text: B.1 (Point ID Poi2a) Scenario M , M , M 124.6 GeV 181.7 GeV 322.6 GeV h Hs H M , M 72.5 GeV 311.7 GeV | As | A| | S 2, P 2 0.90 1 H2hs A1as µ , µ 1.54 1.01 ττ bb µ , µ , µ 0.93 0.93 1.01 ZZ WW γγ tan β, λ, κ 1.9 0.628 0.354 A , A , µ 251.2 GeV 53.8 GeV 158.9 GeV λ κ eﬀ M , M , M 890 GeV 576 GeV 1219 GeV 1 2 3 A , A , A 1555 GeV -1005 GeV -840 GeV t b τ M = M , M 1075 GeV 1 TeV Q3 tR bR M = M , M 530 GeV 2.5 TeV L3 τR SUSY Table 7: The parameters deﬁning scenario B.1, together with the Higgs boson masses, singlet components and reduced signal rates of h. to almost 200 fb. The heavy doublet Higgs H is produced in gluon fusion with 3.2 pb and its branching ratio into hH is 0.51. With the large H decay rate into A A this leads to very s s s s peculiar signatures with up to 6 photons in the ﬁnal state. The largest rate is given by the (4γ)(b¯b) signature with 712 fb. Additionally, we have A production in gluon fusion at 2.5 pb, → which leads via the decay into H A (BR(A H A )=0.21) again to multiphoton ﬁnal states s s s s with or without additional b-quark or τ-pairs. Here the 6γ ﬁnal state even amounts to 302 fb. → With a branching ratio BR(A ZH ) = 0.22 also this decay leads to interesting ﬁnal states s with e.g. (4γ)(b¯b) production at 58 fb. Finally, also the decay into hA may be exploited in its s (γγ)(b¯b) ﬁnal state with a cross section of 16 fb. These and more possible combinations and ﬁnal states are summarised in Table 8. It shows that in this scenario with a rather light overall spectrum, all Higgs bosons can be discovered, and that there are spectacular signatures possible that can be helpful for the discovery. Additionally, in the cascade decays the trilinear couplings λ , λ , λ , λ and λ are accessible. Note ﬁnally, that the multiphoton HsAsAs HHsh HAsAs AAsHs AAsh (plus fermion) ﬁnal states discussed here cannot occur in the MSSM and are unique to an extension beyond, as the NMSSM. B.2) Heavy Higgs spectrum: Scenario B.2, deﬁned in Table 9 is an example for a spectrum where it is challenging to ﬁnd all NMSSM Higgs states. This is because of the heavier H and A with masses around 560 GeV, a not so light A as e.g. in scenario A.2 and a very singlet-like H . The s s latter has a gluon fusion production cross section of 19 fb, leading to a ττ ﬁnal state at only 0.3 fb, cf. Table 10. The only alternative production via Higgs-to-Higgs decays is given by the → H hH decay leading to small 0.5 fb in the (ττ)(b¯b) ﬁnal state. The pseudoscalar singlet A s s on the other hand has a larger gluon fusion production cross section now than in B.1 and can be searched for in standard ﬁnal states. Additionally the decay of A into hA can be used to → → s produce A in the (ττ)(b¯b) ﬁnal state e.g. at 21 fb. Also the H ZA (ττ)(b¯b) production s s leading to 14 fb, may be used. Both H and A are heavy enough to decay into top pairs and may be discovered in these decay channels with rates into tt¯ between 550 fb for A and 623 fb for H. In this scenario only the trilinear couplings λ and λ are accessible. HHsh AAsh B.3) Heavy Higgs spectrum, B.2 reversed: The scenario B.3, cf. Table 11, is reversed compared ≡ ≡ to B.2 in the sense that now H H is somewhat less singlet-like and A A is very singlet- 2 s 1 s like, so that here A production will be challenging. Its gluon fusion production cross section s amounts only to 32 fb leading to a ττ rate of 3 fb as can be inferred from Table 12. It can 30 ##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##

text: (Point

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text: M

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text: this

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text: spectrum,

text: all

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text: bosons

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text: be

text: discovered,

text: and

text: that

text: there

text: are

text: spectacular

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text: be

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text: discovery.

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text: the

text: other

text: hand

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text: a

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text: be

text: searched

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text: decay

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text: hA

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text: be

text: used

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text: (ττ)(b¯b)

text: ﬁnal

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text: production

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text: used.

text: Both

text: H

text: and

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text: heavy

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text: pairs

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text: discovered

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text: accessible.

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text: reversed:

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text: scenario

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text: in

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text: sense

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text: somewhat

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text: singlet-like

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text: A

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text: is

text: very

text: singlet-

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text: like,

text: so

text: that

text: here

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text: production

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text: be

text: challenging.

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text: Table

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text: can

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text: latter is very light with a mass around 62 GeV. The lightest scalar H with a mass of 112 GeV s is close in mass to h, and both mix strongly, so that the gluon fusion production cross section for h only amounts to 27 pb, while the H production is rather large for a singlet-like boson and s reaches 17 pb, as given in Table 18. The h reduced rates are still compatible with the LHC data, although the ﬁnal state rates for ττ and b¯b are somewhat on the lower side. The h cascade decay via an A pair reaches in the (ττ)(b¯b) ﬁnal state a large cross section of 276 fb. This should s make a measurement of the triple Higgs couplings λ possible. Also the 4τ ﬁnal state is hAsAs sizeable with 12 fb. The light A can be directly produced, too, and then searched for in the ττ s ﬁnal state with a rate of 92 fb. The copiously produced H can be searched for in the standard s decay channels. Diﬃcult, however, if not impossible is the production of the doublet-like H and A, as they have masses of 1.3 TeV. Since they are both rather down-component doublet-like, one might consider associated production with a b-quark pair, in view of the not so small tan β = 6.4. But also these cross sections remain below 1 fb. Picking up the idea presented in [95], sum rules for the Higgs couplings can be exploited to give a hint to the missing states, in case the heavy Higgs bosons cannot be discovered. This allows then to disentangle an NMSSM Higgs sector from a supersymmetric Higgs sector with minimal particle content, if e.g. only three Higgs bosons are discovered and not all of them are R scalar. From Eq. (4.26) and the unitarity of the S matrix, it can be derived that the NMSSM CP-even Higgs couplings to vector bos(cid:88)ons with respect to the SM coupling obey the sum rule 3 g2 = 1 . (6.32) HiV V i=1 The H couplings G to top and bottom quarks (normalised to the SM) are given by i Hitt/bb R G ≡ S Hitt g = i2 (6.33) g Hitt sin β HSMtt R G ≡ S Hibb g = i1 . (6.34) g Hibb cos β HSMbb D.2 (Point ID 110) Scenario M , M , M 112.0 GeV 126.3 GeV 1288.2 GeV Hs h H M , M 61.5 GeV 1287.4 GeV | As | A| | S 2, P 2 0.63 1 H1hs A1as µ , µ 0.73 0.62 ττ bb µ , µ , µ 0.90 1.03 1.06 ZZ WW γγ tan β, λ, κ 6.36 0.47 0.14 A , A , µ 1217.1 GeV 19.6 GeV 195.3 GeV λ κ eﬀ A , A , A -1804.6 GeV -1196.8 GeV 1704.8 GeV t b τ M , M , M 417.2 GeV 237.5 GeV 2362.2 GeV 1 2 3 M = M , M 967.8 GeV 3 TeV Q3 tR bR M = M , M 2491.6 GeV 3 TeV L3 τR SUSY Table 17: The parameters deﬁning scenario D.2, together with the Higgs boson masses, singlet components and reduced signal rates of h. 37 ##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##

text: latter

label: paragraph

text: is

text: very

text: light

text: with

text: a

text: mass

text: around

text: GeV.

text: The

text: lightest

text: scalar

text: H

text: with

text: a

text: mass

text: of

text: GeV

text: s

text: is

text: close

text: in

text: mass

text: to

text: h,

text: and

text: both

text: mix

text: strongly,

text: so

text: that

text: the

text: gluon

text: fusion

text: production

text: cross

text: section

text: for

text: h

text: only

text: amounts

text: to

text: pb,

text: while

text: the

text: H

text: production

text: is

text: rather

text: large

text: for

text: a

text: singlet-like

text: boson

text: and

text: s

text: reaches

text: pb,

text: as

text: given

text: in

text: Table

text: The

text: h

text: reduced

text: rates

text: are

text: still

text: compatible

text: with

text: the

text: LHC

text: data,

text: although

text: the

text: ﬁnal

text: state

text: rates

text: for

text: ττ

text: and

text: b¯b

text: are

text: somewhat

text: on

text: the

text: lower

text: side.

text: The

text: h

text: cascade

text: decay

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text: an

text: A

text: pair

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text: the

text: (ττ)(b¯b)

text: ﬁnal

text: state

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text: large

text: cross

text: section

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text: fb.

text: This

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text: the

text: triple

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text: possible.

text: Also

text: the

text: 4τ

text: ﬁnal

text: state

text: is

text: hAsAs

text: sizeable

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text: fb.

text: The

text: light

text: A

text: can

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text: too,

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text: ττ

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text: a

text: rate

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text: fb.

text: The

text: copiously

text: produced

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text: in

text: the

text: standard

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text: decay

text: channels.

text: Diﬃcult,

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text: if

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text: production

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text: the

text: doublet-like

text: H

text: and

text: A,

text: as

text: they

text: have

text: masses

text: of

text: TeV.

text: Since

text: they

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text: rather

text: down-component

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text: one

text: might

text: consider

text: associated

text: production

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text: a

text: b-quark

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text: exploited

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text: missing

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text: case

text: the

text: heavy

text: Higgs

text: bosons

text: cannot

text: be

text: discovered.

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text: an

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text: i

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text: Acknowledgments We thank Alexander Nikitenko and Markus Schumacher for fruitful discussions and advise. RN also thanks Peter Athron, Anthony Thomas and Anthony Williams for useful discussions. MM acknowledges discussions with Conny Beskidt, Ramona Gr¨ober, Rui Santos and Michael Spira. MM is supported by the DFG SFB/TR9 “Computational Particle Physics”. The work by RN was supported by the Australian Research Council through the ARC Center of Excel- lence in Particle Physics at the Terascale. SFK is supported by the STFC Consolidated grant ST/J000396/1 and EU ITN grant INVISIBLES 289442 . KW gratefully acknowledges sup- port of the Graduiertenkolleg GRK1694 “Elementarteilchen bei h¨ochster Energie und h¨ochster Pr¨azision”. Appendix A Higgs Couplings in the Natural NMSSM In the following approximations for the Higgs couplings in the parameter range of the Natural NMSSM will be given. First of all, the 125 GeV Higgs boson has to be produced with SM-like rates. This means that it must have large enough couplings to top quarks and hence a large H component. At the same time, it is advantageous not to have a too large H component, as u d then the coupling to b quarks is suppressed, leading to enhanced branching ratios that account for a possible slight suppression in the production. Due to the unitarity of the mixing matrix rotating the interaction to the Higgs mass eigenstates, this means, that the other doublet-like heavier Higgs boson must be H -like. The light CP-even and CP-odd Higgs bosons, H and A d s s respectively, must be singlet-like in order to avoid the exclusion bounds. In summary, for the natural NMSSM the approximate compositions of the CP-even Higgs mass eigenstates expressed R in terms of the mixing matrix (i = H , h, H, j = h , h , h ), in the notation of Section 5, ij s  d u s are given by   0 0 1 R (H , h, H)T = S(h , h , h )T = 0 1 0 (h , h , h )T . (A.1) s d u s d u s 1 0 0 (cid:18) (cid:19) For the composition of the pseudoscalars we have, R 0 1 (A , A)T = P (a, a )T = (a, a )T , (A.2) s s 1 0 s with a = sin βa + cos βa . Comparing this with the actual composition as result of our scan, d u the actual composition is approximated rather well, with at most 30% deviations. With these assumptions we get for the trilinear Higgs couplings involved in the Higgs-to-Higgs decays, in 40

text: Acknowledgments

label: section

text: We

text: thank

text: Alexander

text: Nikitenko

text: and

text: Markus

text: Schumacher

text: for

text: fruitful

text: discussions

text: and

text: advise.

text: RN

text: also

text: thanks

text: Peter

text: Athron,

text: Anthony

text: Thomas

text: and

text: Anthony

text: Williams

text: for

text: useful

text: discussions.

text: MM

text: acknowledges

text: discussions

text: with

text: Conny

text: Beskidt,

text: Ramona

text: Gr¨ober,

text: Rui

text: Santos

text: and

text: Michael

text: Spira.

text: MM

text: is

text: supported

text: by

text: the

text: DFG

text: “Computational

text: Particle

text: Physics”.

text: The

text: work

text: by

text: RN

text: was

text: supported

text: by

text: the

text: Australian

text: Research

text: Council

text: through

text: the

text: ARC

text: Center

text: of

text: Excel-

text: lence

text: in

text: Particle

text: Physics

text: at

text: the

text: Terascale.

text: SFK

text: is

text: supported

text: by

text: the

text: STFC

text: Consolidated

text: grant

text: and

text: EU

text: ITN

text: grant

text: INVISIBLES

text: KW

text: gratefully

text: acknowledges

text: sup-

text: port

text: of

text: the

text: Graduiertenkolleg

text: “Elementarteilchen

text: bei

text: h¨ochster

text: Energie

text: und

text: h¨ochster

text: Pr¨azision”.

text: Appendix

text: A

text: Higgs

text: Couplings

text: in

text: the

text: Natural

text: NMSSM

text: In

text: the

text: following

text: approximations

text: for

text: the

text: Higgs

text: couplings

text: in

text: the

text: parameter

text: range

text: of

text: the

text: Natural

text: NMSSM

text: will

text: be

text: given.

text: First

text: of

text: all,

text: the

text: GeV

text: Higgs

text: boson

text: has

text: to

text: be

text: produced

text: with

text: SM-like

text: rates.

text: This

text: means

text: that

text: it

text: must

text: have

text: large

text: enough

text: couplings

text: to

text: top

text: quarks

text: and

text: hence

text: a

text: large

text: H

text: component.

text: At

text: the

text: same

text: time,

text: it

text: is

text: advantageous

text: not

text: to

text: have

text: a

text: too

text: large

text: H

text: component,

text: as

text: u

text: d

text: then

text: the

text: coupling

text: to

text: b

text: quarks

text: is

text: suppressed,

text: leading

text: to

text: enhanced

text: branching

text: ratios

text: that

text: account

text: for

text: a

text: possible

text: slight

text: suppression

text: in

text: the

text: production.

text: Due

text: to

text: the

text: unitarity

text: of

text: the

text: mixing

text: matrix

text: rotating

text: the

text: interaction

text: to

text: the

text: Higgs

text: mass

text: eigenstates,

text: this

text: means,

text: that

text: the

text: other

text: doublet-like

text: heavier

text: Higgs

text: boson

text: must

text: be

text: H

text: -like.

text: The

text: light

text: CP-even

text: and

text: CP-odd

text: Higgs

text: bosons,

text: H

text: and

text: A

text: d

text: s

text: s

text: respectively,

text: must

text: be

text: singlet-like

text: in

text: order

text: to

text: avoid

text: the

text: exclusion

text: bounds.

text: In

text: summary,

text: for

text: the

text: natural

text: NMSSM

text: the

text: approximate

text: compositions

text: of

text: the

text: CP-even

text: Higgs

text: mass

text: eigenstates

text: expressed

text: R

text: in

text: terms

text: of

text: the

text: mixing

text: matrix

text: (i

text: H

text: h,

text: H,

text: j

text: h

text: h

text: h

text: in

text: the

text: notation

text: of

text: Section

text: ij

text: s

text: d

text: u

text: s

text: are

text: given

text: by

text: R

text: (H

text: h,

text: H)T

text: S(h

text: h

text: h

text: )T

text: (h

text: h

text: h

text: )T

text: s

text: d

text: u

text: s

text: d

text: u

text: s

text: For

text: the

text: composition

text: of

text: the

text: pseudoscalars

text: we

text: have,

text: R

text: (A

text: A)T

text: P

text: (a,

text: a

text: )T

text: (a,

text: a

text: )T

text: s

text: s

text: s

text: with

text: a

text: sin

text: βa

text: cos

text: βa

text: Comparing

text: this

text: with

text: the

text: actual

text: composition

text: as

text: result

text: of

text: our

text: scan,

text: d

text: u

text: the

text: actual

text: composition

text: is

text: approximated

text: rather

text: well,

text: with

text: at

text: most

text: deviations.

text: With

text: these

text: assumptions

text: we

text: get

text: for

text: the

text: trilinear

text: Higgs

text: couplings

text: involved

text: in

text: the

text: Higgs-to-Higgs

text: decays,

text: in


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text: 8 Fig. (a)]. This violation can be detected experimentally, T. Frey, P. Leek, M. Beck, A. Blais, T. Ihn, K. Ensslin, − as p [ 1] can be extracted from the behaviour of dI/dV and A. Wallraﬀ, noop Phys. Rev. Lett. 108 (2012). tot and ∆S [ω, V ], c.f. Eqs. (20) and (22). I K. D. Petersson, L. W. McFaul, M. D. Schroer, M. Jung, Heisenberg-Langevin equation– A damped, driven cav- J. M. Taylor, A. A. Houck, and J. R. Petta, noop Nature ity can be described using standard input-output theory 490, 380 (2012). [26], with the cavity equation of motion √ M. Delbecq, V. Schmitt, F. Parmentier, N. Roch, J. Vien- − − κ − aˆ˙ = iΩaˆ aˆ κaˆ (t). (25) not, G. Feve, B. Huard, C. Mora, A. Cottet, and T. Kon- 2 in tos, noop Phys Rev Lett 107, 256804 (2011). Here, aˆ (t) = α (t) + ξˆ(t) describes the input ﬁeld on G. Gasse, C. Lupien, and B. Reulet, noop Phys. Rev. in in the cavity: it has an average part α (t) which describes Lett. 111, 136601 (2013). in the classical amplitude of the drive, and a noise part ξˆ(t) A. A. Odintsov, noop Sov. Phys. JETP 67, 1265 (1988). which describes both thermal and quantum noise inci- Y. V. Nazarov, noop Sov. Phys. JETP 68, 561 (1989). dent on the cavity. As with standard input-output the- ory treatments, this noise is taken to be operator-valued M. H. Devoret, D. Esteve, H. Grabert, G.-L. Ingold, Gaussian white noise. For a coherent state drive, ξˆ de- H. Pothier, and C. Urbina, Phys. Rev. Lett. 64, 1824 scribes vacuum noise, and the average cavity amplitude (1990). (cid:104) (cid:105) − is aˆ = αine iωLt. Squeezed input noise can be simply S. M. Girvin, L. I. Glazman, M. Johnson, D. R. Penn, included in(cid:104)the for(cid:48)m(cid:105)alism; it corresponds to anomalous and M. D. Stiles, noop Phys. Rev. Lett. 64, 3183 (1990). correlators ξˆ(t)ξˆ(t ) being non-zero. G.-L. Ingold and Y. Nazarov, edited by H. Grabert and As the input noise is Gaussian and the cavity has no M.H. Devoret 7, 935 (Plenum, 1992). nonlinearities, the cavity will also be in a Gaussian state. As a result, the phase-phase correlator in Eq.(3) is com- P. Delsing, K. K. Likharev, L. S. Kuzmin, and T. Clae- pletely determined by two-point correlation functions, son, Phys. Rev. Lett. 63, 1180 (1989). and is thus easily found from the solution of Eq. (25). L. J. Geerligs, V. F. Anderegg, C. A. v. d. Jeugd, Acknowledgements J. Romijn, and J. E. Mooij, noop Europhys. Lett. We thank F. Portier and F. Marquardt for helpful dis- 10, 79 (1989). cussions. MJW acknowledges funding from an ECR A. N. Cleland, J. M. Schmidt, and J. Clarke, Phys. Rev. Grant of UNSW Canberra.. JG and PS acknowledge Lett. 64, 1565 (1990). partial support from “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM). AC acknowledges C. Altimiras, U. Gennser, A. Cavanna, D. Mailly, and support from NSERC. F. Pierre, noop Phys. Rev. Lett. 99, 256805 (2007). F. Parmentier, A. Anthore, S. Jezouin, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly, and F. Pierre, Nat Phys 7, 935 (2011). ∗ F. Chen, A. J. Sirois, R. W. Simmonds, and A. J. Rim- fxundeﬁned [1]ifx1 fnum [1]1ﬁrstoftwo fx [1]secondoftwo noop [0]secondoftworef[1]@startlink1@hrefhref[1]1@endlinkanitize@urlberg, noop App. Phys. Lett. 98, 132509 (2011). [0]startlink[1]endlink[0]rl [1]href 1bib@innerbibempty A. Armour, M. Blencowe, E. Brahimi, and A. Rimberg, P. K. Tien and J. P. Gordon, Phys. Rev. 129, 647 (1963). noop Phys. Rev. Lett. 111, 247001 (2013). J. R. Tucker and M. J. Feldman, Rev. Mod. Phys. 57, S.-X. Li and J. B. Kycia, noop App. Phys. Lett. 102, 1055 (1985). 242601 (2013). L. P. Kouwenhoven, S. Jauhar, J. Orenstein, P. McEuen, Y. Nazarov and Y. Blanter, noop Quantum Transport: Y. Nagamune, J. Motohisa, and H. Sakaki, noop Phys. Introduction to Nanoscience (Cambridge, Cambridge, Rev. Lett. 73, 3443 (1994). 2009). J. Gabelli and B. Reulet, Phys. Rev. Lett. 100, 026601 A. A. Clerk, M. H. Devoret, S. M. Girvin, F. Marquardt, (2008). and R. J. Schoelkopf, Rev. Mod. Phys. 82, 1155 (2010). T. Holst, D. Esteve, C. Urbina, and M. H. Devoret, C. Altimiras, O. Parlavecchio, P. Joyez, D. Vion, P. Roche, Phys. Rev. Lett. 73, 3455 (1994). D. Esteve, and F. Portier, noop Phys. Rev. Lett. 112, 236803 (2014). J. Basset, H. Bouchiat, and R. Deblock, noop Phys. Rev. Lett. 105, 166801 (2010). J. Poyatos, J. Cirac, and P. Zoller, noop Phys. Rev. Lett. 77, 4728 (1996). M. Hofheinz, F. Portier, Q. Baudouin, P. Joyez, D. Vion, P. Bertet, P. Roche, and D. Esteve, Phys. Rev. Lett. K. Murch, U. Vool, D. Zhou, S. Weber, S. M. Girvin, and 106, 217005 (2011). I. Siddiqi, noop Phys. Rev. Lett. 109, 183602 (2012). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fig.

text: (a)].

text: This

text: violation

text: can

text: be

text: detected

text: experimentally,

text: T.

text: Frey,

text: P.

text: Leek,

text: M.

text: Beck,

text: A.

text: Blais,

text: T.

text: Ihn,

text: K.

text: Ensslin,

text: as

text: p

text: can

text: be

text: extracted

text: from

text: the

text: behaviour

text: of

text: dI/dV

text: and

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text: ##LTLine##

text: ##LTLine##


text: THE EXPONENTIAL MAP IS CHAOTIC 9 H { ∈ C } 1 (a) For the right half plane = z : Re z > 0 , ρH(z) = . Re z 1 (b) For a strip of height 2π: ρ{ ∈C | | }(z) = . z : Imz <π 2 cos(Im(z)/2) (c) For the positively/negatively slit plane: 1 1 ρC\ ∞ (z) = | | ; ρC\ −∞ (z) = | | . [0, ) ( ,0] 2 z sin(arg(z)/2) 2 z cos(Arg(z)/2) (In the ﬁrst case, the branch of the argument should be taken to range between 0 and 2π.) Proof. This can easily be veriﬁed by explicit computation using Theorem 3.3 (c), using the following conformal isomorphisms: − H → D (cid:55)→ 1 z → H (cid:55)→ ϕ : ; z ; ϕ : S ; z ez ; 1 2 2 1 + z H → C \ ∞ (cid:55)→ − C \ ∞ → C \ −∞ (cid:55)→ − ϕ : [0, ); z z2; ϕ : [0, ) ( , 0]; z z, 3 4 { ∈ C | | } where S = z : Im z < π . As an example, we consider the case of ϕ , and leave 2 the remaining calculations to the reader. We have | | · | (cid:48) | ϕ (z) 1 1 (cid:4) 2 ρ (z) = ρH(ϕ (z)) ϕ (z) = = = . S 2 2 2 Re ϕ (z) 2 cos(arg(ϕ (z))) 2 cos(Im(z)/2) 2 2 4. Escaping points and sensitive dependence For the remainder of the article (unless stated otherwise), f will always denote the exponential map C → C (cid:55)→ f : ; z ez. We shall now prove the ﬁrst part of Theorem 1.1, namely that the escaping set { ∈ C → ∞} I(f) ..= z : fn(z) is dense in the complex plane. The proof that we present here was brieﬂy outlined in a paper of Mihaljevi´c-Brandt and the second author [22, Remark on pp. 1583–1584]. 4.1. Theorem (Density of the escaping set). The set I(f), consisting of those points z whose f-orbits converge to inﬁnity, is a dense C 0 subset of the complex plane . ⊂ C The idea of the proof can be outlined as follows. Let D be any small round disc. R ⊂ Since I(f), and any preimage of an escaping point is also escaping, there is nothing to prove if U contains a point z whose orbit contains a point on the real axis. Otherwise, D, f(D) and all forward images fn(D) are contained in the slit plane C \ ∞ U ..= [0, ). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: THE

text: EXPONENTIAL

text: MAP

text: IS

text: CHAOTIC

text: H

label: list

text: C

text: (a)

text: For

text: the

text: right

text: half

text: plane

text: z

text: Re

text: z

text: ρH(z)

text: Re

text: z

text: (b)

text: For

text: a

text: strip

text: of

text: height

text: ρ{

text: ∈C

text: }(z)

text: z

text: Imz

text: <π

text: (c)

text: For

text: the

text: positively/negatively

text: slit

text: plane:

text: ρC\

text: (z)

text: ρC\

text: (z)

text: z

text: z

text: (In

text: the

text: ﬁrst

text: case,

text: the

text: branch

text: of

text: the

text: argument

text: should

text: be

text: taken

text: to

text: range

text: between

text: and

text: Proof.

text: This

text: can

text: easily

text: be

text: veriﬁed

text: by

text: explicit

text: computation

text: using

text: Theorem

text: (c),

text: using

text: the

text: following

text: conformal

text: isomorphisms:

text: H

text: D

text: z

text: H

text: ϕ

text: z

text: ϕ

text: S

text: z

text: ez

text: z

text: H

text: C

text: C

text: C

text: ϕ

text: z

text: ϕ

text: z

text: z,

text: C

text: where

text: S

text: z

text: Im

text: z

text: π

text: As

text: an

text: example,

text: we

text: consider

text: the

text: case

text: of

text: ϕ

text: and

text: leave

text: the

text: remaining

text: calculations

text: to

text: the

text: reader.

text: We

text: have

text: ϕ

text: (z)

text: ρ

text: (z)

text: ρH(ϕ

text: (z))

text: ϕ

text: (z)

text: S

text: Re

text: ϕ

text: (z)

text: cos(arg(ϕ

text: (z)))

text: Escaping

text: points

text: and

text: sensitive

text: dependence

text: For

text: the

text: remainder

text: of

text: the

text: article

text: (unless

text: stated

text: otherwise),

text: f

text: will

text: always

text: denote

text: the

text: exponential

text: map

text: C

text: C

text: f

text: z

text: ez.

text: We

text: shall

text: now

text: prove

text: the

text: ﬁrst

text: part

text: of

text: Theorem

text: namely

text: that

text: the

text: escaping

text: set

text: C

text: I(f)

text: z

text: fn(z)

text: is

text: dense

text: in

text: the

text: complex

text: plane.

text: The

text: proof

text: that

text: we

text: present

text: here

text: was

text: brieﬂy

text: outlined

text: in

text: a

text: paper

text: of

text: Mihaljevi´c-Brandt

text: and

text: the

text: second

text: author

text: Remark

text: on

text: pp.

text: Theorem

text: (Density

text: of

text: the

text: escaping

text: set).

text: The

text: set

text: I(f),

text: consisting

text: of

text: those

text: points

text: z

text: whose

text: f-orbits

text: converge

text: to

text: inﬁnity,

text: is

text: a

text: dense

text: C

text: subset

text: of

text: the

text: complex

text: plane

text: C

text: The

text: idea

text: of

text: the

text: proof

text: can

text: be

text: outlined

text: as

text: follows.

text: Let

text: D

text: be

text: any

text: small

text: round

text: disc.

text: R

text: Since

text: I(f),

text: and

text: any

text: preimage

text: of

text: an

text: escaping

text: point

text: is

text: also

text: escaping,

text: there

text: is

text: nothing

text: to

text: prove

text: if

text: U

text: contains

text: a

text: point

text: z

text: whose

text: orbit

text: contains

text: a

text: point

text: on

text: the

text: real

text: axis.

text: Otherwise,

text: D,

text: f(D)

text: and

text: all

text: forward

text: images

text: fn(D)

text: are

text: contained

text: in

text: the

text: slit

text: plane

text: C

text: U

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: HO¨LDER ESTIMATES AND PROPER MAPPINGS 3 C Here k,α(Γ) is the space of functions on the smooth curve Γ which are k times continuously diﬀerentiable with respect to arc length, and such that the k-th derivative is H¨older continuous (cid:89) of order α. Our second example corresponds to the choice p = ω , where n n − ω (z , . . . , z , t) = (t z ). (1.9) n 1 n j j=1 C\ C \ ∗ Assuming that Γ has κ components, the space n Γ has κn components, and one of these ωn B components is the n-fold cartesian product Un. We deﬁne the operator again by restriction n to Un: B D | φ = ( φ) , (1.10) n ωn Un B T ≥ B Note that = , and for n 2, we will call the Cauchy-Nørlund Transform (cf. [16, 1 A C n ∩O O p. 199]). Denote by k,α (Un) the intersection k,α(Un) (Un), where (Un) is the space sym sym ∈ O sym of symmetric holomorphic functions. Recall that a function f (Un) is symmetric if for each (cid:0) ∈ (cid:1) permutation σ in the symmetric group S and for each z Un, we have f(z) = f(σ(z)), where n σ(z , . . . , z ) = z , . . . , z . (1.11) 1 n σ(1) σ(n) B Our main result regarding is: n ≥ ≥ C Theorem 1.3. Let k 0, n 1 be integers and let the boundary Γ of U be of class k+n+1. Then, B C − for 0 < α < 1, the Cauchy-Nørlund Transform is a continuous linear map from k+n 1,α(Γ) A n to k,α (Un). sym This paper is organized as follows. After reviewing some elementary properties of the operators B E and introduced above in Section , in Section we introduce the key observation (Proposi- n n B tion 4.1), that can be represented in terms of divided diﬀerences, discrete analogs of derivatives n which occur in Newton’s interpolation formula. This, along with a representation (4.4) of the di- vided diﬀerence allow us to prove Theorem 1.3 for convex domains U, and the general case follows in Section by conformal mapping and a covering argument using the fact that bounded locally H¨older functions are H¨older. In Section Theorem 1.2 is deduced from Theorem 1.3 and a dis- tortion estimate for the map π, which we obtain using the results of [12]. Finally, in Section , a representation of proper holomorphic maps between symmetric products as an integral from [4] is used along with Theorem 1.2 to obtain Theorem 1.1. We note here that the statements of Theorem 1.2 and Theorem 1.1 can probably be improved, and slightly better regularity results may be obtained by more work. The non-optimality is because of possible loss of information in the course of the proof. It will be interesting to obtain sharp results in this direction. 2. Acknowledgements This research was conducted in Summer 2014 at the NSF funded Research Experience for Undergraduates program at Central Michigan University, where Chakrabarti served as the faculty advisor and Castle, Gunderman and Lehet as student participants in one of the research teams. We gratefully acknowledge the support of the NSF through their grant DMS 11-56890. Very special thanks are due to Sivaram Narayan and the Mathematics Department of Central Michigan University for the excellent organization of the REU program, and to Ben Salisbury for discussions

text: HO¨LDER

text: ESTIMATES

text: AND

text: PROPER

text: MAPPINGS

text: C

text: Here

text: k,α(Γ)

text: is

text: the

text: space

text: of

text: functions

text: on

text: the

text: smooth

text: curve

text: Γ

text: which

text: are

text: k

text: times

text: continuously

text: diﬀerentiable

text: with

text: respect

text: to

text: arc

text: length,

text: and

text: such

text: that

text: the

text: k-th

text: derivative

text: is

text: H¨older

text: continuous

label: equation

text: of

text: order

text: α.

text: Our

text: second

text: example

text: corresponds

text: to

text: the

text: choice

text: p

text: ω

text: where

text: n

text: n

text: ω

text: (z

text: z

text: t)

text: (t

text: z

text: n

text: n

text: j

text: C\

text: C

text: Assuming

text: that

text: Γ

text: has

text: κ

text: components,

text: the

text: space

text: n

text: Γ

text: has

text: κn

text: components,

text: and

text: one

text: of

text: these

text: ωn

text: B

text: components

text: is

text: the

text: n-fold

text: cartesian

text: product

text: Un.

text: We

text: deﬁne

text: the

text: operator

text: again

text: by

text: restriction

text: n

text: to

text: Un:

text: B

text: D

text: φ

text: φ)

text: n

text: ωn

text: Un

text: B

text: T

text: B

text: Note

text: that

text: and

text: for

text: n

text: we

text: will

text: call

text: the

text: Cauchy-Nørlund

text: Transform

text: (cf.

text: A

text: C

text: n

text: ∩O

text: O

text: p.

text: Denote

text: by

text: k,α

text: (Un)

text: the

text: intersection

text: k,α(Un)

text: (Un),

text: where

text: (Un)

text: is

text: the

text: space

text: sym

text: sym

text: O

text: sym

text: of

text: symmetric

text: holomorphic

text: functions.

text: Recall

text: that

text: a

text: function

text: f

text: (Un)

text: is

text: symmetric

text: if

text: for

text: each

text: permutation

text: σ

text: in

text: the

text: symmetric

text: group

text: S

text: and

text: for

text: each

text: z

text: Un,

text: we

text: have

text: f(z)

text: f(σ(z)),

text: where

text: n

text: σ(z

text: z

text: z

text: z

text: n

text: σ(n)

text: B

text: Our

text: main

text: result

text: regarding

text: is:

text: n

text: C

text: Theorem

text: Let

text: k

text: n

text: be

text: integers

text: and

text: let

text: the

text: boundary

text: Γ

text: of

text: U

text: be

text: of

text: class

text: Then,

text: B

text: C

text: for

text: α

text: the

text: Cauchy-Nørlund

text: Transform

text: is

text: a

text: continuous

text: linear

text: map

text: from

text: k+n

text: A

text: n

text: to

text: k,α

text: (Un).

text: sym

text: This

text: paper

text: is

text: organized

text: as

text: follows.

text: After

text: reviewing

text: some

text: elementary

text: properties

text: of

text: the

text: operators

text: B

text: E

text: and

text: introduced

text: above

text: in

text: Section

text: in

text: Section

text: we

text: introduce

text: the

text: key

text: observation

text: (Proposi-

text: n

text: n

text: B

text: tion

text: that

text: can

text: be

text: represented

text: in

text: terms

text: of

text: divided

text: diﬀerences,

text: discrete

text: analogs

text: of

text: derivatives

text: n

text: which

text: occur

text: in

text: Newton’s

text: interpolation

text: formula.

text: This,

text: along

text: with

text: a

text: representation

text: of

text: the

text: di-

text: vided

text: diﬀerence

text: allow

text: us

text: to

text: prove

text: Theorem

text: for

text: convex

text: domains

text: U,

text: and

text: the

text: general

text: case

text: follows

text: in

text: Section

text: by

text: conformal

text: mapping

text: and

text: a

text: covering

text: argument

text: using

text: the

text: fact

text: that

text: bounded

text: locally

text: H¨older

text: functions

text: are

text: H¨older.

text: In

text: Section

text: Theorem

text: is

text: deduced

text: from

text: Theorem

text: and

text: a

text: dis-

text: tortion

text: estimate

text: for

text: the

text: map

text: π,

text: which

text: we

text: obtain

text: using

text: the

text: results

text: of

text: Finally,

text: in

text: Section

text: a

text: representation

text: of

text: proper

text: holomorphic

text: maps

text: between

text: symmetric

text: products

text: as

text: an

text: integral

text: from

text: is

text: used

text: along

text: with

text: Theorem

text: to

text: obtain

text: Theorem

text: We

text: note

text: here

text: that

text: the

text: statements

text: of

text: Theorem

text: and

text: Theorem

text: can

text: probably

text: be

text: improved,

text: and

text: slightly

text: better

text: regularity

text: results

text: may

text: be

text: obtained

text: by

text: more

text: work.

text: The

text: non-optimality

text: is

text: because

text: of

text: possible

text: loss

text: of

text: information

text: in

text: the

text: course

text: of

text: the

text: proof.

text: It

text: will

text: be

text: interesting

text: to

text: obtain

text: sharp

text: results

text: in

text: this

text: direction.

text: Acknowledgements

text: This

text: research

text: was

text: conducted

text: in

text: Summer

text: at

text: the

text: NSF

text: funded

text: Research

text: Experience

text: for

text: Undergraduates

text: program

text: at

text: Central

text: Michigan

text: University,

text: where

text: Chakrabarti

text: served

text: as

text: the

text: faculty

text: advisor

text: and

text: Castle,

text: Gunderman

text: and

text: Lehet

text: as

text: student

text: participants

text: in

text: one

text: of

text: the

text: research

text: teams.

text: We

text: gratefully

text: acknowledge

text: the

text: support

text: of

text: the

text: NSF

text: through

text: their

text: grant

text: DMS

text: Very

text: special

text: thanks

text: are

text: due

text: to

text: Sivaram

text: Narayan

text: and

text: the

text: Mathematics

text: Department

text: of

text: Central

text: Michigan

text: University

text: for

text: the

text: excellent

text: organization

text: of

text: the

text: REU

text: program,

text: and

text: to

text: Ben

text: Salisbury

text: for

text: discussions


text: 6 EVAN CASTLE, DEBRAJ CHAKRABARTI, DAVID GUNDERMAN, AND ELLEN LEHET This continues to hold when z = z thanks to the fact that both sides are analytic. The result 1 n+1 (cid:3) follows by induction. As a consequence we obtain a property of divided diﬀerences which is far from obvious from the representation (4.1). ∈ O C − ∈ O Corollary 4.2. If f (Ω), where Ω is an open subset of , then f[n 1] (Ωn), the space sym of symmetric holomorphic functions on the Cartesian power Ωn. ∈ C (cid:98) Proof. Given a point (z , . . . , z ) n, let U Ω be a smoothly bounded open set containing 1 n | each of the points z , z , . . . , z and let Γ be the boundary of U. If we set φ = f , by the Cauchy B1 2 n B Γ integral theorem, φ = f on U, where the Cauchy integral φ is taken along Γ. But then, by 1− B B O 1 (cid:3) Proposition 4.1, f[n 1] = φ. Since φ is obviously in (Un), the result follows. n n sym ≥ To state the next result, we rec(cid:12)all some deﬁnitions. For an integer d 0, wedenote by Σ the standard d-simplex inRd+1: (cid:12)(cid:12) (cid:88)  d (cid:12) (cid:12)   (cid:12) d+1 ∈ R ≥ Σ = x d+1 x = 1, and x 0 for j = 1, . . . , d + 1 , d j j j=1  (cid:12)  and let A be the d-simplex in Rd g(cid:12)(cid:12) i(cid:88)ven by  d (cid:12) (cid:12)   (cid:12) d ∈ R ≤ ≥ A = x d x 1, and x 0 for j = 1, . . . , d . d j j (cid:80) j=1  (cid:12)  (cid:55)→ − Then Σd is the graph of the functio(cid:12)(cid:12)(cid:12)n (x1, . . . , xd(cid:88)) 1 dj=1 xj over Ad, i.e., (cid:12)   (cid:12) d ∈ R − ∈ Σ = x d+1 x = 1 x , and (x , . . . , x ) A . d d+1 j 1 d d j=1 Therefore, on Σ , we can take (x , . . . , x ) as coordinates, which we will refer to as the standard d 1 d coordinates of Σ . d H We denote the d-dimensional Hausdorﬀ measure by . Note that since Σ is contained in (cid:80) R d d an aﬃne hyperplane of d+1, the d-dimensional Hausdorﬀ measure on Σ is simply the usual d surface measure on the aﬃne hyperplane. Using the fact that Σ is the graph of the function − d u(x , . . . , x ) = 1 d x , we can represent the Hausdorﬀ measure on Σ in terms of the 1 d j=1 j d standard coordinates: (cid:118) (cid:117) (cid:18) (cid:19) (cid:117) (cid:88) (cid:116) H d ∂u 2 d = 1 + dx dx . . . dx d 1 2 d ∂x √ j=1 j · = 1 + d dx dx . . . dx . (cid:80) (4.3) 1 2 d (cid:104) (cid:105) We denote by z, w the standard Hermitian inner product z w . We prove a complex version j j of a classical representation of divided diﬀerences: ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: EVAN

text: CASTLE,

text: DEBRAJ

text: CHAKRABARTI,

text: DAVID

text: GUNDERMAN,

text: AND

text: ELLEN

text: LEHET

text: This

text: continues

text: to

text: hold

text: when

text: z

text: z

text: thanks

text: to

text: the

text: fact

text: that

text: both

text: sides

text: are

text: analytic.

text: The

text: result

text: follows

text: by

text: induction.

text: As

text: a

text: consequence

text: we

text: obtain

text: a

text: property

text: of

text: divided

text: diﬀerences

text: which

text: is

text: far

text: from

text: obvious

text: from

text: the

text: representation

text: O

text: C

text: O

text: Corollary

text: If

text: f

text: (Ω),

text: where

text: Ω

text: is

text: an

text: open

text: subset

text: of

text: then

text: f[n

text: (Ωn),

text: the

text: space

text: sym

text: of

text: symmetric

text: holomorphic

text: functions

text: on

text: the

text: Cartesian

text: power

text: Ωn.

text: C

text: Proof.

text: Given

text: a

text: point

text: (z

text: z

text: n,

text: let

text: U

text: Ω

text: be

text: a

text: smoothly

text: bounded

text: open

text: set

text: containing

text: n

text: each

text: of

text: the

text: points

text: z

text: z

text: z

text: and

text: let

text: Γ

text: be

text: the

text: boundary

text: of

text: U.

text: If

text: we

text: set

text: φ

text: f

text: by

text: the

text: Cauchy

text: n

text: B

text: Γ

text: integral

text: theorem,

text: φ

text: f

text: on

text: U,

text: where

text: the

text: Cauchy

text: integral

text: φ

text: is

text: taken

text: along

text: Γ.

text: But

text: then,

text: by

text: 1−

text: B

text: B

text: O

text: Proposition

text: f[n

text: φ.

text: Since

text: φ

text: is

text: obviously

text: in

text: (Un),

text: the

text: result

text: follows.

text: n

text: n

text: sym

text: To

text: state

text: the

text: next

text: result,

text: we

text: some

text: deﬁnitions.

text: For

text: an

text: integer

text: d

text: wedenote

text: by

text: Σ

text: the

text: standard

text: d-simplex

text: d

text: R

text: Σ

text: x

text: x

text: and

text: x

text: for

text: j

text: d

text: d

text: j

text: j

text: and

text: let

text: A

text: be

text: the

text: d-simplex

text: in

text: Rd

text: by

text: d

text: d

text: R

text: A

text: x

text: d

text: x

text: and

text: x

text: for

text: j

text: d

text: d

text: j

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text: Secondary Magnetic Reconnection Driven by Kink Instability 2 1. Introduction Flux ropes are bundles of magnetic ﬁeld lines wrapped around an axis. They are ubiquitous structures in space, astrophysical and fusion device plasmas. The presence of ﬂux ropes has been observed in planetary magnetotail [1], dayside magnetopause [2], and in solar corona [3]. Astrophysical jets can be treated as ﬂux ropes [4, 5, 6]. Flux ropes can also reproduce to a ﬁrst approximation the magnetic ﬁeld geometry of fusion devices such as Z-pinch or Tokamak fusion reactors [7]. The kink instability is one of the most studied instabilities of ﬂux ropes [8]. This instability is particularly important in fusion devices, where it can disrupt the ﬂux rope limiting the conﬁnement of plasma [9]. Because of this, many analytical and numerical studies have been devoted to the understanding of its dynamics. The kink instability has been extensively studied using MHD modeling. It has been examined with Particle-in-Cell methods [10, 11, 12] only recently to enlighten possible kinetic eﬀects during the evolution of the kink instability. This paper presents a simulation study of secondary magnetic reconnection triggered by the internal kink instability of a single ﬂux rope in a periodic domain along the axis direction. Secondary magnetic reconnection occurs when magnetic reconnection is driven by a primary instability that forces the magnetic ﬁeld lines from diﬀerent disconnected magnetic regions to be in contact and reconnect. Therefore, secondary magnetic reconnection develops over a dynamic time scale imposed by the primary instability that acts as a driver. For instance, secondary magnetic reconnection might be triggered by Kelvin-Helmholtz (KH) instability [13]. The vortices created by the KH instability force the magnetic ﬁeld lines from diﬀerent magnetic domains to reconnect. An additional example is the magnetic reconnection driven by interchange instability aﬀecting the reconnection fronts in magnetotail magnetic reconnection [14, 15]. Secondary magnetic reconnection has an important role in the overall dynamics of the primary instability: it converts magnetic ﬁeld energy to kinetic energy of the plasma allowing for a local reconﬁguration of the magnetic ﬁeld topology. Several signatures of magnetic reconnection have been identiﬁed in simpliﬁed conﬁgurations using computer simulations. The most famous example of such conﬁgurations is the Harris current-sheet model [16]. The vast majority of simulations starts from the Harris equilibrium. In this conﬁguration, the most famous signature of magnetic reconnection is the quadrupolar structure of the Hall magnetic ﬁeld developing on the reconnection plane [17, 18]. The quadrupolar structure of the Hall magnetic ﬁeld originates because of the decoupling of electron and ion dynamics in a small region in proximity of the reconnection line. Additional signatures of magnetic reconnection are the presence of regions of space with depleted density (density cavities) [19], of bipolar electric ﬁeld structures and electron holes in the phase space [20, 21], of whistler and kinetic Alfv´en waves [22, 23]. In this paper, we focus on signatures of magnetic reconnection in a more complicated initial conﬁguration, such as the single ﬂux rope set- up. In this conﬁguration, we found signatures that are similar to the ones of magnetic reconnection in the Harris sheet [18] and some new features characteristic of magnetic

text: Secondary

text: Magnetic

text: Reconnection

text: Driven

text: by

text: Kink

text: Instability

text: Introduction

text: Flux

text: ropes

text: are

text: bundles

text: of

text: magnetic

text: ﬁeld

text: lines

text: wrapped

text: around

text: an

text: axis.

text: They

text: are

text: ubiquitous

text: structures

text: in

text: space,

text: astrophysical

text: and

text: fusion

text: device

text: plasmas.

text: The

text: presence

text: of

text: ﬂux

text: ropes

text: has

text: been

text: observed

text: in

text: planetary

text: magnetotail

text: dayside

text: magnetopause

text: and

text: in

text: solar

text: corona

text: Astrophysical

text: jets

text: can

text: be

text: treated

text: as

text: ﬂux

text: ropes

text: Flux

text: ropes

text: can

text: also

text: reproduce

text: to

text: a

text: ﬁrst

text: approximation

text: the

text: magnetic

text: ﬁeld

text: geometry

text: of

text: fusion

text: devices

text: such

text: as

text: Z-pinch

text: or

text: Tokamak

text: fusion

text: reactors

text: The

text: kink

text: instability

text: is

text: one

text: of

text: the

text: most

text: studied

text: instabilities

text: of

text: ﬂux

text: ropes

text: This

text: instability

text: is

text: particularly

text: important

text: in

text: fusion

text: devices,

text: where

text: it

text: can

text: disrupt

text: the

text: ﬂux

text: rope

text: limiting

text: the

text: conﬁnement

text: of

text: plasma

text: Because

text: of

text: this,

text: many

text: analytical

text: and

text: numerical

text: studies

text: have

text: been

text: devoted

text: to

text: the

text: understanding

text: of

text: its

text: dynamics.

text: The

text: kink

text: instability

text: has

text: been

text: extensively

text: studied

text: using

text: MHD

text: modeling.

text: It

text: has

text: been

text: examined

text: with

text: Particle-in-Cell

text: methods

text: only

text: recently

text: to

text: enlighten

text: possible

text: kinetic

text: eﬀects

text: during

text: the

text: evolution

text: of

text: the

text: kink

text: instability.

text: This

text: paper

text: presents

text: a

text: simulation

text: study

text: of

text: secondary

text: magnetic

text: reconnection

text: triggered

text: by

text: the

text: internal

text: kink

text: instability

text: of

text: a

text: single

text: ﬂux

text: rope

text: in

text: a

text: periodic

text: domain

text: along

text: the

text: axis

text: direction.

text: Secondary

text: magnetic

text: reconnection

text: occurs

text: when

text: magnetic

text: reconnection

text: is

text: driven

text: by

text: a

text: primary

text: instability

text: that

text: forces

text: the

text: magnetic

text: ﬁeld

text: lines

text: from

text: diﬀerent

text: disconnected

text: magnetic

text: regions

text: to

text: be

text: in

text: contact

text: and

text: reconnect.

text: Therefore,

text: secondary

text: magnetic

text: reconnection

text: develops

text: over

text: a

text: dynamic

text: time

text: scale

text: imposed

text: by

text: the

text: primary

text: instability

text: that

text: acts

text: as

text: a

text: driver.

text: For

text: instance,

text: secondary

text: magnetic

text: reconnection

text: might

text: be

text: triggered

text: by

text: Kelvin-Helmholtz

text: (KH)

text: instability

text: The

text: vortices

text: created

text: by

text: the

text: KH

text: instability

text: force

text: the

text: magnetic

text: ﬁeld

text: lines

text: from

text: diﬀerent

text: magnetic

text: domains

text: to

text: reconnect.

text: An

text: additional

text: example

text: is

text: the

text: magnetic

text: reconnection

text: driven

text: by

text: interchange

text: instability

text: aﬀecting

text: the

text: reconnection

text: fronts

text: in

text: magnetotail

text: magnetic

text: reconnection

text: Secondary

text: magnetic

text: reconnection

text: has

text: an

text: important

text: role

text: in

text: the

text: overall

text: dynamics

text: of

text: the

text: primary

text: instability:

text: it

text: converts

text: magnetic

text: ﬁeld

text: energy

text: to

text: kinetic

text: energy

text: of

text: the

text: plasma

text: allowing

text: for

text: a

text: local

text: reconﬁguration

text: of

text: the

text: magnetic

text: ﬁeld

text: topology.

text: Several

text: signatures

text: of

text: magnetic

text: reconnection

text: have

text: been

text: identiﬁed

text: in

text: simpliﬁed

text: conﬁgurations

text: using

text: computer

text: simulations.

text: The

text: most

text: famous

text: example

text: of

text: such

text: conﬁgurations

text: is

text: the

text: Harris

text: current-sheet

text: model

text: The

text: vast

text: majority

text: of

text: simulations

text: starts

text: from

text: the

text: Harris

text: equilibrium.

text: In

text: this

text: conﬁguration,

text: the

text: most

text: famous

text: signature

text: of

text: magnetic

text: reconnection

text: is

text: the

text: quadrupolar

text: structure

text: of

text: the

text: Hall

text: magnetic

text: ﬁeld

text: developing

text: on

text: the

text: reconnection

text: plane

text: The

text: quadrupolar

text: structure

text: of

text: the

text: Hall

text: magnetic

text: ﬁeld

text: originates

text: because

text: of

text: the

text: decoupling

text: of

text: electron

text: and

text: ion

text: dynamics

text: in

text: a

text: small

text: region

text: in

text: proximity

text: of

text: the

text: reconnection

text: line.

text: Additional

text: signatures

text: of

text: magnetic

text: reconnection

text: are

text: the

text: presence

text: of

text: regions

text: of

text: space

text: with

text: depleted

text: density

text: (density

text: cavities)

text: of

text: bipolar

text: electric

text: ﬁeld

text: structures

text: and

text: electron

text: holes

text: in

text: the

text: phase

text: space

text: of

text: whistler

text: and

text: kinetic

text: Alfv´en

text: waves

text: In

text: this

text: paper,

text: we

text: focus

text: on

text: signatures

text: of

text: magnetic

text: reconnection

text: in

text: a

text: more

text: complicated

text: initial

text: conﬁguration,

text: such

text: as

text: the

text: single

text: ﬂux

text: rope

text: set-

text: up.

text: In

text: this

text: conﬁguration,

text: we

text: found

text: signatures

text: that

text: are

text: similar

text: to

text: the

text: ones

text: of

text: magnetic

text: reconnection

text: in

text: the

text: Harris

text: sheet

text: and

text: some

text: new

text: features

text: characteristic

text: of

text: magnetic