg, 50S 3.403 0.189 365.5 3.440 0.120 360.8 Al, Si, P and S are analyzed. In Fig. 2, the ground state 51S 3.427 0.188 366.4 3.459 -0.090 361.8 quadrupole deformation parameter β is shown as a function 2 52S 3.451 0.183 367.6 3.490 -0.140 362.5 of mass number for Ne, Na, Mg, Al, Si, P and S. The β 2 53S 3.463 0.158 369.1 3.508 -0.113 363.6 value goes on increasing with mass number for Ne, Na and 54S 3.477 0.139 371.0 3.530 0.000 364.7 Mg isotopes near the drip-line. The calculated quadrupole de- 55S 3.494 0.105 371.4 3.541 0.030 365.4 formation parameter β for 34Mg is 0.59 which compares well 2 with the recent experimental measurement of Iwasaki et al [4] ± (β = 0.58 0.06). It found that this superdeformed state is B. Neutron conﬁguration 3.22 MeV above than the ground band. Again, the magnitude of β for the drip nuclei reduces with neutron number N and 2 Analyzing the neutron conﬁguration for these exotic nu- again increases. A region of maximum deformation is found clei, we notice that, for lighter isotopes of Ne, Na, Mg, Al, for almost all the nuclei as shown in the ﬁgure. It so happens Si, P and S the oscillator shell N = 3 is empty in the in cases like, Ne, Na, Mg and Al that the isotopes are maxi- osc [N , n , Λ]Ωπ. However, the N = 3 shell gets occupied mum deformed which may be comparable to superdeformed osc 3 osc ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## 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text: 8 Decay channel µˆ(ggF+t¯tH) µˆ(VBF+VH) ∆µ(ggF+t¯tH) ∆µ(VBF+VH) Ref. γγ (ATLAS) 1.6 1.7 0.25 0.63 [14] ∗ ZZ (ATLAS) 1.8 1.2 0.35 1.30 [15] ∗ WW (ATLAS) 0.82 1.66 0.36 0.79 [16] ττ (ATLAS) 1.1 1.6 1.16 0.75 [17] b¯b (ATLAS) – 0.2 – 0.64 [18] γγ (CMS) 0.52 1.48 0.60 1.33 [19] ∗ ZZ (CMS) 0.9 1.0 0.45 2.35 [20] ∗ WW (CMS) 0.72 0.62 0.37 0.53 [21, 22] ττ (CMS) 1.07 0.94 0.46 0.41 [23] b¯b (CMS) – 1.0 – 0.5 [24] TABLE I: The best-ﬁt signal strengths µˆ(ggF+t¯tH) and µˆ(VBF+VH) reported from the Higgs search at the ATLAS [14–18] ∗ and CMS [19–24] experiments. As for the WW channel (CMS), the value of µˆ(ggF+t¯tH) is taken from Ref.[21], and the value of µˆ(VBF+VH) is from Ref.[22]. The value of µˆ(ggF+t¯tH) for the ττ channel (CMS) is quoted from the one-jet result in Ref.[23]. value of the best-ﬁt signal strength of the 126 GeV Higgs for each production (i) and decay channel (X) reported by the experiments C ∈ {ATLAS,CMS}. We may take into account the next-to-leading-order corrections to the ggF process arising from QCD, the so-called K-factor, for the CP-even scalar [26], Kg = 1 + (215/12)α (m )/π, where h s h α (m ) is the one-loop QCD gauge coupling at the scale µ = m . In the left panel of Fig. 1 (solid curve), we show s h h ∆χ2 ≡ χ2 − χ2 as a function of cos θ, where χ2 = 13.5 at cos θ = 1 for the number of degrees of freedom being min min 18. From the left panel, we ﬁnd the 95% C.L. allowed region for cos θ: 0.97 ≤ cos θ ≤ 1 . (40) Using the mass relation between m(0) and m(0) (cid:39) m given in Eq.(10) with Eq.(26) taken into account, from Eq.(40), h0t A0t A0t we may place an indirect bound on the mass of A0. In the right panel of Fig. 1, we plot m as a function of cos θ, t A0t from which the 95% C.L. allowed region of m (horizontal solid line) reads A0 t m ≥ 923 GeV . (41) A0 t We may take into account the correlation between µˆ(ggF+t¯tH) and µˆ(VBF+VH), which can be read oﬀ from Ref.[27], though it is not in public. The 95% C.L. allowed region in Eqs.(40) and (41) would change to 0.91 ≤ cos θ ≤ 1 , (42) for cos θ and (see the horizontal dashed line in the right panel of Fig. 1) m ≥ 563 GeV , (43) A0 t for m . Thus, the incorporation of the correlation would make the lower bounds on (cos θ, m ) milder. In the A0 A0 t t present study, therefore, we shall explore the LHC phenomenology of A0 by scanning the mass in the high-mass range t [Eq.(41)] and the low-mass range [Eq.(43)]. IV. CP-ODD TOP-MODE PSEUDO (A0) AT THE LHC t In this section, we explore the LHC phenomenologies of A0 to compare the predicted A0 signals in several decay t t channels with the currently available data searching for new resonances provided from LHC Run-I. A. A0 coupling properties t (cid:18) (cid:19) We start with reading oﬀ the relevant A0 interaction terms from the Lagrangian Eq.(23): t sin3 θ m(cid:20) (cid:18)(cid:16) (cid:17) (cid:19) (cid:21) L =−i t A0t¯γ t A0t cos θ v t 5 EW 3 sin θ cos2 θ 2 − (cid:2) z0∂ A0∂µh0 − h0∂ A0(cid:3)∂µz0 − 2 m(0) sin2 θ A0h0z0 4v t µ t t t µ t t A0t t t t EW 3 sin3 θ + A0∂ z0∂µh0 − h0∂ A0∂µz0 . (44) 4v t µ t t t µ t t EW ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: Decay

text: channel

text: µˆ(ggF+t¯tH)

text: µˆ(VBF+VH)

text: ∆µ(ggF+t¯tH)

text: ∆µ(VBF+VH)

text: Ref.

text: γγ

text: (ATLAS)

text: ZZ

text: (ATLAS)

text: WW

text: (ATLAS)

text: ττ

text: (ATLAS)

text: b¯b

text: (ATLAS)

text: γγ

text: (CMS)

text: ZZ

text: (CMS)

text: WW

text: (CMS)

text: ττ

text: (CMS)

text: b¯b

text: (CMS)

text: TABLE

text: I:

text: The

label: caption

text: best-ﬁt

text: signal

text: strengths

text: µˆ(ggF+t¯tH)

text: and

text: µˆ(VBF+VH)

text: reported

text: from

text: the

text: Higgs

text: search

text: at

text: the

text: ATLAS

text: and

text: CMS

text: experiments.

text: As

text: for

text: the

text: WW

text: channel

text: (CMS),

text: the

text: value

text: of

text: µˆ(ggF+t¯tH)

text: is

text: taken

text: from

text: and

text: the

text: value

text: of

text: µˆ(VBF+VH)

text: is

text: from

text: The

text: value

text: of

text: µˆ(ggF+t¯tH)

text: for

text: the

text: ττ

text: channel

text: (CMS)

text: is

text: quoted

text: from

text: the

text: one-jet

text: result

text: in

text: value

text: of

text: the

text: best-ﬁt

text: signal

text: strength

text: of

text: the

text: GeV

text: Higgs

text: for

text: each

text: production

text: (i)

text: and

text: decay

text: channel

text: (X)

text: reported

text: by

text: the

text: experiments

text: C

text: {ATLAS,CMS}.

text: We

text: may

text: take

text: into

text: account

text: the

text: next-to-leading-order

text: corrections

text: to

text: the

text: ggF

text: process

text: arising

text: from

text: QCD,

text: the

text: so-called

text: K-factor,

text: for

text: the

text: CP-even

text: scalar

text: Kg

text: (m

text: )/π,

text: where

text: h

text: s

text: h

text: α

text: (m

text: is

text: the

text: one-loop

text: QCD

text: gauge

text: coupling

text: at

text: the

text: scale

text: µ

text: m

text: In

text: the

text: left

text: panel

text: of

text: Fig.

text: (solid

text: curve),

text: we

text: show

text: s

text: h

text: h

text: ∆χ2

text: χ2

text: χ2

text: as

text: a

text: function

text: of

text: cos

text: θ,

text: where

text: χ2

text: at

text: cos

text: θ

text: for

text: the

text: number

text: of

text: degrees

text: of

text: freedom

text: being

text: min

text: min

text: From

text: the

text: left

text: panel,

text: we

text: ﬁnd

text: the

text: C.L.

text: allowed

text: region

text: for

text: cos

text: θ:

text: cos

text: θ

text: Using

text: the

text: mass

text: relation

text: between

text: and

text: m

text: given

text: in

text: with

text: taken

text: into

text: account,

text: from

text: we

text: may

text: place

text: an

text: indirect

text: bound

text: on

text: the

text: mass

text: of

text: In

text: the

text: right

text: panel

text: of

text: Fig.

text: we

text: plot

text: m

text: as

text: a

text: function

text: of

text: cos

text: θ,

text: t

text: from

text: which

text: the

text: C.L.

text: allowed

text: region

text: of

text: m

text: (horizontal

text: solid

text: line)

text: reads

text: t

text: m

text: GeV

text: t

text: We

text: may

text: take

text: into

text: account

text: the

text: correlation

text: between

text: µˆ(ggF+t¯tH)

text: and

text: µˆ(VBF+VH),

text: which

text: can

text: be

text: read

text: oﬀ

text: from

text: though

text: it

text: is

text: not

text: in

text: public.

text: The

text: C.L.

text: allowed

text: region

text: in

text: and

text: would

text: change

text: to

text: cos

text: θ

text: for

text: cos

text: θ

text: and

text: (see

text: the

text: horizontal

text: dashed

text: line

text: in

text: the

text: right

text: panel

text: of

text: Fig.

text: m

text: GeV

text: t

text: for

text: m

text: Thus,

text: the

text: incorporation

text: of

text: the

text: correlation

text: would

text: make

text: the

text: lower

text: bounds

text: on

text: (cos

text: θ,

text: m

text: milder.

text: In

text: the

text: t

text: t

text: present

text: study,

text: therefore,

text: we

text: shall

text: explore

text: the

text: LHC

text: phenomenology

text: of

text: by

text: scanning

text: the

text: mass

text: in

text: the

text: high-mass

text: range

text: t

text: and

text: the

text: low-mass

text: range

text: IV.

text: CP-ODD

text: TOP-MODE

text: PSEUDO

text: AT

text: THE

text: LHC

text: t

text: In

text: this

text: section,

text: we

text: explore

text: the

text: LHC

text: phenomenologies

text: of

text: to

text: compare

text: the

text: predicted

text: signals

text: in

text: several

text: decay

text: t

text: t

text: channels

text: with

text: the

text: currently

text: available

text: data

text: searching

text: for

text: new

text: resonances

text: provided

text: from

text: LHC

text: Run-I.

text: A.

text: coupling

text: properties

text: t

text: We

text: start

text: with

text: reading

text: oﬀ

text: the

text: relevant

text: interaction

text: terms

text: from

text: the

text: Lagrangian

text: t

text: θ

text: L

text: =−i

text: t

text: t

text: cos

text: θ

text: v

text: t

text: EW

text: sin

text: θ

text: θ

text: θ

text: t

text: µ

text: t

text: t

text: t

text: µ

text: t

text: t

text: t

text: t

text: t

text: EW

text: θ

text: t

text: µ

text: t

text: t

text: t

text: µ

text: t

text: t

text: EW

text: ##LTLine##

text: ##LTLine##

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text: 10 (cid:48) Note the second terms of Eqs.(46) and (47), coming from integrating out the t quark. In Fig. 2, we plot the total decay width (blue-solid curve in the left panel) and branching ratios (right panel) of A0 as a function of the parent t particle mass m = (m , m ) (bottom axis) and cos θ (top axis) where m ((cid:39) m(0)) and cos θ are related to each A0t hSM A0t A0t other by the mass formula Eq.(10) with m(0) (cid:39) 230 GeV ﬁxed. As seen from the left panel, A0 is still a narrow h0 t t resonance even if the mass reaches the scale over 1 TeV, i.e., Γ /m (cid:28) 1, where the tt¯ mode rapidly damps as tot A0 t the mass increases. This happens due to the presence of the mass formula Eq.(10), which signiﬁcantly aﬀects the mass dependence of the total width for the high-mass region; as the mass gets larger, the partial decay width for the tt¯ mode goes like ∼ 1/m5 , due to the high suppression by the overall coupling ∼ (sin3 θ)2 [see Eq.(45)], where A0 t sin θ has been replaced by m(0)/m (cid:39) (230 GeV)/m . The total width is then governed by the gg mode, so that h0t A0t A0t Γ (A0) ∼ Γ(A0 → gg) ∼ m3 tan2 θ ∼ m, according to Eq.(48), which does not grow like ∼ m3 as in the case of the tot t t A0 t typical width of decays to spin-1 bosons. This is the salient feature of A0 closely related to the fact that A0 is the t t Top-Mode Pseudo partner of the Higgs h0. In the left panel, the total width is also compared with that of the SM t Higgs (red-dotted curve). This shows that A0 is indeed a narrower resonance than the SM Higgs boson for the whole t mass range. Note also the prese(cid:32)nce of a dip in(cid:33)the A0 → Zh0 channel; this takes place when the terms in the square t t bracket of Eq.(48) vanish as sin2 θ −(cid:16) m2h0t(cid:17) (cid:18)(cos2 θ −(cid:16)sin2 θ(cid:17)) + m2Z co(cid:19)s2 θ = 0 m2 m2 A0 A0 t t 2(cid:16) (cid:17) 2 m(0) m2 + 2 m(0) − 2m2 h0 Z h0 h0 ↔ m2 = t t t (cid:39) (310 GeV)2 , (51) A0t m(0) 2 − m2 + m2 h0 h0 Z t t where we used the tree-level mass formula Eq.(10) with m(0) (cid:39) 230 GeV, the one-loop mass m = 126 GeV and h0t h0t m (cid:39) 90 GeV. Z The right panel of Fig. 2 combined with the indirect limits on m in Eqs.(41) and (43) imply that the accessible A0 t decay channels of A0 at the LHC can be A0 → tt¯, gg, Zh0: (i) First is the high-mass case with m ≥ 1 TeV indicated t t t A0t from the limit in Eq.(41) (hereafter, we shall call this A0 “high-mass A0”), where the A0 → gg mode will be the t t t expected discovery channel at the LHC, which is accessible in a way similar to searches for new heavy bosons mainly decaying to gluon jets [28]; (ii) Second is the low-mass case with 563 GeV ≤ m ≤ 1 TeV indicated from the limit in A0 t Eq.(43), where the A0 → Zh0 mode will be expected as the discovery channel at the LHC, which can be seen in the t t same way as searches for other CP-odd Higgs boson in the extended Higgs sector as in the MSSM/2HDM through the decay to Zh0 [29] (this A0 will be called “low-mass A0”). Note that the A0 still emerges as the narrow resonance t t t in the Zh channel (see the left panel of Fig. 2), compared to other CP-odd scalars, like A0 in the MSSM/2HDM, which becomes a quite broader resonance with the total width of O(100) GeV in this mass range [30]. Thus, A0 can t be distinguished from A0 in the MSSM/2HDM at the LHC. This is mainly due to the presence of the intrinsic mass formula Eq.(10), which is absent in the MSSM/2HDM, as emphasized above. The predicted signals of A0 through these decay channels compared with current LHC limits will be discussed later. t C. Production cross sections of A0 t The branching fraction in the right panel of Fig. 2 implies that at the LHC A0 is mainly produced through the ggF t or top quark associate process (t¯tA) like the t¯tH production for the SM Higgs. To make a quantitative argument, it is convenient to evaluate the cross section gg/tt¯→ A0 by normalizing it with the corresponding cross section for the t SM Higgs: σ(gg/tt¯→ A0) σ(gg/tt¯→ A0)=σ(gg/tt¯→ h0 ) × t . (52) t SM σ(gg/tt¯→ h0 ) SM In Fig. 3, we plot the ratio σ(gg/tt → A0)/σ(gg/tt → h0 ) as a function of the produced particle mass m = t SM (m , m ). The mass range analyzed here has been restricted to 563 GeV ≤ m ≤ 2000 GeV, which is indicated A0 h t SM from the indirect mass limits in Eqs.(41) and (43). From Fig. 3, one can see that the t¯tA production is suppressed compared to the SM Higgs case for the mass range constrained by the indirect limits in Eqs.(41) and (43). The ggF production is somewhat suppressed compared to the SM Higgs one, which is due to the numerical suppression by the overall coupling, g ∝ 2(2)2 tan2 θ (in the heavy A0gg t quark limit) when we take cos θ ∼ 1 to be consistent with the indirect limits. Since the ggF production is much larger ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: ∈ F (cid:48) (cid:48) F To this end, let (cid:96) (T ). By deﬁnition of T , we have that (cid:96) is also in (T ). In ∈ F (cid:48) addition, (cid:96) is of th∈e Fform (cid:96)1(cid:96)A(cid:96)2 with (cid:96)A (S ). By deﬁnition of Force, (cid:96)A is of the form σ σ with σ (R), that is, consisting only of the edges in A , which are maximal 1 2 2 ∈ ∈ 2 ∩ (cid:48) for the inclusion. Let g A . Let e, f A with e appearing before f in (cid:96) . If e V is ∩ (cid:48) 2 (cid:48) ∈ (cid:48) ∈ (cid:48) ∈A (cid:48) ∈ not included in f V , then there exists a v V such that v e, v / f and v g ⊆ 2 since V g. That would lead to an inconsistent list which is a contradiction. 2 ∈ { ∈ F | ∀ ∈ ⇒ ∩ (cid:48) ⊆ ∩ (cid:48)} Reciprocally, let (cid:96) e . . . e (T ) i < j, e , e A e V e V . (cid:96) is 1 ∈ Fm i j i j (cid:48) of the form (cid:96) (cid:96) (cid:96) with (cid:96) (S). As it is organized by inclusion relative to V , the 1 A 2 A ∈ { | ∈ F } F (cid:48) elements of A should all lie at the end of (cid:96) . Thus (cid:96) σ σ σ (R) = (S ) by 2 ∈ F (cid:48) A A 1 2 2 Proposition 20. It follows that (cid:96) (T ). 4.2 Db-rootable hypergraphs In this section we will prove several structural properties of hypergraphs with disjoint branches decompositions which we will use in the algorithm in the next section. H ∈ H Deﬁnition 22. Let = (V, E) be a hypergraph. For e E, we say that is db-rootable H in e if there exists a disjoint branches decomposition of rooted in e. The algorithm for the construction of disjoint branches decompositions will delete edges of hypergraphs. To this end we (cid:83)introduce the following notation. H ∈ \ Deﬁnition 23. For a hypergraph = (V, E) and an edge e E, we denote H e the \ { } (cid:48) { } ⊆ hypergraph (V , E e ) where V := (cid:48)∈ \{ } e . For a set A = e , . . . , e E, we H \ e e H \ e E\ e \ 1 k deﬁne A to be the hypergraph (( e ) . . .) e . 1 k We make the following observation which will simplify our arguments later. H ∈ H Observation 24. Let = (V, E) be a hypergraph and e E. Then is db-rootable H \ in e if and only if for every connected component C = (V , E ) of e the hypergraph (cid:48) ∪ ∪ { } C C C := (V e, E e ) is db-rootable in e. C C The next lemma shows that deleting the root of a disjont branches decomposition decomposes a hypergraph along the components of the resulting join forest. H H T Lemma 25. Let be a hypergraph with a disjoint branches decomposition that is T T rooted in e. Let and v , v be two vertices that appear in diﬀerent trees and of the T \ { } 1 2 H1 \ 2 forest e . Then v and v lie in diﬀerent connected components of e. 1 2 H \ Proof. Suppose that there is a path e , . . . , e from v to v in e. We show by T 1 k 1 2 ∈ induction on i, that e is in . It is true for i = 1 because v e , the disjointness of T i T1 ∈ T 1 1∩ (cid:54) ∅ prevents e from being in . Now, assume that e . As e e = , it means 1 2 T i 1 i i+1 T that e and e are in a common branch of . If e is an ancestor of e in , then by i i+1 T i T i+1 T induction, e is in . If e is an ancestor of e in , then either e is in either i+1 1 (cid:54) i+1 i H \ ∈ Ti+1 1 e = e. However, e = e, because it is an edge of e, so e . i+1 ∈ T i+1 T i+1 ∈1 Thus e . However, it contradicts the disjointness of since v e . Thus there k 1 H \ 2 k is no path from v to v in e: they are in two diﬀerent connected components. 1 2 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: e

text: V

text: is

text: Fm

text: i

text: j

text: i

text: j

text: of

text: the

text: form

text: with

text: (S).

text: As

text: it

text: is

text: organized

text: by

text: inclusion

text: relative

text: to

text: V

text: the

text: A

text: A

text: F

text: F

text: elements

text: of

text: A

text: should

text: all

text: lie

text: at

text: the

text: end

text: of

text: Thus

text: σ

text: σ

text: σ

text: (R)

text: (S

text: by

text: F

text: A

text: A

text: Proposition

text: It

text: follows

text: that

text: (T

text: Db-rootable

text: hypergraphs

text: In

text: this

text: section

text: we

text: will

text: prove

text: several

text: structural

text: properties

text: of

text: hypergraphs

text: with

text: disjoint

text: branches

text: decompositions

text: which

text: we

text: will

text: use

text: in

text: the

text: algorithm

text: in

text: the

text: next

text: section.

text: H

text: H

text: Deﬁnition

text: Let

text: (V,

text: E)

text: be

text: a

text: hypergraph.

text: For

text: e

text: E,

text: we

text: say

text: that

text: is

text: db-rootable

text: H

text: in

text: e

text: if

text: there

text: exists

text: a

text: disjoint

text: branches

text: decomposition

text: of

text: rooted

text: in

text: e.

text: The

text: algorithm

text: for

text: the

text: construction

text: of

text: disjoint

text: branches

text: decompositions

text: will

text: delete

text: edges

text: of

text: hypergraphs.

text: To

text: this

text: end

text: we

text: the

text: following

text: notation.

text: H

text: Deﬁnition

text: For

text: a

text: hypergraph

text: (V,

text: E)

text: and

text: an

text: edge

text: e

text: E,

text: we

text: denote

text: H

text: e

text: the

text: hypergraph

text: (V

text: E

text: e

text: where

text: V

text: e

text: For

text: a

text: set

text: A

text: e

text: e

text: E,

text: we

text: H

text: e

text: e

text: H

text: e

text: E\

text: e

text: k

text: deﬁne

text: A

text: to

text: be

text: the

text: hypergraph

text: e

text: e

text: k

text: We

text: make

text: the

text: following

text: observation

text: which

text: will

text: simplify

text: our

text: arguments

text: later.

text: H

text: H

text: Observation

text: Let

text: (V,

text: E)

text: be

text: a

text: hypergraph

text: and

text: e

text: E.

text: Then

text: is

text: db-rootable

text: H

text: in

text: e

text: if

text: and

text: only

text: if

text: for

text: every

text: connected

text: component

text: C

text: (V

text: E

text: of

text: e

text: the

text: hypergraph

text: C

text: C

text: C

text: (V

text: e,

text: E

text: e

text: is

text: db-rootable

text: in

text: e.

text: C

text: C

text: The

text: next

text: lemma

text: shows

text: that

text: deleting

text: the

text: root

text: of

text: a

text: disjont

text: branches

text: decomposition

text: decomposes

text: a

text: hypergraph

text: along

text: the

text: components

text: of

text: the

text: resulting

text: join

text: forest.

text: H

text: H

text: T

text: Lemma

text: Let

text: be

text: a

text: hypergraph

text: with

text: a

text: disjoint

text: branches

text: decomposition

text: that

text: is

text: T

text: T

text: rooted

text: in

text: e.

text: Let

text: and

text: v

text: v

text: be

text: two

text: vertices

text: that

text: appear

text: in

text: diﬀerent

text: trees

text: and

text: of

text: the

text: T

text: forest

text: e

text: Then

text: v

text: and

text: v

text: lie

text: in

text: diﬀerent

text: connected

text: components

text: of

text: e.

text: H

text: Proof.

text: Suppose

text: that

text: there

text: is

text: a

text: path

text: e

text: e

text: from

text: v

text: to

text: v

text: in

text: e.

text: We

text: show

text: by

text: T

text: k

text: induction

text: on

text: i,

text: that

text: e

text: is

text: in

text: It

text: is

text: true

text: for

text: i

text: because

text: v

text: e

text: the

text: disjointness

text: of

text: T

text: i

text: T

text: 1∩

text: prevents

text: e

text: from

text: being

text: in

text: Now,

text: assume

text: that

text: e

text: As

text: e

text: e

text: it

text: means

text: T

text: i

text: i

text: T

text: that

text: e

text: and

text: e

text: are

text: in

text: a

text: common

text: branch

text: of

text: If

text: e

text: is

text: an

text: ancestor

text: of

text: e

text: in

text: then

text: by

text: i

text: T

text: i

text: T

text: T

text: induction,

text: e

text: is

text: in

text: If

text: e

text: is

text: an

text: ancestor

text: of

text: e

text: in

text: then

text: either

text: e

text: is

text: in

text: either

text: i

text: H

text: e

text: e.

text: However,

text: e

text: e,

text: because

text: it

text: is

text: an

text: edge

text: of

text: e,

text: so

text: e

text: T

text: T

text: ∈1

text: Thus

text: e

text: However,

text: it

text: contradicts

text: the

text: disjointness

text: of

text: since

text: v

text: e

text: Thus

text: there

text: k

text: H

text: k

text: is

text: no

text: path

text: from

text: v

text: to

text: v

text: in

text: e:

text: they

text: are

text: in

text: two

text: diﬀerent

text: connected

text: components.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Hypertree-width α-acyclicity β-hypertree-width Incidence β-acyclicity clique-width Signed incidence Modular inci- disjoint branches clique-width dent tree-width γ-acyclicity Incidence tree-width Primal tree-width Figure 1: A hierarchy of inclusion of graph and hypergraph classes. Classes not connected by a directed path are incomparable. (cid:48) (cid:48) ∈ (cid:48) (cid:48) ∈ (cid:48) ∈ (cid:48) ∈ x < x and y < y , if (x, y) E and (x , y ) E, then (x, y ) E and (x , y) E. G X Y is called a bipartite permutation graph if it admits a strong ordering. Brandsta¨dt and Lozin showed the following property of bipartite permutation graphs. Lemma 5 ([BL03]). Bipartite permutation graphs have unbounded cliquewidth. It turns out that hypergraphs with a bipartite permutation incidence graph admit a join path. H H Proposition 6. Every hypergraph without empty hyperedges, whose incidence graph I is a bipartite permutation graph, has a join path. H Proof. Let (< , < ) be a strong ordering of = (V, E, A). We claim that for all V ∈E ∩ ∈ I ∈ e < f < g, if v e g then v f. Indeed, as f is not empty, there exists w f. If E E w = v, there is nothing to prove. Otherwise, suppose that w < v. Then, by deﬁnition ∈ ∈ V ∈ ∈ of strong orderings, as (f, w) A and (g, v) A, we have (f, v) A. Thus v f. The ∈ ∈ ∈ case v < w follows symetrically: (f, w) A and (e, v) A implies that (f, v) A so ∈ V H v f. Thus the ordering < has the property of lemma 3 and it follows that has a E join path. By combining Lemma 5 and Proposition 6 we get: Corollary 7. The class of CNF-formulas with join paths has unbounded incidence cliquewidth. The same is thus true for CNF-formulas with disjoint branches decomposi- tions. Again, cycles give an example proving the other direction of imcomparability. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Hypertree-width

text: α-acyclicity

text: β-hypertree-width

text: Incidence

text: β-acyclicity

text: clique-width

text: Signed

text: incidence

text: Modular

text: inci-

text: disjoint

text: branches

text: clique-width

text: dent

text: tree-width

text: γ-acyclicity

text: Incidence

text: tree-width

text: Primal

text: tree-width

text: Figure

text: A

text: hierarchy

text: of

text: inclusion

text: of

text: graph

text: and

text: hypergraph

text: classes.

text: Classes

text: not

text: connected

text: by

text: a

text: directed

text: path

text: are

text: incomparable.

text: x

text: x

text: and

text: y

text: y

text: if

text: (x,

text: y)

text: E

text: and

text: (x

text: y

text: E,

text: then

text: (x,

text: y

text: E

text: and

text: (x

text: y)

text: E.

text: G

text: X

text: Y

text: is

text: called

text: a

text: bipartite

text: permutation

text: graph

text: if

text: it

text: admits

text: a

text: strong

text: ordering.

text: Brandsta¨dt

text: and

text: Lozin

text: showed

text: the

text: following

text: property

text: of

text: bipartite

text: permutation

text: graphs.

text: Lemma

text: Bipartite

text: permutation

text: graphs

text: have

text: unbounded

text: cliquewidth.

text: It

text: turns

text: out

text: that

text: hypergraphs

text: with

text: a

text: bipartite

text: permutation

text: incidence

text: graph

text: admit

text: a

text: join

text: path.

text: H

text: H

text: Proposition

text: Every

text: hypergraph

text: without

text: empty

text: hyperedges,

text: whose

text: incidence

text: graph

text: I

text: is

text: a

text: bipartite

text: permutation

text: graph,

text: has

text: a

text: join

text: path.

text: H

text: Proof.

text: Let

text: be

text: a

text: strong

text: ordering

text: of

text: (V,

text: E,

text: A).

text: We

text: claim

text: that

text: for

text: all

text: V

text: ∈E

text: I

text: e

text: f

text: g,

text: if

text: v

text: e

text: g

text: then

text: v

text: f.

text: Indeed,

text: as

text: f

text: is

text: not

text: empty,

text: there

text: exists

text: w

text: f.

text: If

text: E

text: E

text: w

text: v,

text: there

text: is

text: nothing

text: to

text: prove.

text: Otherwise,

text: suppose

text: that

text: w

text: v.

text: Then,

text: by

text: deﬁnition

text: V

text: of

text: strong

text: orderings,

text: as

text: (f,

text: w)

text: A

text: and

text: (g,

text: v)

text: A,

text: we

text: have

text: (f,

text: v)

text: A.

text: Thus

text: v

text: f.

text: The

text: case

text: v

text: w

text: follows

text: symetrically:

text: (f,

text: w)

text: A

text: and

text: (e,

text: v)

text: A

text: implies

text: that

text: (f,

text: v)

text: A

text: so

text: V

text: H

text: v

text: f.

text: Thus

text: the

text: ordering

text: has

text: the

text: property

text: of

text: lemma

text: and

text: it

text: follows

text: that

text: has

text: a

text: E

text: join

text: path.

text: By

text: combining

text: Lemma

text: and

text: Proposition

text: we

text: get:

text: Corollary

text: The

text: class

text: of

text: CNF-formulas

text: with

text: join

text: paths

text: has

text: unbounded

text: incidence

text: cliquewidth.

text: The

text: same

text: is

text: thus

text: true

text: for

text: CNF-formulas

text: with

text: disjoint

text: branches

text: decomposi-

text: tions.

text: Again,

text: cycles

text: give

text: an

text: example

text: proving

text: the

text: other

text: direction

text: of

text: imcomparability.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: symbols of ∆. Intuitively, for each two-valued interpretation of the parameters, ∆ determines the interpretation of the deﬁned symbols in a unique way. The satisfaction relation of FO can be extended to handle deﬁnitions by means of the well-founded semantics [60]. This semantics formalizes the informal semantics of rule sets as inductive deﬁnitions [20,21]. First, consider deﬁnitions ∆ that only deﬁne predicate symbols and let ∆’ be the deﬁnition constructed ∀ ← ∀ ← ∃ ∧ from ∆ by replacing each rule x : P(t) ϕ with y : P(y) x : t = y ϕ. I I | I The interpretation satisﬁes ∆ ( = ∆) if is a parametrised well-founded I (cid:48) model of ∆, that means that is the well-founded model of ∆ when the open I symbols are interpreted as in . When functions are involved, we transform them away so that we can use parametrised well-founded models as above. Let the graph predicate symbol of a function f/n be the new predicate symbol F/n + 1; given an interpretation I I I , let ’ be with the interpretations of all function symbols f/n replaced by an interpretation of the corresponding graph predicate symbols F/n + 1 such (cid:48) I (cid:48) (cid:48) that F(d, d ) holds in ’ if and only if f(d) = d holds in I. Let ∆ be the deﬁnition constructed from ∆ by replacing rules deﬁning predicate symbols as ∀ (cid:48) ← above and replacing rules x : f(t) = t ϕ as follows: ﬁrst, replace them by ∀ (cid:48) (cid:48) ← ∃ ∧ (cid:48) (cid:48) ∧ y, y : F(y , . . . , y , y ) x : t = y t = y ϕ. Next, replace occurrences of 1 n ∃ ∧ terms f(t) in the bodies. An atom A[f(t)] is replaced by x : A[f(t)/x] F(t, x); { } { ∧∃ } a set expression x : ϕ : f(t) is replaced by x : ϕ x : F(t, x) : x ; concerning priority of application, replacement is done from the leaves of the parse tree I I | I(cid:48) upwards. We say that satisﬁes the deﬁnition ∆ ( = ∆) if is a parametrised (cid:48) well-founded model of deﬁnition ∆ . (cid:87) ∀ ← The completion of ∆ for a symbol P, deﬁned in ∆ by the rules x : P(t ) ∈ ∀ ⇒ i i ϕ with i [1, n], is the set consisting of the sentence x : ϕ P(t ) for each i∈ ∀ i∧ i i i [1, n] and the sentence x : P(x) => ∈ (x = t ϕ ); the completion i [1,n] i i for deﬁned function symbols is deﬁned similarly. This set is denoted as comp , | P,∆ the union of all these sets for ∆ as comp . It is well-known that if I = ∆ then | ∆ I = comp but not always vice-versa (e.g., the inductive deﬁnition expressing ∆ transitive closure is stronger than its completion). 2.3 Types The above language FO(ID, Agg, PF) is already a rich KR language. How- ever, in many real applications, the use of types might be desired. Here, we present a typed extension of this language in terms of a translation to FO(ID, Agg, PF). A vocabulary not only consists of predicate and function symbols, but also contains type symbols. Furthermore it assigns to each of its predicate and func- tion symbols, a tuple of types (of the correct arity). Intuitively, it means that these predicates and functions are only deﬁned on the (tuples of) domain ele- ments within the correct type. Note that, for functions, totality means “deﬁned over every well-typed tuple” instead of over the whole universe as in an untyped logic. Lastly, every quantiﬁed variable is also typed, meaning that the quantiﬁc- ation does not range over the entire domain, but only over that speciﬁc type. Semantics of the typed language are deﬁned in terms of the untyped lan- guage: types are treated as unary predicate symbols. In every structure, the ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: symbols

text: of

text: Intuitively,

text: for

text: each

text: two-valued

text: interpretation

text: of

text: the

text: parameters,

text: determines

text: the

text: interpretation

text: of

text: the

text: deﬁned

text: symbols

text: in

text: a

text: unique

text: way.

text: The

text: satisfaction

text: relation

text: of

text: FO

text: can

text: be

text: extended

text: to

text: handle

text: deﬁnitions

text: by

text: means

text: of

text: the

text: well-founded

text: semantics

text: This

text: semantics

text: formalizes

text: the

text: informal

text: semantics

text: of

text: rule

text: sets

text: as

text: inductive

text: deﬁnitions

text: First,

text: consider

text: deﬁnitions

text: that

text: only

text: deﬁne

text: predicate

text: symbols

text: and

text: let

text: be

text: the

text: deﬁnition

text: constructed

text: from

text: by

text: replacing

text: each

text: rule

text: x

text: P(t)

text: ϕ

text: with

text: y

text: P(y)

text: x

text: t

text: y

text: ϕ.

text: I

text: I

text: I

text: The

text: interpretation

text: satisﬁes

text: if

text: is

text: a

text: parametrised

text: well-founded

text: I

text: model

text: of

text: that

text: means

text: that

text: is

text: the

text: well-founded

text: model

text: of

text: when

text: the

text: open

text: I

text: symbols

text: are

text: interpreted

text: as

text: in

text: When

text: functions

text: are

text: involved,

text: we

text: transform

text: them

text: away

text: so

text: that

text: we

text: can

text: use

text: parametrised

text: well-founded

text: models

text: as

text: above.

text: Let

text: the

text: graph

text: predicate

text: symbol

text: of

text: a

text: function

text: f/n

text: be

text: the

text: new

text: predicate

text: symbol

text: F/n

text: given

text: an

text: interpretation

text: I

text: I

text: I

text: let

text: be

text: with

text: the

text: interpretations

text: of

text: all

text: function

text: symbols

text: f/n

text: replaced

text: by

text: an

text: interpretation

text: of

text: the

text: corresponding

text: graph

text: predicate

text: symbols

text: F/n

text: such

text: I

text: that

text: F(d,

text: d

text: holds

text: in

text: if

text: and

text: only

text: if

text: f(d)

text: d

text: holds

text: in

text: I.

text: Let

text: be

text: the

text: deﬁnition

text: constructed

text: from

text: by

text: replacing

text: rules

text: deﬁning

text: predicate

text: symbols

text: as

text: above

text: and

text: replacing

text: rules

text: x

text: f(t)

text: t

text: ϕ

text: as

text: follows:

text: ﬁrst,

text: replace

text: them

text: by

text: y,

text: y

text: F(y

text: y

text: y

text: x

text: t

text: y

text: t

text: y

text: ϕ.

text: Next,

text: replace

text: occurrences

text: of

text: n

text: terms

text: f(t)

text: in

text: the

text: bodies.

text: An

text: atom

text: A[f(t)]

text: is

text: replaced

text: by

text: x

text: A[f(t)/x]

text: F(t,

text: x);

text: a

text: set

text: expression

text: x

text: ϕ

text: f(t)

text: is

text: replaced

text: by

text: x

text: ϕ

text: x

text: F(t,

text: x)

text: x

text: concerning

text: priority

text: of

text: application,

text: replacement

text: is

text: done

text: from

text: the

text: leaves

text: of

text: the

text: parse

text: tree

text: I

text: I

text: upwards.

text: We

text: say

text: that

text: satisﬁes

text: the

text: deﬁnition

text: if

text: is

text: a

text: parametrised

text: well-founded

text: model

text: of

text: deﬁnition

text: The

text: completion

text: of

text: for

text: a

text: symbol

text: P,

text: deﬁned

text: in

text: by

text: the

text: rules

text: x

text: P(t

text: i

text: i

text: ϕ

text: with

text: i

text: n],

text: is

text: the

text: set

text: consisting

text: of

text: the

text: sentence

text: x

text: ϕ

text: P(t

text: for

text: each

text: i∈

text: i∧

text: i

text: i

text: i

text: n]

text: and

text: the

text: sentence

text: x

text: P(x)

text: (x

text: t

text: ϕ

text: the

text: completion

text: i

text: i

text: i

text: for

text: deﬁned

text: function

text: symbols

text: is

text: deﬁned

text: similarly.

text: This

text: set

text: is

text: denoted

text: as

text: comp

text: P,∆

text: the

text: union

text: of

text: all

text: these

text: sets

text: for

text: as

text: comp

text: It

text: is

text: well-known

text: that

text: if

text: I

text: then

text: I

text: comp

text: but

text: not

text: always

text: vice-versa

text: (e.g.,

text: the

text: inductive

text: deﬁnition

text: expressing

text: transitive

text: closure

text: is

text: stronger

text: than

text: its

text: completion).

text: Types

text: The

text: above

text: language

text: FO(ID,

text: Agg,

text: PF)

text: is

text: already

text: a

text: rich

text: KR

text: language.

text: How-

text: ever,

text: in

text: many

text: real

text: applications,

text: the

text: use

text: of

text: types

text: might

text: be

text: desired.

text: Here,

text: we

text: present

text: a

text: typed

text: extension

text: of

text: this

text: language

text: in

text: terms

text: of

text: a

text: translation

text: to

text: FO(ID,

text: Agg,

text: PF).

text: A

text: vocabulary

text: not

text: only

text: consists

text: of

text: predicate

text: and

text: function

text: symbols,

text: but

text: also

text: contains

text: type

text: symbols.

text: Furthermore

text: it

text: assigns

text: to

text: each

text: of

text: its

text: predicate

text: and

text: func-

text: tion

text: symbols,

text: a

text: tuple

text: of

text: types

text: (of

text: the

text: correct

text: arity).

text: Intuitively,

text: it

text: means

text: that

text: these

text: predicates

text: and

text: functions

text: are

text: only

text: deﬁned

text: on

text: the

text: (tuples

text: of)

text: domain

text: ele-

text: ments

text: within

text: the

text: correct

text: type.

text: Note

text: that,

text: for

text: functions,

text: totality

text: means

text: “deﬁned

text: over

text: every

text: well-typed

text: tuple”

text: instead

text: of

text: over

text: the

text: whole

text: universe

text: as

text: in

text: an

text: untyped

text: logic.

text: Lastly,

text: every

text: quantiﬁed

text: variable

text: is

text: also

text: typed,

text: meaning

text: that

text: the

text: quantiﬁc-

text: ation

text: does

text: not

text: range

text: over

text: the

text: entire

text: domain,

text: but

text: only

text: over

text: that

text: speciﬁc

text: type.

text: Semantics

text: of

text: the

text: typed

text: language

text: are

text: deﬁned

text: in

text: terms

text: of

text: the

text: untyped

text: lan-

text: guage:

text: types

text: are

text: treated

text: as

text: unary

text: predicate

text: symbols.

text: In

text: every

text: structure,

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 2. The ampliﬁcation mechanism is the crucial property of collapse models: the collapse rate for the center of mass of a composite object is a function of the size and number of constituents of the system. The larger the system, the stronger the collapse rate. This is how collapse models can accommodate, within one single dynamical equation, both the quantum properties of microscopic systems and the classical properties of macroscopic objects. particles: if the noise tries to collapse a particle, it has to collapse all other particles identical to it, in order to preserve the symmetry of the wave function. This quadratic increase is not present in the original GRW model, which deals only with distinguishable particles. Given this, the relevant question is: which values can one reasonably assign to λ and r ? C The question is not easy to answer from the theoretical point of view, as there are only very rough guidelines one can follow. As we explained here above, r sets the division between micro C and macro distances. A reasonable value to take, ﬁrst suggested by GRW [17], is: (cid:39) − r 10 7m, (6) C ∼ − which is much bigger than typical inter-atomic distances in matter ( 10 10m), but much ∼ smaller than human-size distances ( 0.1mm). Of course, this value could ﬂuctuate a few orders of magnitude, but this is the best one can say at the moment. There is much less of a consensus regarding the value λ should have. In particular, two very diﬀerent values were proposed, The ﬁrst one was suggested by GRW: ##LTFigure##

text: Figure

text: The

text: ampliﬁcation

text: mechanism

text: is

text: the

text: crucial

text: property

text: of

text: collapse

text: models:

text: the

text: collapse

text: rate

text: for

text: the

text: center

text: of

text: mass

text: of

text: a

text: composite

text: object

text: is

text: a

text: function

text: of

text: the

text: size

text: and

text: number

text: of

text: constituents

text: of

text: the

text: system.

text: The

text: larger

text: the

text: system,

text: the

text: stronger

text: the

text: collapse

text: rate.

text: This

text: is

text: how

text: collapse

text: models

text: can

text: accommodate,

text: within

text: one

text: single

text: dynamical

text: equation,

text: both

text: the

text: quantum

text: properties

text: of

text: microscopic

text: systems

text: and

text: the

text: classical

text: properties

text: of

text: macroscopic

text: objects.

text: particles:

text: if

text: the

text: noise

text: tries

text: to

text: collapse

text: a

text: particle,

text: it

text: has

text: to

text: collapse

text: all

text: other

text: particles

text: identical

text: to

text: it,

text: in

text: order

text: to

text: preserve

text: the

text: symmetry

text: of

text: the

text: wave

text: function.

text: This

text: quadratic

text: increase

text: is

text: not

text: present

text: in

text: the

text: original

text: GRW

text: model,

text: which

text: deals

text: only

text: with

text: distinguishable

text: particles.

text: Given

text: this,

text: the

text: relevant

text: question

text: is:

text: which

text: values

text: can

text: one

text: reasonably

text: assign

text: to

text: λ

text: and

text: r

text: C

text: The

text: question

text: is

text: not

text: easy

text: to

text: answer

text: from

text: the

text: theoretical

text: point

text: of

text: view,

text: as

text: there

text: are

text: only

text: very

text: rough

text: guidelines

text: one

text: can

text: follow.

text: As

text: we

text: explained

text: here

text: above,

text: r

text: sets

text: the

text: division

text: between

text: micro

text: C

text: and

text: macro

text: distances.

text: A

text: reasonable

text: value

text: to

text: take,

text: ﬁrst

text: suggested

text: by

text: GRW

text: is:

text: r

text: C

text: which

text: is

text: much

text: bigger

text: than

text: typical

text: inter-atomic

text: distances

text: in

text: matter

text: but

text: much

text: smaller

text: than

text: human-size

text: distances

text: Of

text: course,

text: this

text: value

text: could

text: ﬂuctuate

text: a

text: few

text: orders

text: of

text: magnitude,

text: but

text: this

text: is

text: the

text: best

text: one

text: can

text: say

text: at

text: the

text: moment.

text: There

text: is

text: much

text: less

text: of

text: a

text: consensus

text: regarding

text: the

text: value

text: λ

text: should

text: have.

text: In

text: particular,

text: two

text: very

text: diﬀerent

text: values

text: were

text: proposed,

text: The

text: ﬁrst

text: one

text: was

text: suggested

text: by

text: GRW:

text: ##LTFigure##


text: ∗ Explicit schemes for time propagating many-body wavefunctions Ana Laura Frapiccini1b, Aliou Hamido1, Sebastian Schr¨oter2, Dean Pyke3, Francisca Mota-Furtado3, Patrick F. O’Mahony3, Javier Madron˜ero4, Johannes Eiglsperger5 and Bernard Piraux1 ‡ § (Dated: May 26, 2020) Accurate theoretical data on many time-dependent processes in atomic and molecular physics and in chemistry require the direct numerical ab initio solution of the time-dependent Schr¨odinger equation, thereby motivating the development of very eﬃcient time propagators. These usually involve the solution of very large systems of ﬁrst order diﬀerential equations that are characterized by a high degree of stiﬀness. In this contribution, we analyze and compare the performance of the explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have exactly the same stability function, therefore sharing the same stability properties that turn out to be optimum. Their respective accuracy however diﬀers signiﬁcantly and depends on the physical situation involved. In order to test this accuracy, we use a predictor-corrector scheme in which the predictor is either Fatunla’s or Arnoldi’s algorithm and the corrector, a fully implicit four-stage Radau IIA method of order 7. In this contribution, we consider two physical processes. The ﬁrst one is the ionization of an atomic system by a short and intense electromagnetic pulse; the atomic systems include a one-dimensional Gaussian model potential as well as atomic hydrogen and helium, both in full dimensionality. The second process is the decoherence of two-electron quantum states when a time independent perturbation is applied to a planar two-electron quantum dot where both electrons are conﬁned in an anharmonic potential. Even though the Hamiltonian of this system is time independent the corresponding diﬀerential equation shows a striking stiﬀness which makes the time integration extremely diﬃcult. In the case of the one-dimensional Gaussian potential we discuss in detail the possibility of monitoring the time step for both explicit algorithms. In the other physical situations that are much more demanding in term of computations, we show that the accuracy of both algorithms depends strongly on the degree of stiﬀness of the problem. PACS numbers: b∗ On leave of absence from CONICET and Departamento de F´ısica, Universidad Nacional del Sur, 8000 Bahia Blanca, Argentina. ‡ § ##LTLine##

text: Explicit

label: title

text: schemes

text: for

text: time

text: propagating

text: many-body

text: wavefunctions

text: Ana

label: author

text: Laura

text: Aliou

text: Sebastian

text: Dean

text: Francisca

text: Patrick

text: F.

text: Javier

text: Johannes

text: and

text: Bernard

text: (Dated:

text: May

label: date

text: Accurate

label: abstract

text: theoretical

text: data

text: on

text: many

text: time-dependent

text: processes

text: in

text: atomic

text: and

text: molecular

text: physics

text: and

text: in

text: chemistry

text: require

text: the

text: direct

text: numerical

text: ab

text: initio

text: solution

text: of

text: the

text: time-dependent

text: Schr¨odinger

text: equation,

text: thereby

text: motivating

text: the

text: development

text: of

text: very

text: eﬃcient

text: time

text: propagators.

text: These

text: usually

text: involve

text: the

text: solution

text: of

text: very

text: large

text: systems

text: of

text: ﬁrst

text: order

text: diﬀerential

text: equations

text: that

text: are

text: characterized

text: by

text: a

text: high

text: degree

text: of

text: stiﬀness.

text: In

text: this

text: contribution,

text: we

text: analyze

text: and

text: compare

text: the

text: performance

text: of

text: the

text: explicit

text: one-step

text: algorithms

text: of

text: Fatunla

text: and

text: Arnoldi.

text: Both

text: algorithms

text: have

text: exactly

text: the

text: same

text: stability

text: function,

text: therefore

text: sharing

text: the

text: same

text: stability

text: properties

text: that

text: turn

text: out

text: to

text: be

text: optimum.

text: Their

text: respective

text: accuracy

text: however

text: diﬀers

text: signiﬁcantly

text: and

text: depends

text: on

text: the

text: physical

text: situation

text: involved.

text: In

text: order

text: to

text: test

text: this

text: accuracy,

text: we

text: use

text: a

text: predictor-corrector

text: scheme

text: in

text: which

text: the

text: predictor

text: is

text: either

text: Fatunla’s

text: or

text: Arnoldi’s

text: algorithm

text: and

text: the

text: corrector,

text: a

text: fully

text: implicit

text: four-stage

text: Radau

text: IIA

text: method

text: of

text: order

text: In

text: this

text: contribution,

text: we

text: consider

text: two

text: physical

text: processes.

text: The

text: ﬁrst

text: one

text: is

text: the

text: ionization

text: of

text: an

text: atomic

text: system

text: by

text: a

text: short

text: and

text: intense

text: electromagnetic

text: pulse;

text: the

text: atomic

text: systems

text: include

text: a

text: one-dimensional

text: Gaussian

text: model

text: potential

text: as

text: well

text: as

text: atomic

text: hydrogen

text: and

text: helium,

text: both

text: in

text: full

text: dimensionality.

text: The

text: second

text: process

text: is

text: the

text: decoherence

text: of

text: two-electron

text: quantum

text: states

text: when

text: a

text: time

text: independent

text: perturbation

text: is

text: applied

text: to

text: a

text: planar

text: two-electron

text: quantum

text: dot

text: where

text: both

text: electrons

text: are

text: conﬁned

text: in

text: an

text: anharmonic

text: potential.

text: Even

text: though

text: the

text: Hamiltonian

text: of

text: this

text: system

text: is

text: time

text: independent

text: the

text: corresponding

text: diﬀerential

text: equation

text: shows

text: a

text: striking

text: stiﬀness

text: which

text: makes

text: the

text: time

text: integration

text: extremely

text: diﬃcult.

text: In

text: the

text: case

text: of

text: the

text: one-dimensional

text: Gaussian

text: potential

text: we

text: discuss

text: in

text: detail

text: the

text: possibility

text: of

text: monitoring

text: the

text: time

text: step

text: for

text: both

text: explicit

text: algorithms.

text: In

text: the

text: other

text: physical

text: situations

text: that

text: are

text: much

text: more

text: demanding

text: in

text: term

text: of

text: computations,

text: we

text: show

text: that

text: the

text: accuracy

text: of

text: both

text: algorithms

text: depends

text: strongly

text: on

text: the

text: degree

text: of

text: stiﬀness

text: of

text: the

text: problem.

text: PACS

text: numbers:

text: b∗

text: On

text: leave

text: of

text: absence

text: from

text: CONICET

text: and

text: Departamento

text: de

text: F´ısica,

text: Universidad

text: Nacional

text: del

text: Sur,

text: Bahia

text: Blanca,

text: Argentina.

text: ##LTLine##


text: 13 1.0 bicgstab preconditioned bicgstab not preconditioned 0.8 5 0 1 x 0.6 y cit pli ulti 0.4 M 0.2 0 0 10 20 30 40 50 Number of iterations in biconjugate FIG. 4. (Color online) Number of iterations in the Bi-CGSTAB and their multiplicities during the propagation with the electromagnetic pulse. The parameters of the Gaussian model problem are V = 4 a.u., β = 0.1 a.u., with a pulse of peak 0 intensity I = 1016 Watt/cm2, photon energy ω = 0.5 a.u. and duration of 8 optical cyles. needed by 25%. However, care must be taken when including a pre-conditioner, since, by increasing the number of operations, it may increase the computational times even though it accelerates convergence. As mentioned above, the corrector scheme is iterated up to j =max where convergence is achieved. In Fig.5 we plot the time evolution of the cor time step during the propagation for diﬀerent values of the maximum of iterations max in the corrector . We see cor here that, as we increase this maximum number of iterations, the value of the time step becomes larger. The relative − error in the norm which is the same for all the calculations is of the order of 10 11. Moreover, the computational 0.03 0.025 0.02 u.) max =10 p (a.0.015 maxccoorr=15 e max =20 e st maxcor=100 m 0.01 cor Ti 0.005 0 -40 -20 0 20 40 Time (a.u.) FIG. 5. (Color online) Time step evolution for P-C method of time propagation during the propagation with the electromagnetic pulse. The parameters of the Gaussian model problem are the same as in Fig.4. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: bicgstab

text: preconditioned

text: bicgstab

text: not

text: preconditioned

text: x

text: y

text: cit

text: pli

text: ulti

text: M

text: Number

text: of

text: iterations

text: in

text: biconjugate

text: FIG.

text: (Color

text: online)

text: Number

text: of

text: iterations

text: in

text: the

text: Bi-CGSTAB

text: and

text: their

text: multiplicities

text: during

text: the

text: propagation

text: with

text: the

text: electromagnetic

text: pulse.

text: The

text: parameters

text: of

text: the

text: Gaussian

text: model

text: problem

text: are

text: V

text: a.u.,

text: β

text: a.u.,

text: with

text: a

text: pulse

text: of

text: peak

text: intensity

text: I

text: photon

text: energy

text: ω

text: a.u.

text: and

text: duration

text: of

text: optical

text: cyles.

text: needed

text: by

text: However,

text: care

text: must

text: be

text: taken

text: when

text: including

text: a

text: pre-conditioner,

text: since,

text: by

text: increasing

text: the

text: number

text: of

text: operations,

text: it

text: may

text: increase

text: the

text: computational

text: times

text: even

text: though

text: it

text: accelerates

text: convergence.

text: As

text: mentioned

text: above,

text: the

text: corrector

text: scheme

text: is

text: iterated

text: up

text: to

text: j

text: =max

text: where

text: convergence

text: is

text: achieved.

text: In

text: we

text: plot

text: the

text: time

text: evolution

text: of

text: the

text: cor

text: time

text: step

text: during

text: the

text: propagation

text: for

text: diﬀerent

text: values

text: of

text: the

text: maximum

text: of

text: iterations

text: max

text: in

text: the

text: corrector

text: We

text: see

text: cor

text: here

text: that,

text: as

text: we

text: increase

text: this

text: maximum

text: number

text: of

text: iterations,

text: the

text: value

text: of

text: the

text: time

text: step

text: becomes

text: larger.

text: The

text: relative

text: error

text: in

text: the

text: norm

text: which

text: is

text: the

text: same

text: for

text: all

text: the

text: calculations

text: is

text: of

text: the

text: order

text: of

text: Moreover,

text: the

text: computational

text: u.)

text: max

text: p

text: e

text: max

text: e

text: st

text: m

text: cor

text: Ti

text: Time

text: (a.u.)

text: FIG.

text: (Color

text: online)

text: Time

text: step

text: evolution

text: for

text: P-C

text: method

text: of

text: time

text: propagation

text: during

text: the

text: propagation

text: with

text: the

text: electromagnetic

text: pulse.

text: The

text: parameters

text: of

text: the

text: Gaussian

text: model

text: problem

text: are

text: the

text: same

text: as

text: in

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##