text: 13 Replacing H with λV we see that to 5’th order in λ 0 (cid:18) (cid:19) − eiSλV e iS = λV + λ2 (i[S , V ]) 1 (cid:16) − 1 +λ3 i[S , V ] [S , [S , V ]] 2 2 1 1 The fj can be constructed in a straightforward manner − i (cid:17) which allows for the computation of the perturbation to +λ4 i[S3, V ] 6[S1, [S1, [S1, V ]]] any order. First, write all decompositions of k > 0 into a sum of non-negative integers as follows −1 ([S , [S , V ]] + [S , [S , V ]]) . 2 1 2 2 1 Hence in total (cid:18) (cid:19) − eiS(H + λV )e iS = H + λ(i[S , H ] + V ) 0 0 1 0 (cid:16) − 1 k : (k), +λ2 i[S , H ] [S , [S , H ]] + i[S , V ] − − − 2 0 2 1 1 0 1 k 1 : (k 1, 1), (1, k 1), − − − − − i k 2 : (k 2, 1, 1), (1, k 2, 1), (1, 1, k 2), +λ3 i[S , H ] [S , [S , [S , H ]]] − − 3 0 6 1 1 1 0 (k 2, 2), (2, k 2), −1 (cid:17) − − − − ([S , [S , H ]] + [S , [S , H ]]) k 3 : (k 3, 1, 1, 1), (1, k 3, 1, 1), (1, 1, k 3, 1), 2 1 2 0 2 1 0 − − − (cid:32) (1, 1, 1, k 3), (k 3, 1, 2), (k 3, 2, 1), (cid:16) − 1 − − − − +i[S , V ] [S , [S , V ]] 2 2 1 1 (1, k 3, 2), (2, k 3, 1), (1, 2, k 3), (2, 1, k 3), . +λ4 i[S , H ] − i [S (cid:17), [S , [S , H ]]] + [S , [S , [S , H ]]] 4 0 6 1 1 2 0 1 2 1 0 . . 1 +[S , [S , [S , H ]]] + [S , [S , [S , [S , H ]]]] 2 1 1 0 24 1 1 1 1 0 0 : (1, 1, 1, ..., 1), −1 − 1 [S , [S , H ]] ([S , [S , H ]] + [S , [S , H ]]) 2 2 2 0 2 1 3 0 3 1 0 (cid:33) − i +i[S , V ] [S , [S , [S , V ]]] 3 6 1 1 1 −1 (cid:0) (cid:1) ([S , [S , V ]] + [S , [S , V ]]) where (1, 1, 1, ..., 1) has k indices. We now take each 2 1 2 2 1 (j , ..., j ) from the above expression and make the as- 1 b signment +O λ5 . (A5) (cid:110) (cid:16) (cid:17)(cid:111) Eq. A5 can be written in a more com- p(cid:110)act fashio(cid:16)n by (cid:17)(cid:111) deﬁning two ∞ sequences { } of functions f = f A j+1 and j j i i=1 ∞ j=0 { } → ib H(j) = H(j) Ai ji=+12 j=0 where the Ai are in- (j1, ..., jb) b![Sj1, [Sj2, ..., [Sjn−1, [Sjn, H0]]...]]. determinate variables indicating the number of inputs to each function, (cid:88) (cid:0) (cid:1) ∞ As an example we compute the ﬁfth order expression. eiS(H + λV )e−iS = λkH(k) {S }k , H , V We have 0 j j=1 0 (cid:2) (cid:3) k=0 (5), = λ0(cid:2)[f0(H{ 0)}] (cid:3) (4, 1), (1, 4), + λ1 (cid:2)f1( Sj 1j=1, H0) + f0(V ) (cid:3) (3, 1, 1), (1, 3, 1), (1, 1, 3), { } { } + λ2 (cid:2)f2( Sj 2j=1, H0) + f1( Sj 1j=1, V )(cid:3) (3, 2), (2, 3), { } { } + λ3(cid:0) f3(cid:1)( Sj 3j=1, H0) + f2( Sj 2j=1, V ) (2, 1, 1, 1), (1, 2, 1, 1), (1, 1, 2, 1), (1, 1, 1, 2), { } { } + λ4 f ( S 4 , H ) + f ( S 3 , V ) (2, 2, 1), (2, 1, 2), (1, 2, 2), 4 j j=1 0 3 j j=1 + O λ5 . (1, 1, 1, 1, 1). ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Replacing

text: H

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text: λV

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text: V

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text: 5     ﬁrst dress H(0) in Eq. 2.12 and then rotate the drive term √ sys        √  into this frame. The system is then moved into the frame  2   0      rotating at the target qubit frequency on both qubits and  √0   2  an RWA is performed. In this rotating frame the con- −√ 2Ω , √0 , trol |0(cid:105) and |1(cid:105) states deﬁne two subspaces that are far  (∆+ ∆02+Ω2)   −√ 2Ω  detuned by ∼ ∆ and an eﬀective block-diagonal Hamil- (∆+ ∆2+Ω2) √ tonian is obtained via the perturbative analysis of Sec. .        √  Unlike the qubit case, exact analytical expressions are  2   0      not straightforward to obtain and so realistic parameters  √0   2  (∆−−√∆22Ω+Ω2) , −√√02Ω . (3.11) aetreerusswede fﬁonrdthtehaetxatchtempeetrhtuordboaftiSveec.ex.pFroersstihonesseapnadratmhe- 0 − exact block-diagonalization agree up to medium power (∆ ∆2+Ω2) ∼ drives of Ω 50 MHz with the exact method holding for much larger values of Ω. (cid:28) To start we assume that | −J | 1 and obtain an Finally computing the block-diagonal of HR and mov- ω˜1 ω˜2 ing back to the physical frame consisting of the trans- eﬀective diagonal Hamiltonian for H(0). Letting U be sys mons rotating at their respective frequencies gives the the diagonalizing (dressing) unitary the eﬀective diagonal × block-diagonal Hamiltonian HCR with 2 2 blocks given Hamiltonian is given by by (cid:34) (cid:35) √ † 1 ∆ − ∆2 + Ω2 −√√ JΩ H˜s(y0s) = U Hs(y0s)U, (4.1) 2 −√ JΩ ∆ − ∆∆22+Ω+2Ω2 . (3.12) ∆2+Ω2 where to second order in the two-qubit subspace    √   −∆ + ∆2 + Ω2 √√JΩ  H˜s(y0s) = ω1 Z211 + ω2 112Z + ξ Z2Z , (4.2) 1 − ∆2+Ω2 √ JΩ ∆ + ∆2 + Ω2 . (3.13) 2 ∆2+Ω2 with The ZX(cid:18)term i(cid:20)s thus(cid:21)(cid:19)given by − − J2 − ω = ω˜ ξ, (4.3) 1 1 ∆ ZX −√ JΩ − J2 − tr HCR 2 = ∆2 + Ω2 , (3.14) ω2 = ω˜2 + ∆ ξ, (4.4) − J2(δ + δ ) ξ = 1 −2 . (4.5) where by virtue of the system Hamiltonian deﬁnition, (∆ + δ1)(δ2 ∆) the two-qubit Pauli operators are scaled by 1 (in an n- 2 qubit system they are scaled by 1− ). The Stark-shift The presence of higher levels has produced an eﬀec- 2n 1 term on th(cid:18)e cont(cid:20)rol q(cid:21)u(cid:19)bit is given by tive ZZ interaction in the two-qubit subspace. The drive (cid:112) term of Eq. 2.14 is rotated into this frame by applying the diagonalizing unitary U, ZI − (cid:88) (cid:2) (cid:3) tr H = ∆ ∆2 + Ω2, (3.15) CR 2 2 H˜ = Ω (t) cos(ω t) + Ω (t) sin(ω t) B˜ , and so i(cid:16)n tota(cid:112)l (cid:17) (cid:18) (cid:19) d j=1 Xj dj Yj dj j (4.6) HCR = ∆ − ∆2 + Ω2 Z21 − √∆J2 Ω+ Ω2 Z2X . where B˜j = U†(b†j + bj)U for j = 1, 2. We set ωd1 = ω = ω and the Hamiltonian in the dressed frame is (3.16) d2 d given by IV. EFFECTIVE CR HAMILTONIAN FOR A HIGHER-LEVEL MODEL H(t) = H˜ (0) + H˜ (t), (4.7) sys d For a model including higher levels the approach is to ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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

label: equation

text: ﬁrst

text: dress

text: in

text: Eq.

text: and

text: then

text: rotate

text: the

text: drive

text: term

text: sys

text: into

text: this

text: frame.

text: The

text: system

text: is

text: then

text: moved

text: into

text: the

text: frame

text: rotating

text: at

text: the

text: target

text: qubit

text: frequency

text: on

text: both

text: qubits

text: and

text: √0

text: an

text: RWA

text: is

text: performed.

text: In

text: this

text: rotating

text: frame

text: the

text: con-

text: 2Ω

text: √0

text: trol

text: and

text: states

text: deﬁne

text: two

text: subspaces

text: that

text: are

text: far

text: 2Ω

text: detuned

text: by

text: and

text: an

text: eﬀective

text: block-diagonal

text: Hamil-

text: tonian

text: is

text: obtained

text: via

text: the

text: perturbative

text: analysis

text: of

text: Sec.

text: Unlike

text: the

text: qubit

text: case,

text: exact

text: analytical

text: expressions

text: are

text: not

text: straightforward

text: to

text: obtain

text: and

text: so

text: realistic

text: parameters

text: √0

text: (∆−−√∆22Ω+Ω2)

text: −√√02Ω

text: aetreerusswede

text: exact

text: block-diagonalization

text: agree

text: up

text: to

text: medium

text: power

text: drives

text: of

text: Ω

text: MHz

text: with

text: the

text: exact

text: method

text: holding

text: for

text: much

text: larger

text: values

text: of

text: Ω.

text: To

text: start

text: we

text: assume

text: that

text: −J

text: and

text: obtain

text: an

text: Finally

text: computing

text: the

text: block-diagonal

text: of

text: HR

text: and

text: mov-

text: ω˜1

text: ω˜2

text: ing

text: back

text: to

text: the

text: physical

text: frame

text: consisting

text: of

text: the

text: trans-

text: eﬀective

text: diagonal

text: Hamiltonian

text: for

text: Letting

text: U

text: be

text: sys

text: mons

text: rotating

text: at

text: their

text: respective

text: frequencies

text: gives

text: the

text: the

text: diagonalizing

text: (dressing)

text: unitary

text: the

text: eﬀective

text: diagonal

text: block-diagonal

text: Hamiltonian

text: HCR

text: with

text: blocks

text: given

text: Hamiltonian

text: is

text: given

text: by

text: by

text: ∆2

text: Ω2

text: JΩ

text: U

text: JΩ

text: where

text: to

text: second

text: order

text: in

text: the

text: two-qubit

text: subspace

text: ∆2

text: Ω2

text: √√JΩ

text: ω1

text: ω2

text: ξ

text: JΩ

text: ∆2

text: Ω2

text: with

text: The

text: by

text: ω

text: ω˜

text: ξ,

text: ZX

text: JΩ

text: tr

text: HCR

text: ∆2

text: Ω2

text: ω2

text: ω˜2

text: ξ,

text: δ

text: ξ

text: −2

text: where

text: by

text: virtue

text: of

text: the

text: system

text: Hamiltonian

text: deﬁnition,

text: the

text: two-qubit

text: Pauli

text: operators

text: are

text: scaled

text: by

text: (in

text: an

text: n-

text: qubit

text: system

text: they

text: are

text: scaled

text: by

text: 1−

text: The

text: Stark-shift

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text: presence

text: of

text: higher

text: levels

text: has

text: produced

text: an

text: eﬀec-

text: term

text: on

text: is

text: given

text: by

text: tive

text: ZZ

text: interaction

text: in

text: the

text: two-qubit

text: subspace.

text: The

text: drive

text: term

text: of

text: Eq.

text: is

text: rotated

text: into

text: this

text: frame

text: by

text: applying

text: the

text: diagonalizing

text: unitary

text: U,

text: ZI

text: tr

text: H

text: ∆2

text: CR

text: H˜

text: Ω

text: (t)

text: cos(ω

text: t)

text: Ω

text: (t)

text: sin(ω

text: t)

text: B˜

text: and

text: so

text: d

text: Xj

text: dj

text: Yj

text: dj

text: j

text: HCR

text: ∆2

text: Ω2

text: √∆J2

text: Ω+

text: Ω2

text: where

text: B˜j

text: U†(b†j

text: bj)U

text: for

text: j

text: We

text: set

text: ω

text: ω

text: and

text: the

text: Hamiltonian

text: in

text: the

text: dressed

text: frame

text: is

text: d

text: given

text: by

text: IV.

text: EFFECTIVE

text: CR

text: HAMILTONIAN

text: FOR

text: A

text: HIGHER-LEVEL

text: MODEL

text: H(t)

text: H˜

text: H˜

text: (t),

text: sys

text: d

text: For

text: a

text: model

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text: the

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text: 10 {| (cid:105)} In the experiment of Ref. [6] an IY term was found to q . H is uniquely deﬁned by a unitary matrix T a eﬀ | (cid:105) be present in contrast to what is predicted from our anal- that maps the eigenvectors of H, s , to the eigenvectors | (cid:105) a ysis with a single CR drive on the control transmon. We r of H with the eigenvalues being preserved since T a eﬀ propose this discrepancy is a result of classical cross-talk is unitary. From the desired form of H having support K eﬀ | (cid:105) between the two transmons and generalized the model to only on the , the sole restriction on the r is that the a K a include this eﬀect via an additional phase-shifted drive ﬁrst dK vectors have support only on , the next dK 1 K 1 2 term on the target transmon. For a set of realistic model have support only on , and so on. 2 parameters we found good agreement between the the- Let us now discuss how to actually compute T. The {| (cid:105)} oretical results here and those of Ref. [6] which implies ﬁrst step is to map the eigenvalues of H onto the q j classical cross-talk may be a signiﬁcant issue in real sys- basis via the eigenvector matrix X of H so that all of the (cid:80) tems. Understanding the cross-talk channels leading to freedom in computing T comes from choosing a block- {| (cid:105)} drive terms on the target transmon is an important area diagonal (with respect to q ) unitary matrix F. Since | (cid:105)(cid:104) | j | (cid:105) of further research. H = E s s , the columns of X are equal to s a a a a {| (cid:105)} j Various interesting questions remain as directions for when written with respec(cid:88)t to the basis qk , future research. First, it is useful to understand whether the perturbative construction converges to that of the | (cid:105)(cid:104) | principle of least action. For the standard two-block X = s q , (A1) j j Schrieﬀer-Wolﬀ transformation the perturbative con- j struction does converge to exact unitary rotation and ideally this property holds for the multi-block case as and so   (cid:32) (cid:33) (cid:32) (cid:33) well. It will also be interesting to apply these methods (cid:88) (cid:88) (cid:88) to larger multi-qubit systems, especially in the context of   ﬁnding points to avoid in frequency space when dealing † | (cid:105)(cid:104) | | (cid:105)(cid:104) | | (cid:105)(cid:104) | with ﬁxed-frequency transmons in a circuit-QED archi- X HX = (cid:88) q s E s s s q j j a a a k k tecture. The results from a multi-qubit analysis will have j a k an impact on future design considerations in supercon- | (cid:105)(cid:104) | = E q q . ducting circuit systems. a a a VII. ACKNOWLEDGMENTS a The unitary matrix F now rotates into the desired The authors thank Andrew Cross and Sarah Sheldon {| (cid:105)} | (cid:105) eigenbasis r and since the r only have support on for helpful discussions and comments. This work was jK j | (cid:105) the subspaces , F represented in q is a unitary block- supported by the Army Research Oﬃce under contract a j diagonal matrix. The total block-diagonalizing unitary T W911NF-14-1-0124. can be written as the composition of F with X where X Appendix A: Eﬀective Hamiltonians is given in Eq. A1 and (cid:88) 1. Eﬀective Hamiltonian from principle of least action | (cid:105)(cid:104) | F = r q . (A2) j j j Suppose one is given a Hamiltonian H on the Hilbert H | (cid:105) space with eigenvalue(cid:88)s E and eigenvectors s ; a a It is clear the freedom in choosing H comes entirely eﬀ from choosing F. Ideally, one would like to obtain a | (cid:105)(cid:104) | unique H given H. The approach given in Ref. [14] is H = E s s . eﬀ a a a to solve the following optimization problem: a A Hermitian matrix H is said to be an eﬀective (cid:107) − I(cid:107) eﬀ argmin ( T ) , (A3) Hamiltonian for H with respect to the orthogonal sub- F 2 {K } ∪ K H spaces ( = ) if the following are satisﬁed, a a a 1. H has the same energy spectrum as H, eﬀ which means to ﬁnd the unitary matrix F that mini- K I 2. H only has support on the . mizes the 2-norm (Euclidean) distance between T and . eﬀ a K The unique solution of this problem is given by Suppose each subspace has dimension dK and let aK a (cid:113) PK be the projector onto . We set an orthonormal a K {|aK (cid:105)} basis for each a, denoted qb a , b = 1, ..., dKa, to be F = XBD , the standard basis for working in coordinates. Note that † K | K (cid:105) X X for each any linear combination of the q a is still BD BD a K b supported only on . The full orthonormal basis for H a comprised of the union of these bases will be denoted where X is the projection of X onto the subspaces BD ##LTLine## ##LTLine##

text: In

text: the

text: experiment

text: of

text: Ref.

text: an

text: IY

text: term

text: was

text: found

text: to

text: q

text: H

text: is

text: uniquely

text: deﬁned

text: by

text: a

text: unitary

text: matrix

text: T

text: a

text: eﬀ

text: be

text: present

text: in

text: contrast

text: to

text: what

text: is

text: predicted

text: from

text: our

text: anal-

text: that

text: maps

text: the

text: eigenvectors

text: of

text: H,

text: s

text: to

text: the

text: eigenvectors

text: a

text: ysis

text: with

text: a

text: single

text: CR

text: drive

text: on

text: the

text: control

text: transmon.

text: We

text: r

text: of

text: H

text: with

text: the

text: eigenvalues

text: being

text: preserved

text: since

text: T

text: a

text: eﬀ

text: propose

text: this

text: discrepancy

text: is

text: a

text: result

text: of

text: classical

text: cross-talk

text: is

text: unitary.

text: From

text: the

text: desired

text: form

text: of

text: H

text: having

text: support

text: K

text: eﬀ

text: between

text: the

text: two

text: transmons

text: and

text: generalized

text: the

text: model

text: to

text: only

text: on

text: the

text: the

text: sole

text: restriction

text: on

text: the

text: r

text: is

text: that

text: the

text: a

text: K

text: a

text: include

text: this

text: eﬀect

text: via

text: an

text: additional

text: phase-shifted

text: drive

text: ﬁrst

text: dK

text: vectors

text: have

text: support

text: only

text: on

text: the

text: next

text: dK

text: K

text: term

text: on

text: the

text: target

text: transmon.

text: For

text: a

text: set

text: of

text: realistic

text: model

text: have

text: support

text: only

text: on

text: and

text: so

text: on.

text: parameters

text: we

text: found

text: good

text: agreement

text: between

text: the

text: the-

text: Let

text: us

text: now

text: discuss

text: how

text: to

text: actually

text: compute

text: T.

text: The

text: oretical

text: results

text: here

text: and

text: those

text: of

text: Ref.

text: which

text: implies

text: ﬁrst

text: step

text: is

text: to

text: map

text: the

text: eigenvalues

text: of

text: H

text: onto

text: the

text: q

text: j

text: classical

text: cross-talk

text: may

text: be

text: a

text: signiﬁcant

text: issue

text: in

text: real

text: sys-

text: basis

text: via

text: the

text: eigenvector

text: matrix

text: X

text: of

text: H

text: so

text: that

text: all

text: of

text: the

text: tems.

text: Understanding

text: the

text: cross-talk

text: channels

text: leading

text: to

text: freedom

text: in

text: computing

text: T

text: comes

text: from

text: choosing

text: a

text: block-

text: drive

text: terms

text: on

text: the

text: target

text: transmon

text: is

text: an

text: important

text: area

text: diagonal

text: (with

text: respect

text: to

text: q

text: unitary

text: matrix

text: F.

text: Since

text: j

text: of

text: further

text: research.

text: H

text: E

text: s

text: s

text: the

text: columns

text: of

text: X

text: are

text: equal

text: to

text: s

text: a

text: a

text: a

text: a

text: j

text: Various

text: interesting

text: questions

text: remain

text: as

text: directions

text: for

text: when

text: written

text: with

text: to

text: the

text: basis

text: qk

text: future

text: research.

text: First,

text: it

text: is

text: useful

text: to

text: understand

text: whether

text: the

text: perturbative

text: construction

text: converges

text: to

text: that

text: of

text: the

text: principle

text: of

text: least

text: action.

text: For

text: the

text: standard

text: two-block

text: X

text: s

text: q

text: j

text: j

text: Schrieﬀer-Wolﬀ

text: transformation

text: the

text: perturbative

text: con-

text: j

text: struction

text: does

text: converge

text: to

text: exact

text: unitary

text: rotation

text: and

text: ideally

text: this

text: property

text: holds

text: for

text: the

text: multi-block

text: case

text: as

text: and

text: so

text: well.

text: It

text: will

text: also

text: be

text: interesting

text: to

text: apply

text: these

text: methods

text: to

text: larger

text: multi-qubit

text: systems,

text: especially

text: in

text: the

text: context

text: of

text: ﬁnding

text: points

text: to

text: avoid

text: in

text: frequency

text: space

text: when

text: dealing

text: with

text: ﬁxed-frequency

text: transmons

text: in

text: a

text: circuit-QED

text: archi-

text: X

text: HX

text: q

text: s

text: E

text: s

text: s

text: s

text: q

text: j

text: j

text: a

text: a

text: a

text: k

text: k

text: tecture.

text: The

text: results

text: from

text: a

text: multi-qubit

text: analysis

text: will

text: have

text: j

text: a

text: k

text: an

text: impact

text: on

text: future

text: design

text: considerations

text: in

text: supercon-

text: E

text: q

text: q

text: ducting

text: circuit

text: systems.

text: a

text: a

text: a

text: VII.

text: ACKNOWLEDGMENTS

text: a

text: The

text: unitary

text: matrix

text: F

text: now

text: rotates

text: into

text: the

text: desired

text: The

text: authors

text: thank

text: Andrew

text: Cross

text: and

text: Sarah

text: Sheldon

text: eigenbasis

text: r

text: and

text: since

text: the

text: r

text: only

text: have

text: support

text: on

text: for

text: helpful

text: discussions

text: and

text: comments.

text: This

text: work

text: was

text: jK

text: j

text: the

text: subspaces

text: F

text: represented

text: in

text: q

text: is

text: a

text: unitary

text: block-

text: supported

text: by

text: the

text: Army

text: Research

text: Oﬃce

text: under

text: contract

text: a

text: j

text: diagonal

text: matrix.

text: The

text: total

text: block-diagonalizing

text: unitary

text: T

text: can

text: be

text: written

text: as

text: the

text: composition

text: of

text: F

text: with

text: X

text: where

text: X

text: Appendix

text: A:

text: Eﬀective

text: Hamiltonians

text: is

text: given

text: in

text: Eq.

text: and

text: Eﬀective

text: Hamiltonian

text: from

text: principle

text: of

text: least

text: action

text: F

text: r

text: q

text: j

text: j

text: j

text: Suppose

text: one

text: is

text: given

text: a

text: Hamiltonian

text: H

text: on

text: the

text: Hilbert

text: H

text: space

text: with

text: E

text: and

text: eigenvectors

text: s

text: a

text: a

text: It

text: is

text: clear

text: the

text: freedom

text: in

text: choosing

text: H

text: comes

text: entirely

text: eﬀ

text: from

text: choosing

text: F.

text: Ideally,

text: one

text: would

text: like

text: to

text: obtain

text: a

text: unique

text: H

text: given

text: H.

text: The

text: approach

text: given

text: in

text: Ref.

text: is

text: H

text: E

text: s

text: s

text: eﬀ

text: a

text: a

text: a

text: to

text: solve

text: the

text: following

text: optimization

text: problem:

text: a

text: A

text: Hermitian

text: matrix

text: H

text: is

text: said

text: to

text: be

text: an

text: eﬀective

text: eﬀ

text: argmin

text: T

text: Hamiltonian

text: for

text: H

text: with

text: respect

text: to

text: the

text: orthogonal

text: sub-

text: F

text: {K

text: K

text: H

text: spaces

text: if

text: the

text: following

text: are

text: satisﬁed,

text: a

text: a

text: a

text: H

text: has

text: the

text: same

text: energy

text: spectrum

text: as

text: H,

text: eﬀ

text: which

text: means

text: to

text: ﬁnd

text: the

text: unitary

text: matrix

text: F

text: that

text: mini-

text: K

text: I

text: H

text: only

text: has

text: support

text: on

text: the

text: mizes

text: the

text: (Euclidean)

text: distance

text: between

text: T

text: and

text: eﬀ

text: a

text: K

text: The

text: unique

text: solution

text: of

text: this

text: problem

text: is

text: given

text: by

text: Suppose

text: each

text: subspace

text: has

text: dimension

text: dK

text: and

text: let

text: aK

text: a

text: PK

text: be

text: the

text: projector

text: onto

text: We

text: set

text: an

text: orthonormal

text: a

text: K

text: {|aK

text: basis

text: for

text: each

text: a,

text: denoted

text: qb

text: a

text: b

text: dKa,

text: to

text: be

text: F

text: XBD

text: the

text: standard

text: basis

text: for

text: working

text: in

text: coordinates.

text: Note

text: that

text: K

text: K

text: X

text: X

text: for

text: each

text: any

text: linear

text: combination

text: of

text: the

text: q

text: a

text: is

text: still

text: BD

text: BD

text: a

text: K

text: b

text: supported

text: only

text: on

text: The

text: full

text: orthonormal

text: basis

text: for

text: H

text: a

text: comprised

text: of

text: the

text: union

text: of

text: these

text: bases

text: will

text: be

text: denoted

text: where

text: X

text: is

text: the

text: projection

text: of

text: X

text: onto

text: the

text: subspaces

text: BD

text: ##LTLine##

text: ##LTLine##


text: As discussed in the previous section, we focus on the RH neutrino decaying to leptonic ﬁnal states. In particular, we focus on states with diﬀerent multiplicity. This can be µ obtained assuming a diagonal mixing or a mixing with hierarchies . V V (cid:29) V , V lN µN eN τN Below, we ﬁrst present a brief discussion for the LHC and its future upgrades. A common, simpliﬁed detector layout and the details of the relevant detector parameters are shown in Fig. 4 and Table. 1, respectively [17, 36–38]. The diﬀerent variables we use for the geometry of detectors are as follows: is the transverse distance between the heavy neutrino and d N 0 , µ (5.1) |d | = |xp − yp |/p , 0 y x T where and are the position where the right handed neutrino decayed, and , , x y p p x y are the components of momentum and transverse momentum of the ﬁnal particles , p µ T and L / L are the transverse / longitudinal decay lengths of the RH neutrino, and σt is xy z d the resolution of the detector in transverse distance. Previous searches that analyzed the displaced decays of RH neutrinos are for example given in [14, 17]. In [14], the authors propose to detect the RH neutrino having mass through a prompt lepton and a displaced lepton jet arising from the RH M < M N W neutrino decay whereas in [17], the authors propose to detect the RH neutrino through its displaced decays to at least a di-muon ﬁnal state. Displaced decays of other exotic states have also been searched for at CMS and ATLAS in various physics scenarios, e.g. long-lived neutralinos [39] via one-muon and multi-jet ﬁnal state; Higgs decaying to two long-lived particles [9–11] producing two muons. Below, we describe brieﬂy the proposed triggers and the estimated background. 5.1 LHC and Upgrade We consider the RH neutrino decaying inside the tracker or muon chamber in the CMS detector [18]. We categorize the signal as pp → h → NN → Nµ±jj) (denoted as 1 Channel 1) and pp → h → NN → Nµ+µ−ν ) (denoted as Channel 2), respectively. We 1 µ analyze the sensitivity reach in probing the active-sterile neutrino mixing at the 13 TeV V µN LHC using these channels. For the event simulation, we use 2.3 [40] with the F eynRules model ﬁle [41]. The resulting Universal FeynRules Output (UFO) [42] is fed to the Monte Carlo event generator v2.5.5 [34]. MadGraph5aMC@NLO Triggers and Background We diﬀerentiate between two regions, as shown in Fig. 4 in the detector that can potentially detect displaced vertices [17]. Region 1 is chosen to probe long-lived RH neutrinos that decay within the inner tracker. The inner (grey) area of Fig. 4 represents Region 1 which approximately consists of the inner tracker near the vertex detector such that RH neutrinos decaying within the inner tracker are registered via the tracks of ﬁnal state muons. Region 2 is represented by the outer (orange) area which approximately consists of the electromagnetic (ECAL) and hadronic calorimeter (HCAL), and the inner region of the muon chamber. RH neutrino decays are registered in this region via tracks of the ﬁnal state muons in the muon chamber [17]. – 11 –

text: As

text: discussed

text: in

text: the

text: previous

text: section,

text: we

text: focus

text: on

text: the

text: RH

text: neutrino

text: decaying

text: to

text: leptonic

text: ﬁnal

text: states.

text: In

text: particular,

text: we

text: focus

text: on

text: states

text: with

text: diﬀerent

text: multiplicity.

text: This

text: can

text: be

text: µ

text: obtained

text: assuming

text: a

text: diagonal

text: mixing

text: or

text: a

text: mixing

text: with

text: hierarchies

text: V

text: V

text: V

text: V

text: lN

text: µN

text: eN

text: τN

text: Below,

text: we

text: ﬁrst

text: present

text: a

text: brief

text: discussion

text: for

text: the

text: LHC

text: and

text: its

text: future

text: upgrades.

text: A

text: common,

text: simpliﬁed

text: detector

text: layout

text: and

text: the

text: details

text: of

text: the

text: relevant

text: detector

text: parameters

text: are

text: shown

text: in

text: Fig.

text: and

text: Table.

text: respectively

text: The

text: diﬀerent

text: variables

text: we

text: use

text: for

text: the

text: geometry

text: of

text: detectors

text: are

text: as

text: follows:

text: is

text: the

text: transverse

text: distance

text: between

text: the

text: heavy

text: neutrino

text: and

text: d

text: N

text: µ

text: |d

text: |xp

text: yp

text: |/p

text: y

text: x

text: T

text: where

text: and

text: are

text: the

text: position

text: where

text: the

text: right

text: handed

text: neutrino

text: decayed,

text: and

text: x

text: y

text: p

text: p

text: x

text: y

text: are

text: the

text: components

text: of

text: momentum

text: and

text: transverse

text: momentum

text: of

text: the

text: ﬁnal

text: particles

text: p

text: µ

text: T

text: and

text: L

text: L

text: are

text: the

text: transverse

text: longitudinal

text: decay

text: lengths

text: of

text: the

text: RH

text: neutrino,

text: and

text: σt

text: is

text: xy

text: z

text: d

text: the

text: resolution

text: of

text: the

text: detector

text: in

text: transverse

text: distance.

text: Previous

text: searches

text: that

text: analyzed

text: the

text: displaced

text: decays

text: of

text: RH

text: neutrinos

text: are

text: for

text: example

text: given

text: in

text: In

text: the

text: authors

text: propose

text: to

text: detect

text: the

text: RH

text: neutrino

text: having

text: mass

text: through

text: a

text: prompt

text: lepton

text: and

text: a

text: displaced

text: lepton

text: jet

text: arising

text: from

text: the

text: RH

text: M

text: M

text: N

text: W

text: neutrino

text: decay

text: whereas

text: in

text: the

text: authors

text: propose

text: to

text: detect

text: the

text: RH

text: neutrino

text: through

text: its

text: displaced

text: decays

text: to

text: at

text: least

text: a

text: di-muon

text: ﬁnal

text: state.

text: Displaced

text: decays

text: of

text: other

text: exotic

text: states

text: have

text: also

text: been

text: searched

text: for

text: at

text: CMS

text: and

text: ATLAS

text: in

text: various

text: physics

text: scenarios,

text: e.g.

text: long-lived

text: neutralinos

text: via

text: one-muon

text: and

text: multi-jet

text: ﬁnal

text: state;

text: Higgs

text: decaying

text: to

text: two

text: long-lived

text: particles

text: producing

text: two

text: muons.

text: Below,

text: we

text: describe

text: brieﬂy

text: the

text: proposed

text: triggers

text: and

text: the

text: estimated

text: background.

text: LHC

text: and

text: Upgrade

text: We

text: consider

text: the

text: RH

text: neutrino

text: decaying

text: inside

text: the

text: tracker

text: or

text: muon

text: chamber

text: in

text: the

text: CMS

text: detector

text: We

text: categorize

text: the

text: signal

text: as

text: pp

text: h

text: NN

text: Nµ±jj)

text: (denoted

text: as

text: Channel

text: and

text: pp

text: h

text: NN

text: Nµ+µ−ν

text: (denoted

text: as

text: Channel

text: respectively.

text: We

text: µ

text: analyze

text: the

text: sensitivity

text: reach

text: in

text: probing

text: the

text: active-sterile

text: neutrino

text: mixing

text: at

text: the

text: TeV

text: V

text: µN

text: LHC

text: using

text: these

text: channels.

text: For

text: the

text: event

text: simulation,

text: we

text: use

text: with

text: the

text: F

text: eynRules

text: model

text: ﬁle

text: The

text: resulting

text: Universal

text: FeynRules

text: Output

text: (UFO)

text: is

text: fed

text: to

text: the

text: Monte

text: Carlo

text: event

text: generator

text: Triggers

text: and

text: Background

text: We

text: diﬀerentiate

text: between

text: two

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We scale this rate up for the 100 fb−1 LHC and 3000 fb−1 HL-LHC, giving 15 and 450 potential background events, respectively. However, we still consider leptonic colliders and MATHUSLA to be free from background. Therefore, we consider a parameter point to be excluded on non-observation if χ2 = (Ntot − N )2/N > 3.84 at 95 % C.L. In the case of two separate displaced vertices, since this is B B a highly rare process, we think only the optimistic view is necessary. In Fig. 7 we show the resulting sensitivities at the LHC, HL-LHC (without and with MATHUSLA) in the optimistic view assuming no background (left) and the pessimistic view with a scaling of the background (right). We consider that one of the heavy neutrinos decays to muons in one displaced vertex. To estimate the sensitivity reach, we consider 100 fb−1 luminosity for the LHC, 3000 fb−1 for HL-LHC. Taking the optimistic and rather naive view of negligible background, it is evident that the 13 TeV LHC has a sensitivity reach down to V ≈ 10−7 for RH neutrino masses around M ≈ 55 GeV. The high-luminosity run of µN N LHC can further probe smaller mixing angle, as low as 10−8 for a similar RH neutrino mass value. The sensitivity reach of MATHUSLA is similar to the HL-LHC; despite the much longer decay length being probed, the geometric coverage and hence eﬀective cross section is smaller. We should stress again that in deriving these limit we considered a Higgs mixing , close to the experimental limit. 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text: [20] G. Cottin, J. C. Helo and M. Hirsch, Phys. Rev. D 97, no. 5, 055025 (2018) doi:10.1103/PhysRevD.97.055025 [arXiv:1801.02734 [hep-ph]]. [21] J. C. Helo, M. Hirsch and Z. S. Wang, arXiv:1803.02212 [hep-ph]. [22] S. Mandal, M. Mitra and N. Sinha, Phys. Rev. D 96, no. 3, 035023 (2017) doi:10.1103/PhysRevD.96.035023 [arXiv:1705.01932 [hep-ph]]. [23] M. Nemevsek, F. Nesti and G. Popara, arXiv:1801.05813 [hep-ph]. [24] J. P. Chou, D. Curtin and H. J. Lubatti, Phys. Lett. B 767 (2017) 29 doi:10.1016/j.physletb.2017.01.043 [arXiv:1606.06298 [hep-ph]]. [25] T. Robens and T. Stefaniak, Eur. Phys. J. C 75 (2015) 104 doi:10.1140/epjc/s10052-015-3323-y [arXiv:1501.02234 [hep-ph]]. [26] P. A. R. Ade et al. [Planck Collaboration], Astron. Astrophys. 594 (2016) A13 doi:10.1051/0004-6361/201525830 [arXiv:1502.01589 [astro-ph.CO]]. [27] F. F. Deppisch, P. S. Bhupal Dev and A. Pilaftsis, New J. Phys. 17 (2015) no.7, 075019 doi:10.1088/1367-2630/17/7/075019 [arXiv:1502.06541 [hep-ph]]. [28] P. S. Bhupal Dev and A. Pilaftsis, Phys. Rev. D 87 (2013) no.5, 053007 doi:10.1103/PhysRevD.87.053007 [arXiv:1212.3808 [hep-ph]]. [29] G. M. Pruna, arXiv:1106.4691 [hep-ph]. [30] [CMS Collaboration], CMS-PAS-HIG-13-002. [31] V. Khachatryan et al. [CMS Collaboration], Eur. Phys. J. C 75, no. 5, 212 (2015) doi:10.1140/epjc/s10052-015-3351-7 [arXiv:1412.8662 [hep-ex]]. [32] S. Banerjee, M. Mitra and M. Spannowsky, Phys. Rev. D 92, no. 5, 055013 (2015) doi:10.1103/PhysRevD.92.055013 [arXiv:1506.06415 [hep-ph]]. [33] D. López-Val and T. Robens, Phys. Rev. D 90 (2014) 114018 doi:10.1103/PhysRevD.90.114018 [arXiv:1406.1043 [hep-ph]]. [34] J. Alwall et al., JHEP 1407 (2014) 079 doi:10.1007/JHEP07(2014)079 [arXiv:1405.0301 [hep-ph]]. [35] G. Aad et al. [ATLAS Collaboration], Phys. Rev. Lett. 108 (2012) 251801 doi:10.1103/PhysRevLett.108.251801 [arXiv:1203.1303 [hep-ex]]. [36] T. Behnke et al., arXiv:1306.6329 [physics.ins-det]. [37] H. Aihara et al., arXiv:0911.0006 [physics.ins-det]. [38] CEPC-SPPC Study Group, IHEP-CEPC-DR-2015-01, IHEP-TH-2015-01, IHEP-EP-2015-01. [39] G. Aad et al. [ATLAS Collaboration], Phys. Lett. B 719 (2013) 280 doi:10.1016/j.physletb.2013.01.042 [arXiv:1210.7451 [hep-ex]]. [40] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks, Comput. Phys. Commun. 185 (2014) 2250 doi:10.1016/j.cpc.2014.04.012 [arXiv:1310.1921 [hep-ph]]. [41] L. Basso, A. Belyaev, S. Moretti and C. H. Shepherd-Themistocleous, Phys. Rev. D 80 (2009) 055030 doi:10.1103/PhysRevD.80.055030 [arXiv:0812.4313 [hep-ph]]. [42] C. Degrande, C. Duhr, B. Fuks, D. Grellscheid, O. Mattelaer and T. Reiter, Comput. Phys. Commun. 183 (2012) 1201 doi:10.1016/j.cpc.2012.01.022 [arXiv:1108.2040 [hep-ph]]. – 22 –

label: reference

text: G.

text: Cottin,

text: J.

text: C.

text: Helo

text: and

text: M.

text: Hirsch,

text: Phys.

text: Rev.

text: D

text: no.

text: [hep-ph]].

text: J.

text: C.

text: Helo,

text: M.

text: Hirsch

text: and

text: Z.

text: S.

text: Wang,

text: [hep-ph].

text: S.

text: Mandal,

text: M.

text: Mitra

text: and

text: N.

text: Sinha,

text: Phys.

text: Rev.

text: D

text: no.

text: [hep-ph]].

text: M.

text: Nemevsek,

text: F.

text: Nesti

text: and

text: G.

text: Popara,

text: [hep-ph].

text: J.

text: P.

text: Chou,

text: D.

text: Curtin

text: and

text: H.

text: J.

text: Lubatti,

text: Phys.

text: Lett.

text: B

text: [hep-ph]].

text: T.

text: Robens

text: and

text: T.

text: Stefaniak,

text: Eur.

text: Phys.

text: J.

text: C

text: [hep-ph]].

text: P.

text: A.

text: R.

text: Ade

text: et

text: al.

text: [Planck

text: Collaboration],

text: Astron.

text: Astrophys.

text: [astro-ph.CO]].

text: F.

text: F.

text: Deppisch,

text: P.

text: S.

text: Bhupal

text: Dev

text: and

text: A.

text: Pilaftsis,

text: New

text: J.

text: Phys.

text: [hep-ph]].

text: P.

text: S.

text: Bhupal

text: Dev

text: and

text: A.

text: Pilaftsis,

text: Phys.

text: Rev.

text: D

text: [hep-ph]].

text: G.

text: M.

text: Pruna,

text: [hep-ph].

text: [CMS

text: Collaboration],

text: V.

text: Khachatryan

text: et

text: al.

text: [CMS

text: Collaboration],

text: Eur.

text: Phys.

text: J.

text: C

text: no.

text: [hep-ex]].

text: S.

text: Banerjee,

text: M.

text: Mitra

text: and

text: M.

text: Spannowsky,

text: Phys.

text: Rev.

text: D

text: no.

text: [hep-ph]].

text: D.

text: López-Val

text: and

text: T.

text: Robens,

text: Phys.

text: Rev.

text: D

text: [hep-ph]].

text: J.

text: Alwall

text: et

text: al.,

text: JHEP

text: [hep-ph]].

text: G.

text: Aad

text: et

text: al.

text: [ATLAS

text: Collaboration],

text: Phys.

text: Rev.

text: Lett.

text: [hep-ex]].

text: T.

text: Behnke

text: et

text: al.,

text: [physics.ins-det].

text: H.

text: Aihara

text: et

text: al.,

text: [physics.ins-det].

text: CEPC-SPPC

text: Study

text: Group,

text: G.

text: Aad

text: et

text: al.

text: [ATLAS

text: Collaboration],

text: Phys.

text: Lett.

text: B

text: [hep-ex]].

text: A.

text: Alloul,

text: N.

text: D.

text: Christensen,

text: C.

text: Degrande,

text: C.

text: Duhr

text: and

text: B.

text: Fuks,

text: Comput.

text: Phys.

text: Commun.

text: [hep-ph]].

text: L.

text: Basso,

text: A.

text: Belyaev,

text: S.

text: Moretti

text: and

text: C.

text: H.

text: Shepherd-Themistocleous,

text: Phys.

text: Rev.

text: D

text: [hep-ph]].

text: C.

text: Degrande,

text: C.

text: Duhr,

text: B.

text: Fuks,

text: D.

text: Grellscheid,

text: O.

text: Mattelaer

text: and

text: T.

text: Reiter,

text: Comput.

text: Phys.

text: Commun.

text: [hep-ph]].


text: In the present work, we consider relatively low mass RH neutrinos, 1 GeV < < M N GeV in order to pair-produce them from SM Higgs decays. The RH neutrino M /2 = 62.5 h1 mass and its mixing with the active neutrinos are tightly constrained assuming successful neutrino mass measurements. In a pure Type-I scenario with gauge symmetry, the B − L light neutrino mass m ∼ m2D ∼ V 2 M with V ∼ m /M . The sub-eV scale light neutrino mass constrainνts frMomR lNandNTritium lbNeta decDay eRxperiments as well as from 0νββ cosmological observations such as Planck [26] ﬁxes the active-sterile mixing, (cid:115) eV m /(0.1 ) V ≈ 10−6 ν . (3.1) lN GeV M /(50 ) N However, note that for models such as an inverse seesaw, the active-sterile mixing can be signiﬁcantly larger, , only limited by direct searches (see e.g. [27] and V ∼ 0.01 lN references therein), while still satisfying the light neutrino mass constraint [28]. The SM singlet Higgs and its mixin(cid:112)g angle with the SM Higgs is constrained by α perturbativity and unitarity considerations [29], setting an upper limit on as a function M (cid:46) h2 of the breaking scale , . In our subsequent analysis, we consider B − L x M 2 2π/3x h2 a mass of GeV, although it does not directly enter into our considerations. M = 450 h2 Additionally, direct searches at the LHC for a BSM Higgs signal further constrains the (cid:46) mixing in the aforementioned mass range [30]. An indirect constraint on the sin α 0.35 (cid:46) Higgs mixing angle sin2 α 0.31 can also be obtained from the measurement of SM Higgs decays into a number of SM ﬁnal states [31, 32]. The bound coming from SM Higgs signal strength measurement is valid for all masses of the BSM Higgs . Precision measurements h 2 (cid:46) of the mass give a competitive bound on the mixing angle for a wide mass range W α 0.3 (cid:38) GeV [33]. In the present work, we consider the value to determine M 300 sin α = 0.3 h2 the maximal sensitivity on the neutrino mixing. Searches at LEP-II [5–8] for a resonance constrain Z(cid:48) mass and gauge coupling, and thus the B − L breaking scale x ≡ MZ(cid:48)(cid:48) ≥ 3.45 TeV. Resonance searches in pp → Z(cid:48) → l+l− bound the Z(cid:48) mass to M (cid:48) (cid:38) 4.5 TeV2g1[4] with a SM-valued gauge coupling. Thus we choose Z M (cid:48) TeV, in agreement with the LEP and LHC bounds, where we consider x = Z(cid:48) = 3.75 2g M (cid:48) = 61 TeV, g(cid:48) = 0.8 for deﬁniteness. Z 1 In summary, we consider the following model parameters: M = 1 − 60 GeV, V = 10−9 − 10−2, N lN M (cid:48) = 6 TeV, g(cid:48) = 0.8, x = 3.75 TeV, (3.2) Z 1 GeV M = 450 , sin α = 0.3. h2 4 Right-handed Neutrino Production and Decay 4.1 Pair-Production through Higgs Resonance – 6 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: In

text: the

text: present

text: work,

text: we

text: consider

text: relatively

text: low

text: mass

text: RH

text: neutrinos,

text: GeV

text: M

text: N

text: GeV

text: in

text: order

text: to

text: pair-produce

text: them

text: from

text: SM

text: Higgs

text: decays.

text: The

text: RH

text: neutrino

text: M

text: mass

text: and

text: its

text: mixing

text: with

text: the

text: active

text: neutrinos

text: are

text: tightly

text: constrained

text: assuming

text: successful

text: neutrino

text: mass

text: measurements.

text: In

text: a

text: pure

text: Type-I

text: scenario

text: with

text: gauge

text: symmetry,

text: the

text: B

text: L

text: light

text: neutrino

text: mass

text: m

text: V

text: M

text: with

text: V

text: m

text: /M

text: The

text: sub-eV

text: scale

text: light

text: neutrino

text: mass

text: constrainνts

text: frMomR

text: lNandNTritium

text: lbNeta

text: decDay

text: eRxperiments

text: as

text: well

text: as

text: from

text: 0νββ

text: cosmological

text: observations

text: such

text: as

text: Planck

text: ﬁxes

text: the

text: active-sterile

text: mixing,

text: eV

text: m

text: V

text: ν

text: lN

text: GeV

text: M

text: N

text: However,

text: note

text: that

text: for

text: models

text: such

text: as

text: an

text: inverse

text: seesaw,

text: the

text: active-sterile

text: mixing

text: can

text: be

text: signiﬁcantly

text: larger,

text: only

text: limited

text: by

text: direct

text: searches

text: (see

text: e.g.

text: and

text: V

text: lN

text: references

text: therein),

text: while

text: still

text: satisfying

text: the

text: light

text: neutrino

text: mass

text: constraint

text: The

text: SM

text: singlet

text: Higgs

text: and

text: its

text: angle

text: with

text: the

text: SM

text: Higgs

text: is

text: constrained

text: by

text: α

text: perturbativity

text: and

text: unitarity

text: considerations

text: setting

text: an

text: upper

text: limit

text: on

text: as

text: a

text: function

text: M

text: of

text: the

text: breaking

text: scale

text: In

text: our

text: subsequent

text: analysis,

text: we

text: consider

text: B

text: L

text: x

text: M

text: a

text: mass

text: of

text: GeV,

text: although

text: it

text: does

text: not

text: directly

text: enter

text: into

text: our

text: considerations.

text: M

text: Additionally,

text: direct

text: searches

text: at

text: the

text: LHC

text: for

text: a

text: BSM

text: Higgs

text: signal

text: further

text: constrains

text: the

text: mixing

text: in

text: the

text: aforementioned

text: mass

text: range

text: An

text: indirect

text: constraint

text: on

text: the

text: sin

text: α

text: Higgs

text: mixing

text: angle

text: α

text: can

text: also

text: be

text: obtained

text: from

text: the

text: measurement

text: of

text: SM

text: Higgs

text: decays

text: into

text: a

text: number

text: of

text: SM

text: ﬁnal

text: states

text: The

text: bound

text: coming

text: from

text: SM

text: Higgs

text: signal

text: strength

text: measurement

text: is

text: valid

text: for

text: all

text: masses

text: of

text: the

text: BSM

text: Higgs

text: Precision

text: measurements

text: h

text: of

text: the

text: mass

text: give

text: a

text: competitive

text: bound

text: on

text: the

text: mixing

text: angle

text: for

text: a

text: wide

text: mass

text: range

text: W

text: α

text: GeV

text: In

text: the

text: present

text: work,

text: we

text: consider

text: the

text: value

text: to

text: determine

text: M

text: sin

text: α

text: the

text: maximal

text: sensitivity

text: on

text: the

text: neutrino

text: mixing.

text: Searches

text: at

text: LEP-II

text: for

text: a

text: resonance

text: constrain

text: mass

text: and

text: gauge

text: coupling,

text: and

text: thus

text: the

text: B

text: L

text: breaking

text: scale

text: x

text: TeV.

text: Resonance

text: searches

text: in

text: pp

text: l+l−

text: bound

text: the

text: mass

text: to

text: M

text: with

text: a

text: SM-valued

text: gauge

text: coupling.

text: Thus

text: we

text: choose

text: Z

text: M

text: TeV,

text: in

text: agreement

text: with

text: the

text: LEP

text: and

text: LHC

text: bounds,

text: where

text: we

text: consider

text: x

text: M

text: TeV,

text: for

text: deﬁniteness.

text: Z

text: In

text: summary,

text: we

text: consider

text: the

text: following

text: model

text: parameters:

text: M

text: GeV,

text: V

text: N

text: lN

text: M

text: TeV,

text: x

text: TeV,

text: Z

text: GeV

text: M

text: sin

text: α

text: Right-handed

text: Neutrino

text: Production

text: and

text: Decay

text: Pair-Production

text: through

text: Higgs

text: Resonance

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ��� ��� � �� ��� ��� ��� ��� ��� �� ��� �� ��� ��� �� �� ��� ��� ��� ��� ��� �� �� �� ���� �� [ ] [ ] � �� ��� ��� � �� �� �� �� �� �� � �� �� �� �� �� �� � ��� � ��� � � Figure 1: Left: Cross section as a function of the heavy neutrino mass σ(pp → h → NN) 1 √ and the Higgs mixing at the LHC with a center-of-mass energy of TeV. M sin α s = 13 N Right: Cross section σ(e+e− → Z → Zh → Z +NN) as a function of the same parameters 1 √ at an electron-positron collider with a center-of-mass energy of GeV. s = 250 We ﬁrst consider the production of RH neutrinos through an -channel SM-like Higgs, s . We will assume that only one species of RH neutrinos is suﬃciently light pp → h → NN 1 and that it dominantly couples to one lepton generation only. The leading-order coupling of the SM-like Higgs h with the two RH neutrino states is κ = yM sin α cos2 θ (cid:39) 1 h1NN ν yM sin α = MN sin α, where yM is the associated Yukawa coupling that can be written in x terms of the RH neutrino mass and breaking vev . We have considered a M B − L x N small active-sterile neutrino mixing, that leads to . In such a parametrization, cos θ ∼ 1 ν the production cross-section is inversely proportional to x2 and proportional to M2 , with N the latter being bounded by the kinematic threshold , and independent of the 2M ≤ M N h1 active-sterile mixing. Note that the SM-like Higgs couples with the SM fermions and gauge bosons as h 1 , while the BSM Higgs interacts with the same ﬁnal states as . We do not cos α h sin α 2 consider the decays h → Z(cid:48)Z(cid:48) and h → Z(cid:48)Z(cid:48)∗, as we take the Z(cid:48) to be much heavier than 1 1 the SM gauge bosons. The branching ratio of SM-like Higgs can then be approximated as [17], Γ(h → NN) Br 1 (4.1) (h → NN) = , 1 Γ(h)SM cos α2 + Γ(h1 → NN) where Γ(h)SM ≈ 4.2 × 10−3 GeV is the total decay width of the SM Higgs and – 7 – α ��� α ��� ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: Figure

text: Left:

text: Cross

text: section

text: as

text: a

text: function

text: of

text: the

text: heavy

text: neutrino

text: mass

text: σ(pp

text: h

text: NN)

text: and

text: the

text: Higgs

text: mixing

text: at

text: the

text: LHC

text: with

text: a

text: center-of-mass

text: energy

text: of

text: TeV.

text: M

text: sin

text: α

text: s

text: N

text: Right:

text: Cross

text: section

text: σ(e+e−

text: Z

text: Zh

text: Z

text: +NN)

text: as

text: a

text: function

text: of

text: the

text: same

text: parameters

text: at

text: an

text: electron-positron

text: collider

text: with

text: a

text: center-of-mass

text: energy

text: of

text: GeV.

text: s

text: We

text: ﬁrst

text: consider

text: the

text: production

text: of

text: RH

text: neutrinos

text: through

text: an

text: -channel

text: SM-like

text: Higgs,

text: s

text: We

text: will

text: assume

text: that

text: only

text: one

text: species

text: of

text: RH

text: neutrinos

text: is

text: suﬃciently

text: light

text: pp

text: h

text: NN

text: and

text: that

text: it

text: dominantly

text: couples

text: to

text: one

text: lepton

text: generation

text: only.

text: The

text: leading-order

text: coupling

text: of

text: the

text: SM-like

text: Higgs

text: h

text: with

text: the

text: two

text: RH

text: neutrino

text: states

text: is

text: κ

text: yM

text: sin

text: α

text: θ

text: ν

text: yM

text: sin

text: α

text: MN

text: sin

text: α,

text: where

text: yM

text: is

text: the

text: associated

text: Yukawa

text: coupling

text: that

text: can

text: be

text: written

text: in

text: x

text: terms

text: of

text: the

text: RH

text: neutrino

text: mass

text: and

text: breaking

text: vev

text: We

text: have

text: considered

text: a

text: M

text: B

text: L

text: x

text: N

text: small

text: active-sterile

text: neutrino

text: mixing,

text: that

text: leads

text: to

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text: such

text: a

text: parametrization,

text: cos

text: θ

text: ν

text: the

text: production

text: cross-section

text: is

text: inversely

text: proportional

text: to

text: and

text: proportional

text: to

text: with

text: N

text: the

text: latter

text: being

text: bounded

text: by

text: the

text: kinematic

text: threshold

text: and

text: independent

text: of

text: the

text: M

text: N

text: active-sterile

text: mixing.

text: Note

text: that

text: the

text: SM-like

text: Higgs

text: couples

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text: the

text: SM

text: fermions

text: and

text: gauge

text: bosons

text: as

text: h

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text: interacts

text: with

text: the

text: same

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text: states

text: as

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text: do

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text: cos

text: α

text: h

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text: α

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text: the

text: decays

text: h

text: and

text: h

text: as

text: we

text: take

text: the

text: to

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text: much

text: heavier

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text: the

text: SM

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text: The

text: branching

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text: as

text: Γ(h

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text: Br

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text: where

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text: (cid:32) (cid:33) 3/2 2 M2 m 4M2 Γ(h → NN) = sin2 α N h1 1 − N (4.2) 1 3 x2 8π m h2 1 is the partial width of the SM-like Higgs to the new exotic channel. Similarly, the 13 TeV LHC cross section for the production of the SM-like is h 1 σ(pp → h1) = σ(pp → h)SM cos2 α ≈ cos2 α (44 ± 4) pb. (4.3) In Fig. 1 (left) we show the total production cross section as a function of the heavy N neutrino mass and the Higgs mixing angle. As noted before, as long as the active-sterile neutrino mixing is suﬃciently small, the production cross section is not sensitive to it. For a particular RH neutrino mass , the cross section rises with increasing value of the mixing M N angle . On the other hand, for a higher value of within the kinematic threshold, a α M N cross section as large as fb can be obtained for , in accordance with the 50 sin α ∼ 0.3 LHC bound [33], where the limit was derived for the Higgs singlet extension, however also applicable for the model. The cross section rapidly drops for small . B − L M N √ For the electron-positron collider case, we consider a center-of-mass energy s = 250 GeV. The dominant Higgs production process is Higgs-Strahlung, e+e− → Z∗ → Zh . √ 1 In the SM, the cross section is fb for GeV, reduced by the Higgs mixing σ ∼ 240 s = 250 angle, ∝ cos2 α, in our scenario. Hence, as shown in Fig. 1 right, the total N production cross section is about 200 times smaller than at the LHC. 4.2 Right-handed Neutrino Decay The heavy neutrino, once produced, will decay to diﬀerent SM states. For our mass (cid:46) region of interest, GeV, and assuming there are no lighter exotic particles, M 62.5 N the RH neutrino decays via three body processes such as N → µ±qq¯ and N → µ+µ−ν µ for ﬁnal states including muons. In this case and with our assumption that the heavy neutrino only mixes with one SM lepton generation, the branching ratios do not depend on the active-sterile mixing. In Fig. 2, we show the branching ratios of the diﬀerent decay modes as a function of the RH neutrino mass. In the region of interest, the heavy neutrino can decay to ﬁnal states with one, two or three leptons. For relatively higher masses (cid:38) (cid:46) GeV, predominantly decays to , while for lower masses GeV, the M 1 N µqq¯ M 0.1 N N branching ratio of becomes dominant. The given branching ratios, calculated N → ννν using v2.5.5 [34] and denoted by parton states correctly take into MadGraph5aMC@NLO account decays to mesons for small RH neutrino masses. Approximately, the resulting decay (cid:46) length for can be expressed as M m N Z (cid:18) (cid:19) (cid:18) (cid:19) 10−6 2 100 GeV 5 m (4.4) L ≈ 0.025 · · , N V M µN N – 8 – ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: m

text: Γ(h

text: NN)

text: α

text: N

text: N

text: 8π

text: m

text: is

text: the

text: partial

text: width

text: of

text: the

text: SM-like

text: Higgs

text: to

text: the

text: new

text: exotic

text: channel.

text: Similarly,

text: the

text: TeV

text: LHC

text: cross

text: section

text: for

text: the

text: production

text: of

text: the

text: SM-like

text: is

text: h

text: σ(pp

text: σ(pp

text: h)SM

text: α

text: α

text: pb.

text: In

text: Fig.

text: (left)

text: we

text: show

text: the

text: total

text: production

text: cross

text: section

text: as

text: a

text: function

text: of

text: the

text: heavy

text: N

text: neutrino

text: mass

text: and

text: the

text: Higgs

text: mixing

text: angle.

text: As

text: noted

text: before,

text: as

text: long

text: as

text: the

text: active-sterile

text: neutrino

text: mixing

text: is

text: suﬃciently

text: small,

text: the

text: production

text: cross

text: section

text: is

text: not

text: sensitive

text: to

text: it.

text: For

text: a

text: particular

text: RH

text: neutrino

text: mass

text: the

text: cross

text: section

text: rises

text: with

text: increasing

text: value

text: of

text: the

text: mixing

text: M

text: N

text: angle

text: On

text: the

text: other

text: hand,

text: for

text: a

text: higher

text: value

text: of

text: within

text: the

text: kinematic

text: threshold,

text: a

text: α

text: M

text: N

text: cross

text: section

text: as

text: large

text: as

text: fb

text: can

text: be

text: obtained

text: for

text: in

text: accordance

text: with

text: the

text: sin

text: α

text: LHC

text: bound

text: where

text: the

text: limit

text: was

text: derived

text: for

text: the

text: Higgs

text: singlet

text: extension,

text: however

text: also

text: applicable

text: for

text: the

text: model.

text: The

text: cross

text: section

text: rapidly

text: drops

text: for

text: small

text: B

text: L

text: M

text: N

text: For

text: the

text: electron-positron

text: collider

text: case,

text: we

text: consider

text: a

text: center-of-mass

text: energy

text: s

text: GeV.

text: The

text: dominant

text: Higgs

text: production

text: process

text: is

text: Higgs-Strahlung,

text: e+e−

text: Z∗

text: Zh

text: In

text: the

text: SM,

text: the

text: cross

text: section

text: is

text: fb

text: for

text: GeV,

text: reduced

text: by

text: the

text: Higgs

text: mixing

text: σ

text: s

text: angle,

text: α,

text: in

text: our

text: scenario.

text: Hence,

text: as

text: shown

text: in

text: Fig.

text: right,

text: the

text: total

text: N

text: production

text: cross

text: section

text: is

text: about

text: times

text: smaller

text: than

text: at

text: the

text: LHC.

text: Right-handed

text: Neutrino

text: Decay

text: The

text: heavy

text: neutrino,

text: once

text: produced,

text: will

text: decay

text: to

text: diﬀerent

text: SM

text: states.

text: For

text: our

text: mass

text: region

text: of

text: interest,

text: GeV,

text: and

text: assuming

text: there

text: are

text: no

text: lighter

text: exotic

text: particles,

text: M

text: N

text: the

text: RH

text: neutrino

text: decays

text: via

text: three

text: body

text: processes

text: such

text: as

text: N

text: µ±qq¯

text: and

text: N

text: µ+µ−ν

text: µ

text: for

text: ﬁnal

text: states

text: including

text: muons.

text: In

text: this

text: case

text: and

text: with

text: our

text: assumption

text: that

text: the

text: heavy

text: neutrino

text: only

text: mixes

text: with

text: one

text: SM

text: lepton

text: generation,

text: the

text: branching

text: ratios

text: do

text: not

text: depend

text: on

text: the

text: active-sterile

text: mixing.

text: In

text: Fig.

text: we

text: show

text: the

text: branching

text: ratios

text: of

text: the

text: diﬀerent

text: decay

text: modes

text: as

text: a

text: function

text: of

text: the

text: RH

text: neutrino

text: mass.

text: In

text: the

text: region

text: of

text: interest,

text: the

text: heavy

text: neutrino

text: can

text: decay

text: to

text: ﬁnal

text: states

text: with

text: one,

text: two

text: or

text: three

text: leptons.

text: For

text: relatively

text: higher

text: masses

text: GeV,

text: predominantly

text: decays

text: to

text: while

text: for

text: lower

text: masses

text: GeV,

text: the

text: M

text: N

text: µqq¯

text: M

text: N

text: N

text: branching

text: ratio

text: of

text: becomes

text: dominant.

text: The

text: given

text: branching

text: ratios,

text: calculated

text: N

text: ννν

text: using

text: and

text: denoted

text: by

text: parton

text: states

text: correctly

text: take

text: into

text: account

text: decays

text: to

text: mesons

text: for

text: small

text: RH

text: neutrino

text: masses.

text: Approximately,

text: the

text: resulting

text: decay

text: length

text: for

text: can

text: be

text: expressed

text: as

text: M

text: m

text: N

text: Z

text: GeV

text: m

text: L

text: N

text: V

text: M

text: µN

text: N

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: 100 νµνiνi µ±qq¯ ν e+e− νµqq¯ µ l l¯ν or l¯l ν i j i i j i 10−1 ν µ+µ− µ 10−2 ν τ+τ− µ 10−2 10−1 100 101 M [GeV] N Figure 2: Decay branching ratios Br of the RH neutrino into the denoted (N → X) N channels as a function of . Here, denotes lepton ﬂavour with X M i, j = e, µ, τ i (cid:54)= j N and the heavy neutrino is assumed to mix only with the light muon neutrino, , V (cid:54)= 0 µN . In this case, the branching ratios are independent of . V = 0 V e,τN µN where corrections due to the boost of parent will be discussed below. For a RH h 1 neutrino mass M ≈ 30 GeV and a mixing V ≈ 10−4, the decay length is of the order N µN L ≈ 1 mm. For smaller mixing, such as in the naive seesaw estimate V ≈ 10−6, the N µN decay length can be of the order of meters, potentially detectable through a displaced vertex at high energy colliders. For even smaller mixing and thus longer decay lengths, the decay products will escape the detector volume resulting in a missing energy signature. 4.3 Estimate of Displaced Vertex Event Rate Before embarking on a detailed event simulation, we ﬁrst estimate the rate of displaced vertex events from the production and decay of heavy neutrinos. As discussed in the previous subsection, we are interested in the decay length of the RH neutrino varying in the millimeter to several meters range, corresponding to active-sterile neutrino mixing in the relevant range for light neutrino mass generation. For long decay lengths, the heavy neutrino decay products will predominantly not register as prompt objects. Leptonic ﬁnal states, such as a muon, or hadronic ﬁnal states originating from such a RH neutrino decay can still be detected with displaced vertex searches at LHC [14, 35]. In the following, we estimate the number of such displaced events and thus the sensitivity of the 13 TeV LHC in probing the active-sterile mixing . V µN We take into account the probability of the heavy neutrino decaying inside the detector and estimate the event rate corresponding to the observe displaced vertex events, – 9 – )X → N(rB ##LTFigure## ##LTFigure##

text: νµνiνi

text: µ±qq¯

text: ν

text: e+e−

text: νµqq¯

text: µ

text: l

text: l¯ν

text: or

text: l¯l

text: ν

text: i

text: j

text: i

text: i

text: j

text: i

text: ν

text: µ+µ−

text: µ

text: ν

text: τ+τ−

text: µ

text: M

text: [GeV]

text: N

text: Figure

text: Decay

label: caption

text: branching

text: ratios

text: Br

text: of

text: the

text: RH

text: neutrino

text: into

text: the

text: denoted

text: (N

text: X)

text: N

text: channels

text: as

text: a

text: function

text: of

text: Here,

text: denotes

text: lepton

text: ﬂavour

text: with

text: X

text: M

text: i,

text: j

text: e,

text: µ,

text: τ

text: i

text: j

text: N

text: and

text: the

text: heavy

text: neutrino

text: is

text: assumed

text: to

text: mix

text: only

text: with

text: the

text: light

text: muon

text: neutrino,

text: V

text: µN

text: In

text: this

text: case,

text: the

text: branching

text: ratios

text: are

text: independent

text: of

text: V

text: V

text: e,τN

text: µN

text: where

text: corrections

text: due

text: to

text: the

text: boost

text: of

text: parent

text: will

text: be

text: discussed

text: below.

text: For

text: a

text: RH

text: h

text: neutrino

text: mass

text: M

text: GeV

text: and

text: a

text: mixing

text: V

text: the

text: decay

text: length

text: is

text: of

text: the

text: order

text: N

text: µN

text: L

text: mm.

text: For

text: smaller

text: mixing,

text: such

text: as

text: in

text: the

text: naive

text: seesaw

text: estimate

text: V

text: the

text: N

text: µN

text: decay

text: length

text: can

text: be

text: of

text: the

text: order

text: of

text: meters,

text: potentially

text: detectable

text: through

text: a

text: displaced

text: vertex

text: at

text: high

text: energy

text: colliders.

text: For

text: even

text: smaller

text: mixing

text: and

text: thus

text: longer

text: decay

text: lengths,

text: the

text: decay

text: products

text: will

text: escape

text: the

text: detector

text: volume

text: resulting

text: in

text: a

text: missing

text: energy

text: signature.

text: Estimate

text: of

text: Displaced

text: Vertex

text: Event

text: Rate

text: Before

text: embarking

text: on

text: a

text: detailed

text: event

text: simulation,

text: we

text: ﬁrst

text: estimate

text: the

text: rate

text: of

text: displaced

text: vertex

text: events

text: from

text: the

text: production

text: and

text: decay

text: of

text: heavy

text: neutrinos.

text: As

text: discussed

text: in

text: the

text: previous

text: subsection,

text: we

text: are

text: interested

text: in

text: the

text: decay

text: length

text: of

text: the

text: RH

text: neutrino

text: varying

text: in

text: the

text: millimeter

text: to

text: several

text: meters

text: range,

text: corresponding

text: to

text: active-sterile

text: neutrino

text: mixing

text: in

text: the

text: relevant

text: range

text: for

text: light

text: neutrino

text: mass

text: generation.

text: For

text: long

text: decay

text: lengths,

text: the

text: heavy

text: neutrino

text: decay

text: products

text: will

text: predominantly

text: not

text: register

text: as

text: prompt

text: objects.

text: Leptonic

text: ﬁnal

text: states,

text: such

text: as

text: a

text: muon,

text: or

text: hadronic

text: ﬁnal

text: states

text: originating

text: from

text: such

text: a

text: RH

text: neutrino

text: decay

text: can

text: still

text: be

text: detected

text: with

text: displaced

text: vertex

text: searches

text: at

text: LHC

text: In

text: the

text: following,

text: we

text: estimate

text: the

text: number

text: of

text: such

text: displaced

text: events

text: and

text: thus

text: the

text: sensitivity

text: of

text: the

text: TeV

text: LHC

text: in

text: probing

text: the

text: active-sterile

text: mixing

text: V

text: µN

text: We

text: take

text: into

text: account

text: the

text: probability

text: of

text: the

text: heavy

text: neutrino

text: decaying

text: inside

text: the

text: detector

text: and

text: estimate

text: the

text: event

text: rate

text: corresponding

text: to

text: the

text: observe

text: displaced

text: vertex

text: events,

text: )X

text: N(rB

text: ##LTFigure##

text: ##LTFigure##