text: ∗ Continuous-variable dense coding via a general Gaussian state: Monogamy relation Jaehak Lee, Se-Wan Ji, Jiyong Park, and Hyunchul Nha † ‡ We study a continuous variable (CV) dense-coding protocol, originally proposed to employ a two-mode squeezed state, using a general two-mode Gaussian state as a quantum channel. We par- ticularly obtain conditions to manifest quantum advantage by beating two well-known single-mode schemes, namely, the squeezed-state scheme (best Gaussian scheme) and the number-state scheme (optimal scheme achieving the Holevo bound). We then extend our study to a multipartite Gaussian state and investigate the monogamy of operational entanglement measured by the communication capacity under the dense-coding protocol. We show that this operational entanglement represents a strict monogamy relation, by means of Heisenberg’s uncertainty principle among diﬀerent par- ties, i.e., the quantum advantage for communication can be possible for only one pair of two-mode systems among many parties. I. INTRODUCTION and show a strict monogamy relation in this regime. Ever since quantum dense coding was ﬁrst developed in qubit Quantum correlation, especially entanglement [1], is a systems [4], the dense coding capacity has been derived key resource for quantum information processing, e.g., for arbitrary ﬁnite-dimensional systems [12] with arbi- quantum teleportation [2, 3] and quantum dense coding trary number of parties [13] for discrete-variable systems. [4, 5]. Performance in such protocols, quantiﬁed by the On the other hand, there have been few studies on CV output ﬁdelity of teleportation and the communication dense coding with a general Gaussian state. The CV capacity of dense coding, respectively, can be used as an dense coding protocol was ﬁrst proposed by Braunstein operational measure of entanglement. When entangle- and Kimble [5] in which a pure two-mode squeezed vac- ment is shared by more than two parties, we may look uum (TMSV) state is employed as a quantum resource. into the multipartite entanglement structure and its use- A controlled dense coding was later developed [14], sim- fulness by selecting a few parties among all and inves- ilar to the network quantum teleportation [15], in which tigating their performance in quantum protocols for all a three-mode entangled state is prepared and informa- such selected subsystems. tion is transferred between two parties with the help of Numerous studies have so far demonstrated that a classical communication provided by a third party. monogamy relation is one of the fundamental proper- We here study the dense coding protocol employing an ties of multipartite entanglement, that is, entanglement arbitrary two-mode Gaussian state. We then compare between some subsystems puts limitation on their cor- this two-mode scheme with two single-mode schemes, relation with the other parties. For instance, if two namely, the squeezed-state scheme known to be best quantum systems are maximally entangled, they cannot among Gaussian schemes and the number-state scheme be correlated with a third party at all, even classically [6]. known to be the ultimate optimal scheme achieving the The monogamy relation has been formulated quantita- Holevo bound. We derive conditions to beat those two tively with a proper measure of entanglement for discrete schemes, i.e. to detect two-mode Gaussian states that variables [6, 7], which was also extended to Gaussian CV can manifest quantum advantage in CV dense coding. systems [8, 9]. On the other hand, only a few studies These conditions can be expressed in terms of the vari- have attempted to ﬁnd such monogamy relations in view ances of two correlated quadratures as of useful entanglement for quantum communication pro- tocols. In Ref. [10], it was shown that teleportation ﬁdelity satisﬁes a monogamy relation for three-qutrit Vx−Vp+ < Bc, (1) pure states and for N-qubit pure states, but not in gen- eral. Recently, it was proved that a strict monogamy − relation exists in dense coding for discrete variables [11], where Vx− and Vp+ are the variances of x1 x2 and i.e., if a sender has some quantum advantage in dense p +p , respectively. The bound B turns out to be (1/4)2 1 2 c coding with one receiver, (s)he can not have quantum and (1/2e)2 to beat the squeezed-state scheme and the advantage with any other receivers. number-state scheme, respectively. These are more strin- In this paper, we study a CV dense coding protocol gent than the suﬃcient condition to verify entanglement [16], where the bound is 1/4. We also extend our study to the case of multipartite ∗ Gaussian states. We particularly show that, by means † of Heisenberg’s uncertainty principle, the criterion (1) ‡ can be satisﬁed for only one pair of two-modes among ##LTLine##

text: Continuous-variable

label: title

text: dense

text: coding

text: via

text: a

text: general

text: Gaussian

text: state:

text: Monogamy

text: relation

text: Jaehak

label: author

text: Lee,

text: Se-Wan

text: Ji,

text: Jiyong

text: Park,

text: and

text: Hyunchul

text: Nha

text: We

label: abstract

text: study

text: a

text: continuous

text: variable

text: (CV)

text: dense-coding

text: protocol,

text: originally

text: proposed

text: to

text: employ

text: a

text: two-mode

text: squeezed

text: state,

text: using

text: a

text: general

text: two-mode

text: Gaussian

text: state

text: as

text: a

text: quantum

text: channel.

text: We

text: par-

text: ticularly

text: obtain

text: conditions

text: to

text: manifest

text: quantum

text: advantage

text: by

text: beating

text: two

text: well-known

text: single-mode

text: schemes,

text: namely,

text: the

text: squeezed-state

text: scheme

text: (best

text: Gaussian

text: scheme)

text: and

text: the

text: number-state

text: scheme

text: (optimal

text: scheme

text: achieving

text: the

text: Holevo

text: bound).

text: We

text: then

text: extend

text: our

text: study

text: to

text: a

text: multipartite

text: Gaussian

text: state

text: and

text: investigate

text: the

text: monogamy

text: of

text: operational

text: entanglement

text: measured

text: by

text: the

text: communication

text: capacity

text: under

text: the

text: dense-coding

text: protocol.

text: We

text: show

text: that

text: this

text: operational

text: entanglement

text: represents

text: a

text: strict

text: monogamy

text: relation,

text: by

text: means

text: of

text: Heisenberg’s

text: uncertainty

text: principle

text: among

text: diﬀerent

text: par-

text: ties,

text: i.e.,

text: the

text: quantum

text: advantage

text: for

text: communication

text: can

text: be

text: possible

text: for

text: only

text: one

text: pair

text: of

text: two-mode

text: systems

text: among

text: many

text: parties.

text: I.

text: INTRODUCTION

text: and

text: show

text: a

text: strict

text: monogamy

text: relation

text: in

text: this

text: regime.

text: Ever

text: since

text: quantum

text: dense

text: coding

text: was

text: ﬁrst

text: developed

text: in

text: qubit

text: Quantum

text: correlation,

text: especially

text: entanglement

text: is

text: a

text: systems

text: the

text: dense

text: coding

text: capacity

text: has

text: been

text: derived

text: key

text: resource

text: for

text: quantum

text: information

text: processing,

text: e.g.,

text: for

text: arbitrary

text: ﬁnite-dimensional

text: systems

text: with

text: arbi-

text: quantum

text: teleportation

text: and

text: quantum

text: dense

text: coding

text: trary

text: number

text: of

text: parties

text: for

text: discrete-variable

text: systems.

text: Performance

text: in

text: such

text: protocols,

text: quantiﬁed

text: by

text: the

text: On

text: the

text: other

text: hand,

text: there

text: have

text: been

text: few

text: studies

text: on

text: CV

text: output

text: ﬁdelity

text: of

text: teleportation

text: and

text: the

text: communication

text: dense

text: coding

text: with

text: a

text: general

text: Gaussian

text: state.

text: The

text: CV

text: capacity

text: of

text: dense

text: coding,

text: respectively,

text: can

text: be

text: used

text: as

text: an

text: dense

text: coding

text: protocol

text: was

text: ﬁrst

text: proposed

text: by

text: Braunstein

text: operational

text: measure

text: of

text: entanglement.

text: When

text: entangle-

text: and

text: Kimble

text: in

text: which

text: a

text: pure

text: two-mode

text: squeezed

text: vac-

text: ment

text: is

text: shared

text: by

text: more

text: than

text: two

text: parties,

text: we

text: may

text: look

text: uum

text: (TMSV)

text: state

text: is

text: employed

text: as

text: a

text: quantum

text: resource.

text: into

text: the

text: multipartite

text: entanglement

text: structure

text: and

text: its

text: use-

text: A

text: controlled

text: dense

text: coding

text: was

text: later

text: developed

text: sim-

text: fulness

text: by

text: selecting

text: a

text: few

text: parties

text: among

text: all

text: and

text: inves-

text: ilar

text: to

text: the

text: network

text: quantum

text: teleportation

text: in

text: which

text: tigating

text: their

text: performance

text: in

text: quantum

text: protocols

text: for

text: all

text: a

text: three-mode

text: entangled

text: state

text: is

text: prepared

text: and

text: informa-

text: such

text: selected

text: subsystems.

text: tion

text: is

text: transferred

text: between

text: two

text: parties

text: with

text: the

text: help

text: of

text: Numerous

text: studies

text: have

text: so

text: far

text: demonstrated

text: that

text: a

text: classical

text: communication

text: provided

text: by

text: a

text: third

text: party.

text: monogamy

text: relation

text: is

text: one

text: of

text: the

text: fundamental

text: proper-

text: We

text: here

text: study

text: the

text: dense

text: coding

text: protocol

text: employing

text: an

text: ties

text: of

text: multipartite

text: entanglement,

text: that

text: is,

text: entanglement

text: arbitrary

text: two-mode

text: Gaussian

text: state.

text: We

text: then

text: compare

text: between

text: some

text: subsystems

text: puts

text: limitation

text: on

text: their

text: cor-

text: this

text: two-mode

text: scheme

text: with

text: two

text: single-mode

text: schemes,

text: relation

text: with

text: the

text: other

text: parties.

text: For

text: instance,

text: if

text: two

text: namely,

text: the

text: squeezed-state

text: scheme

text: known

text: to

text: be

text: best

text: quantum

text: systems

text: are

text: maximally

text: entangled,

text: they

text: cannot

text: among

text: Gaussian

text: schemes

text: and

text: the

text: number-state

text: scheme

text: be

text: correlated

text: with

text: a

text: third

text: party

text: at

text: all,

text: even

text: classically

text: known

text: to

text: be

text: the

text: ultimate

text: optimal

text: scheme

text: achieving

text: the

text: The

text: monogamy

text: relation

text: has

text: been

text: formulated

text: quantita-

text: Holevo

text: bound.

text: We

text: derive

text: conditions

text: to

text: beat

text: those

text: two

text: tively

text: with

text: a

text: proper

text: measure

text: of

text: entanglement

text: for

text: discrete

text: schemes,

text: i.e.

text: to

text: detect

text: two-mode

text: Gaussian

text: states

text: that

text: variables

text: which

text: was

text: also

text: extended

text: to

text: Gaussian

text: CV

text: can

text: manifest

text: quantum

text: advantage

text: in

text: CV

text: dense

text: coding.

text: systems

text: On

text: the

text: other

text: hand,

text: only

text: a

text: few

text: studies

text: These

text: conditions

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: vari-

text: have

text: attempted

text: to

text: ﬁnd

text: such

text: monogamy

text: relations

text: in

text: view

text: ances

text: of

text: two

text: correlated

text: quadratures

text: as

text: of

text: useful

text: entanglement

text: for

text: quantum

text: communication

text: pro-

text: tocols.

text: In

text: Ref.

text: it

text: was

text: shown

text: that

text: teleportation

text: ﬁdelity

text: satisﬁes

text: a

text: monogamy

text: relation

text: for

text: three-qutrit

text: Vx−Vp+

text: Bc,

text: pure

text: states

text: and

text: for

text: N-qubit

text: pure

text: states,

text: but

text: not

text: in

text: gen-

text: eral.

text: Recently,

text: it

text: was

text: proved

text: that

text: a

text: strict

text: monogamy

text: relation

text: exists

text: in

text: dense

text: coding

text: for

text: discrete

text: variables

text: where

text: Vx−

text: and

text: Vp+

text: are

text: the

text: variances

text: of

text: and

text: i.e.,

text: if

text: a

text: sender

text: has

text: some

text: quantum

text: advantage

text: in

text: dense

text: p

text: +p

text: respectively.

text: The

text: bound

text: B

text: turns

text: out

text: to

text: be

text: c

text: coding

text: with

text: one

text: receiver,

text: (s)he

text: can

text: not

text: have

text: quantum

text: and

text: to

text: beat

text: the

text: squeezed-state

text: scheme

text: and

text: the

text: advantage

text: with

text: any

text: other

text: receivers.

text: number-state

text: scheme,

text: respectively.

text: These

text: are

text: more

text: strin-

text: In

text: this

text: paper,

text: we

text: study

text: a

text: CV

text: dense

text: coding

text: protocol

text: gent

text: than

text: the

text: suﬃcient

text: condition

text: to

text: verify

text: entanglement

text: where

text: the

text: bound

text: is

text: We

text: also

text: extend

text: our

text: study

text: to

text: the

text: case

text: of

text: multipartite

text: Gaussian

text: states.

text: We

text: particularly

text: show

text: that,

text: by

text: means

text: of

text: Heisenberg’s

text: uncertainty

text: principle,

text: the

text: criterion

text: can

text: be

text: satisﬁed

text: for

text: only

text: one

text: pair

text: of

text: two-modes

text: among

text: ##LTLine##


text: [7] Z. Fu¨redi and P. Hajnal, Davenport-Schinzel theory of matrices, Dis- crete Math. 103 (1992), 233–251. [8] M. Klazar, Extremal problems for ordered (hyper) graphs: applications of Davenport-Schinzel sequences, European J. Combin. 25 (2004), 125– 140. [9] J. Pach and G. Tardos, Forbidden paths and cycles in ordered graphs and matrices, Israel J. Math. 155 (2006), 359–380. 13

label: reference

text: Z.

text: Fu¨redi

text: and

text: P.

text: Hajnal,

text: Davenport-Schinzel

text: theory

text: of

text: matrices,

text: Dis-

text: crete

text: Math.

text: M.

text: Klazar,

text: Extremal

text: problems

text: for

text: ordered

text: (hyper)

text: graphs:

text: applications

text: of

text: Davenport-Schinzel

text: sequences,

text: European

text: J.

text: Combin.

text: J.

text: Pach

text: and

text: G.

text: Tardos,

text: Forbidden

text: paths

text: and

text: cycles

text: in

text: ordered

text: graphs

text: and

text: matrices,

text: Israel

text: J.

text: Math.

label: paragraph


text: b x 1 x b 1 y y a1 b0 a1 2 b0 b 2 a 2 a 2 2 b b 2 3 a a b 3 3 3 b 4 a0 a b0 3 b 4 4 4 z t z a t b 4 5 Figure 2: The graphs R and S . 5 4 be reducible. If d(a ) = 1 and the neighbor b of a is adjacent to a if i j i i−1 i > 1 and is adjacent to a if i < p, then a is also said to be reducible. i+1 ∈ i Similarly we deﬁne when a vertex b B is reducible. Clearly, if a (or b ) j − i − j is reducible, then G has a K -interval minor if and only if G a (G b ) 2,2 i j has one. Therefore, we may assume that we remove all reducible vertices from G. When G has no reducible vertices, we say that G is reduced, which we assume henceforth. { } { } Let X = a , a if d(a ) = 1 and X = a , otherwise. Similarly, { 1 }2 1 { } 1 { } let Y = a , a if d(a ) = 1 and Y = a , otherwise; Z = b , b p−1 p { p } p{ } 1 2 if d(b ) = 1 and Z = b , otherwise; T = b , b if d(b ) = 1 and 1{ } 1 q−1 q q T = b , otherwise. We may assume that all these sets are mutually q disjoint. Otherwise G has a simple structure – it is equivalent to a subgraph of a graph shown in Figure 3 and any such graph has no K as interval 2,2 minor. Note that each such subgraph becomes equivalent to a subgraph of R after removing reducible vertices. 2 Claim 3.2. There is an edge from X to b or b . 1 q { } Proof. Suppose that there is no edge from X to b , b . Since G is reduced, 1 q there are two distinct vertices b and b (1 < i < j < q) connected to i j X. Assume that b and b are adjacent to a and a , respectively. Note (cid:54)∈ 1 q \ {k l } { } that a , a X. Consider the sets X, A X, b , . . . , b and b , . . . , b k l 1 i i+1 q and identify them to single vertices to get K as an interval minor, a 2,2 contradiction. Note that Claim 3.2 also applies to Y, Z and T . Hence, considering an 8 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: b

font: default

text: x

text: x

text: b

text: y

text: y

text: b

text: a

text: a

text: b

text: b

text: a

text: a

text: b

text: b

text: a

text: b

text: z

text: t

text: z

text: a

text: t

text: b

text: Figure

text: The

label: caption

text: graphs

text: R

text: and

text: S

text: be

text: reducible.

text: If

text: d(a

text: and

text: the

text: neighbor

text: b

text: of

text: a

text: is

text: adjacent

text: to

text: a

text: if

text: i

text: j

text: i

text: i

text: and

text: is

text: adjacent

text: to

text: a

text: if

text: i

text: p,

text: then

text: a

text: is

text: also

text: said

text: to

text: be

text: reducible.

text: i

text: Similarly

text: we

text: deﬁne

text: when

text: a

text: vertex

text: b

text: B

text: is

text: reducible.

text: Clearly,

text: if

text: a

text: (or

text: b

text: j

text: i

text: j

text: is

text: reducible,

text: then

text: G

text: has

text: a

text: K

text: -interval

text: minor

text: if

text: and

text: only

text: if

text: G

text: a

text: (G

text: b

text: i

text: j

text: has

text: one.

text: Therefore,

text: we

text: may

text: assume

text: that

text: we

text: remove

text: all

text: reducible

text: vertices

text: from

text: G.

text: When

text: G

text: has

text: no

text: reducible

text: vertices,

text: we

text: say

text: that

text: G

text: is

text: reduced,

text: which

text: we

text: assume

text: henceforth.

text: Let

text: X

text: a

text: a

text: if

text: d(a

text: and

text: X

text: a

text: otherwise.

text: Similarly,

text: let

text: Y

text: a

text: a

text: if

text: d(a

text: and

text: Y

text: a

text: otherwise;

text: Z

text: b

text: b

text: p

text: p

text: p{

text: if

text: d(b

text: and

text: Z

text: b

text: otherwise;

text: T

text: b

text: b

text: if

text: d(b

text: and

text: q

text: q

text: T

text: b

text: otherwise.

text: We

text: may

text: assume

text: that

text: all

text: these

text: sets

text: are

text: mutually

text: q

text: disjoint.

text: Otherwise

text: G

text: has

text: a

text: simple

text: structure

text: it

text: is

text: equivalent

text: to

text: a

text: subgraph

text: of

text: a

text: graph

text: shown

text: in

text: Figure

text: and

text: any

text: such

text: graph

text: has

text: no

text: K

text: as

text: interval

text: minor.

text: Note

text: that

text: each

text: such

text: subgraph

text: becomes

text: equivalent

text: to

text: a

text: subgraph

text: of

text: R

text: after

text: removing

text: reducible

text: vertices.

text: Claim

text: There

text: is

text: an

text: edge

text: from

text: X

text: to

text: b

text: or

text: b

text: q

text: Proof.

text: Suppose

text: that

text: there

text: is

text: no

text: edge

text: from

text: X

text: to

text: b

text: b

text: Since

text: G

text: is

text: reduced,

text: q

text: there

text: are

text: two

text: distinct

text: vertices

text: b

text: and

text: b

text: i

text: j

text: q)

text: connected

text: to

text: i

text: j

text: X.

text: Assume

text: that

text: b

text: and

text: b

text: are

text: adjacent

text: to

text: a

text: and

text: a

text: respectively.

text: Note

text: q

text: {k

text: l

text: that

text: a

text: a

text: X.

text: Consider

text: the

text: sets

text: X,

text: A

text: X,

text: b

text: b

text: and

text: b

text: b

text: k

text: l

text: i

text: q

text: and

text: identify

text: them

text: to

text: single

text: vertices

text: to

text: get

text: K

text: as

text: an

text: interval

text: minor,

text: a

text: contradiction.

text: Note

text: that

text: Claim

text: also

text: applies

text: to

text: Y,

text: Z

text: and

text: T

text: Hence,

text: considering

text: an

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Figure 3: X and Y intersect only in special situations. equivalent graph of G instead of G if necessary, we may assume that there is an edge from X to Z. If there is no edge from Y to T , then there are edges from Y to Z and from T to X. By reversing the order of B, we obtain an equivalent graph that has edges from X to Z and from Y to T . Thus we may assume henceforth that the following claim holds: Claim 3.3. The graph G has edges from X to Z and from Y to T . Claim 3.4. Every vertex of G has degree at most 2, except possibly one of a , b and/or one of a , b , which may be of degree 3. If d(a ) = 3, 2 2 p−1 q−1 2 then it has neighbors b , b , b , we have d(a ) = d(b ) = d(b ) = 1 and ∈ 1 3 4 1 1 2 a b E(G). Similar situations occur when b , a , or b are of degree 1 2 2 p−1 q−1 3. ≥ Proof. Suppose that d(a ) 3. We claim that a has at most one neighbour | | ≥ i i ∈ in Z. Otherwise, Z 2 and hence d(b ) = 1 and a b , a b E(G). This 1 i 1 i 2 is a contradiction since G is reduced. Similarly we see that a has at most i one neighbour in T . \ ∪ Suppose now that a middle neighbor b of a is in B (Z T ). Let b j i j1 and b be neighbors of a with j < j < j . If d(b ) > 1, then an edge a b (cid:54) j2 i 1 2 j k j (k = i), the edges joining X and Z and joining Y and T , and the edge a b i j1 (if k < i) or a b (if k > i) can be used to obtain a K -interval minor. i j2 2,2 Thus, d(b ) = 1. j Let us now consider b . Suppose that b is not adjacent to a . Then − j−1 j−1 i j < j 1. If b is adjacent to a vertex a , where k < i, then the edges 1 j−1 k a b , a b and the edges joining X with Z and Y with T give rise to a i j1 k j−1 K -interval minor in G (which is excluded), unless the following situation 2,2 occurs: the edge a b is equal to the edge joining X and Z. This is only k j−1 | | possible if j = 1, j = 3 and Z = 2, i.e., d(b ) = 1. If a is adjacent to b 1 1 1 1 or to some other b with t > 2, we obtain a K -interval minor again. So, t 2,2 9 ##LTFigure##

text: Figure

text: X

text: and

text: Y

text: intersect

text: only

text: in

text: special

text: situations.

text: equivalent

text: graph

text: of

text: G

text: instead

text: of

text: G

text: if

text: necessary,

text: we

text: may

text: assume

text: that

text: there

text: is

text: an

text: edge

text: from

text: X

text: to

text: Z.

text: If

text: there

text: is

text: no

text: edge

text: from

text: Y

text: to

text: T

text: then

text: there

text: are

text: edges

text: from

text: Y

text: to

text: Z

text: and

text: from

text: T

text: to

text: X.

text: By

text: reversing

text: the

text: order

text: of

text: B,

text: we

text: obtain

text: an

text: equivalent

text: graph

text: that

text: has

text: edges

text: from

text: X

text: to

text: Z

text: and

text: from

text: Y

text: to

text: T

text: Thus

text: we

text: may

text: assume

text: henceforth

text: that

text: the

text: following

text: claim

text: holds:

text: Claim

text: The

text: graph

text: G

text: has

text: edges

text: from

text: X

text: to

text: Z

text: and

text: from

text: Y

text: to

text: T

text: Claim

text: Every

text: vertex

text: of

text: G

text: has

text: degree

text: at

text: most

text: except

text: possibly

text: one

text: of

text: a

text: b

text: and/or

text: one

text: of

text: a

text: b

text: which

text: may

text: be

text: of

text: degree

text: If

text: d(a

text: then

text: it

text: has

text: neighbors

text: b

text: b

text: b

text: we

text: have

text: d(a

text: d(b

text: d(b

text: and

text: a

text: b

text: E(G).

text: Similar

text: situations

text: occur

text: when

text: b

text: a

text: or

text: b

text: are

text: of

text: degree

text: Proof.

text: Suppose

text: that

text: d(a

text: We

text: claim

text: that

text: a

text: has

text: at

text: most

text: one

text: neighbour

text: i

text: i

text: in

text: Z.

text: Otherwise,

text: Z

text: and

text: hence

text: d(b

text: and

text: a

text: b

text: a

text: b

text: E(G).

text: This

text: i

text: i

text: is

text: a

text: contradiction

text: since

text: G

text: is

text: reduced.

text: Similarly

text: we

text: see

text: that

text: a

text: has

text: at

text: most

text: i

text: one

text: neighbour

text: in

text: T

text: Suppose

text: now

text: that

text: a

text: middle

text: neighbor

text: b

text: of

text: a

text: is

text: in

text: B

text: (Z

text: T

text: Let

text: b

text: j

text: i

text: and

text: b

text: be

text: neighbors

text: of

text: a

text: with

text: j

text: j

text: j

text: If

text: d(b

text: then

text: an

text: edge

text: a

text: b

text: i

text: j

text: k

text: j

text: (k

text: i),

text: the

text: edges

text: joining

text: X

text: and

text: Z

text: and

text: joining

text: Y

text: and

text: T

text: and

text: the

text: edge

text: a

text: b

text: i

text: (if

text: k

text: i)

text: or

text: a

text: b

text: (if

text: k

text: i)

text: can

text: be

text: used

text: to

text: obtain

text: a

text: K

text: -interval

text: minor.

text: i

text: Thus,

text: d(b

text: j

text: Let

text: us

text: now

text: consider

text: b

text: Suppose

text: that

text: b

text: is

text: not

text: adjacent

text: to

text: a

text: Then

text: i

text: j

text: j

text: If

text: b

text: is

text: adjacent

text: to

text: a

text: vertex

text: a

text: where

text: k

text: i,

text: then

text: the

text: edges

text: k

text: a

text: b

text: a

text: b

text: and

text: the

text: edges

text: joining

text: X

text: with

text: Z

text: and

text: Y

text: with

text: T

text: give

text: rise

text: to

text: a

text: i

text: k

text: K

text: -interval

text: minor

text: in

text: G

text: (which

text: is

text: excluded),

text: unless

text: the

text: following

text: situation

text: occurs:

text: the

text: edge

text: a

text: b

text: is

text: equal

text: to

text: the

text: edge

text: joining

text: X

text: and

text: Z.

text: This

text: is

text: only

text: k

text: possible

text: if

text: j

text: j

text: and

text: Z

text: i.e.,

text: d(b

text: If

text: a

text: is

text: adjacent

text: to

text: b

text: or

text: to

text: some

text: other

text: b

text: with

text: t

text: we

text: obtain

text: a

text: K

text: -interval

text: minor

text: again.

text: So,

text: t

text: ##LTFigure##


text: 2 may distort the spectrum. A most relevant question is “How hot a laboratory was it” or phrased in the context of Inﬂation “What are the energy scale Λ and the reheat temperature T of the last relevant inﬂation”. A rele- R vant inﬂation is operationally deﬁned as one with many e- fold expansion suﬃcient to signiﬁcantly dilute the number densities of particles or topological structures inherited from previous epochs. Recent observation by the BICEP collaboration if not due to dust, suggest “B-like” patterns in the CMB polarization. [3]. To generate it one needs gravity waves emitted early on and with suﬃcient intensity to be able to imprint at recombination this polarization pattern. The ratio r of tensor to scalar ﬂuctuations r(Bicep) = 0.2 suggested a high mass scale for the primordial inﬂation: ≈ Λ(I) = V 1/4 21016GeV , only two orders of magnitude below the Planck mass and consistent − with the scales of GUT (grand uniﬁed theories). Even smaller r = 0.05 0.1 values along with WMAP and PLANCK measurement of a density ﬂuctuation index n close to unity suggest large s ≥ Λ(I) 1015GeV . ∼ The simplest interpretation of these results involves an inﬂanton of mass M Λ(I) with inlaton initial Planckian oﬀset away from zero of the inﬂaton ﬁeld driving the inﬂation. We choose a high inﬂanton mass in order to avoid extreme super-planckian deviations ∆(φ) which would be required ∼ ∼ in order to have V (φ) m2φ2 Λ4 . While one can have the desired sixty e-fold horizon expansion and n close to unity in many scenarios this would be among the simpler ones. Unless the decaying s inﬂanton prefers certain sectors of particle theory this suggests that all forms of matter have been early on in thermal equilibrium at a reheat temperature T (R) which is not much smaller than m(Inﬂaton) or Λ(I). A universal couplings are natural if the inﬂaton is an alias for quantum gravity [4]. We can choose a very small m(Inflaton)/M(P lanck) - or small ﬁne-tuned quartic dimensionless couplings λ. While we will then need very large, superplanckian ∆(φ) to achieve the energy density required, the reheat temperature can be much smaller than Λ. Still in the following we would like to discuss what would happen if : (i) the scale Λ of the last relevant inﬂation is high? or (ii) The reheat temperature T (R) after this inﬂation is high? The second assumption is much stronger as lower reheat temperatures can be achieved if the coupling of the inﬂaton to all other forms of matter is very small. While much of the material below is not new, it is useful to emphasize the limitations on many particle physics models which the above assumptions imply. In Sec II we show how the high inﬂation scale can constrain models of symmetry breaking

text: may

text: distort

text: the

text: spectrum.

text: A

text: most

text: relevant

text: question

text: is

text: “How

text: hot

text: a

text: laboratory

text: was

text: it”

text: or

text: phrased

text: in

text: the

text: context

text: of

text: Inﬂation

text: “What

text: are

text: the

text: energy

text: scale

text: Λ

text: and

text: the

text: reheat

text: temperature

text: T

text: of

text: the

text: last

text: relevant

text: inﬂation”.

text: A

text: rele-

text: R

text: vant

text: inﬂation

text: is

text: operationally

text: deﬁned

text: as

text: one

text: with

text: many

text: e-

text: fold

text: expansion

text: suﬃcient

text: to

text: signiﬁcantly

text: dilute

text: the

text: number

text: densities

text: of

text: particles

text: or

text: topological

text: structures

text: inherited

text: from

text: previous

text: epochs.

text: Recent

text: observation

text: by

text: the

text: BICEP

text: collaboration

text: if

text: not

text: due

text: to

text: dust,

text: suggest

text: “B-like”

text: patterns

text: in

text: the

text: CMB

text: polarization.

text: To

text: generate

text: it

text: one

text: needs

text: gravity

text: waves

text: emitted

text: early

text: on

text: and

text: with

text: suﬃcient

text: intensity

text: to

text: be

text: able

text: to

text: imprint

text: at

text: recombination

text: this

text: polarization

text: pattern.

text: The

text: ratio

text: r

text: of

text: tensor

text: to

text: scalar

text: ﬂuctuations

text: r(Bicep)

text: suggested

text: a

text: high

text: mass

text: scale

text: for

text: the

text: primordial

text: inﬂation:

text: Λ(I)

text: V

text: only

text: two

text: orders

text: of

text: magnitude

text: below

text: the

text: Planck

text: mass

text: and

text: consistent

text: with

text: the

text: scales

text: of

text: GUT

text: (grand

text: uniﬁed

text: theories).

text: Even

text: smaller

text: r

text: values

text: along

text: with

text: WMAP

text: and

text: PLANCK

text: measurement

text: of

text: a

text: density

text: ﬂuctuation

text: index

text: n

text: close

text: to

text: unity

text: suggest

text: large

text: s

text: Λ(I)

text: The

text: simplest

text: interpretation

text: of

text: these

text: results

text: involves

text: an

text: inﬂanton

text: of

text: mass

text: M

text: Λ(I)

text: with

text: inlaton

text: initial

text: Planckian

text: oﬀset

text: away

text: from

text: zero

text: of

text: the

text: inﬂaton

text: ﬁeld

text: driving

text: the

text: inﬂation.

text: We

text: choose

text: a

text: high

text: inﬂanton

text: mass

text: in

text: order

text: to

text: avoid

text: extreme

text: super-planckian

text: deviations

text: ∆(φ)

text: which

text: would

text: be

text: required

text: in

text: order

text: to

text: have

text: V

text: (φ)

text: Λ4

text: While

text: one

text: can

text: have

text: the

text: desired

text: sixty

text: e-fold

text: horizon

text: expansion

text: and

text: n

text: close

text: to

text: unity

text: in

text: many

text: scenarios

text: this

text: would

text: be

text: among

text: the

text: simpler

text: ones.

text: Unless

text: the

text: decaying

text: s

text: inﬂanton

text: prefers

text: certain

text: sectors

text: of

text: particle

text: theory

text: this

text: suggests

text: that

text: all

text: forms

text: of

text: matter

text: have

text: been

text: early

text: on

text: in

text: thermal

text: equilibrium

text: at

text: a

text: reheat

text: temperature

text: T

text: (R)

text: which

text: is

text: not

text: much

text: smaller

text: than

text: m(Inﬂaton)

text: or

text: Λ(I).

text: A

text: universal

text: couplings

text: are

text: natural

text: if

text: the

text: inﬂaton

text: is

text: an

text: alias

text: for

text: quantum

text: gravity

text: We

text: can

text: choose

text: a

text: very

text: small

text: m(Inflaton)/M(P

text: lanck)

text: or

text: small

text: ﬁne-tuned

text: quartic

text: dimensionless

text: couplings

text: λ.

text: While

text: we

text: will

text: then

text: need

text: very

text: large,

text: superplanckian

text: ∆(φ)

text: to

text: achieve

text: the

text: energy

text: density

text: required,

text: the

text: reheat

text: temperature

text: can

text: be

text: much

text: smaller

text: than

text: Λ.

text: Still

text: in

text: the

text: following

text: we

text: would

text: like

text: to

text: discuss

text: what

text: would

text: happen

text: if

text: (i)

text: the

text: scale

text: Λ

text: of

text: the

text: last

text: relevant

text: inﬂation

text: is

text: high?

text: or

text: (ii)

text: The

text: reheat

text: temperature

text: T

text: (R)

text: after

text: this

text: inﬂation

text: is

text: high?

text: The

text: second

text: assumption

text: is

text: much

text: stronger

text: as

text: lower

text: reheat

text: temperatures

text: can

text: be

text: achieved

text: if

text: the

text: coupling

text: of

text: the

text: inﬂaton

text: to

text: all

text: other

text: forms

text: of

text: matter

text: is

text: very

text: small.

text: While

text: much

text: of

text: the

text: material

text: below

text: is

text: not

text: new,

text: it

text: is

text: useful

text: to

text: emphasize

text: the

text: limitations

text: on

text: many

text: particle

text: physics

text: models

text: which

text: the

text: above

text: assumptions

text: imply.

text: In

text: Sec

text: II

text: we

text: show

text: how

text: the

text: high

text: inﬂation

text: scale

text: can

text: constrain

text: models

text: of

text: symmetry

text: breaking


text: 11 ≤ suppressed by 1/t(min) + µ2 1/M(X)2. The annihilation cross-section is in fact smaller by another factor 4. The reason is that the lightest particle that can be exchanged has a mass ≈ µ M(X) (see Fig. 1b) as otherwise the X quickly decays into the ﬁnal i and the exchanged µ particles and would not be long-lived contrary to our assumption. Note that the annihilation: (cid:0) (cid:1) → (cid:48) (cid:48) (cid:48) X¯X X¯ X where X is just slightly lighter than X and with a light µ exchanged in the t channel ∼ − (cid:48) and a large cross-section: σ 1/ µ2 + (M(X) M(X )2) is also excluded. In this case the heavy (cid:48) X is the lighter particle in the new sector and therefore would be stable. Another X” will have (cid:48) then to be invoked to have the X annihilate quickly etc. More importantly all non zero l terms in the partial wave expansion of the annihilation cross-section are - because of the short range t ≈ channel induced interactions r 1/M(X)- drastically suppressed at threshold by the factor of 0 (k/M(X))2l. This makes the contribution of all none-zero l states negligible leaving us only with − − (cid:48) JP (R) = 1 or 0 for spin 1/2, X particles or with JP (R) = 0+ for spin zero X s and we cannot use t channel exchanges to enhance the rate of annihilation. All of this just elaborates on and accords with KM. There are however several new points which we expound next. (1) The t channel exchanges can provide a potential in the X¯X channel tuned to produce a near threshold X¯X resonance/ bound state, in the l=0 partial wave. However we still cannot saturate the upper bound σ = π/k2 in this partial wave and annihilation cross-section enhanced by ann (M(X)/k)2 = f relative to the nominal 1/M(X)2 are not obtained. ∼ We argued above that annihilation requires that a particle of mass M(X) be exchanged in the t channel and the range of the “annihilation potential” r = 1/M(X), which is deﬁnitely smaller 0 than the eﬀective wavelength size 1/k of the resonance (See Fig 3). This suggests a geometric reduction of the cross-section by k2/M(X)2 = 1/f which certainly would hold at high energies, It is however also the case at low near threshold energies as the following argument indicates. During the lifetime τ(R) = 1/Γ = f/M(X) of the resonance R, the X¯, say, travels a distance of d = vτ − relative to X which we assume to be ﬁxed at the origin. Using v = k/M(X) = f 1/2 we ﬁnd that d = f1/2/M(X) and the eﬀective annihilation volume swept by the X¯ is V ol(ann) = πr2d = 0 πf1/2/M(X)3. The full volume of the resonance (or the corresponding bound state obtained → − by k ik) is V ol(tot) = 1/k3 = f3/2M(X) 3 namely f times larger. The probability of actual annihilation rather than scattering back into the initial X¯X state is then the ratio of these volumes namely 1/f . ∼ Thus we are back to the original σ(ann) 1/M(X)2 estimate and upper bounds on M(X) ∼ larger by f1/2 5 ensue.

text: suppressed

text: by

text: µ2

text: The

text: annihilation

text: cross-section

text: is

text: in

text: fact

text: smaller

text: by

text: another

text: factor

text: The

text: reason

text: is

text: that

text: the

text: lightest

text: particle

text: that

text: can

text: be

text: exchanged

text: has

text: a

text: mass

text: µ

text: M(X)

text: (see

text: Fig.

text: as

text: otherwise

text: the

text: X

text: quickly

text: decays

text: into

text: the

text: ﬁnal

text: i

text: and

text: the

text: exchanged

text: µ

text: particles

text: and

text: would

text: not

text: be

text: long-lived

text: contrary

text: to

text: our

text: assumption.

text: Note

text: that

text: the

text: annihilation:

text: X¯X

text: X¯

text: X

text: where

text: X

text: is

text: just

text: slightly

text: lighter

text: than

text: X

text: and

text: with

text: a

text: light

text: µ

text: exchanged

text: in

text: the

text: t

text: channel

text: and

text: a

text: large

text: cross-section:

text: σ

text: µ2

text: (M(X)

text: M(X

text: is

text: also

text: excluded.

text: In

text: this

text: case

text: the

text: heavy

text: X

text: is

text: the

text: lighter

text: particle

text: in

text: the

text: new

text: sector

text: and

text: therefore

text: would

text: be

text: stable.

text: Another

text: X”

text: will

text: have

text: then

text: to

text: be

text: invoked

text: to

text: have

text: the

text: X

text: annihilate

text: quickly

text: etc.

text: More

text: importantly

text: all

text: non

text: zero

text: l

text: terms

text: in

text: the

text: partial

text: wave

text: expansion

text: of

text: the

text: annihilation

text: cross-section

text: are

text: because

text: of

text: the

text: short

text: range

text: t

text: channel

text: induced

text: interactions

text: r

text: drastically

text: suppressed

text: at

text: threshold

text: by

text: the

text: factor

text: of

text: This

text: makes

text: the

text: contribution

text: of

text: all

text: none-zero

text: l

text: states

text: negligible

text: leaving

text: us

text: only

text: with

text: JP

text: (R)

text: or

text: for

text: spin

text: X

text: particles

text: or

text: with

text: JP

text: (R)

text: for

text: spin

text: zero

text: X

text: s

text: and

text: we

text: cannot

text: use

text: t

text: channel

text: exchanges

text: to

text: enhance

text: the

text: rate

text: of

text: annihilation.

text: All

text: of

text: this

text: just

text: elaborates

text: on

text: and

text: accords

text: with

text: KM.

text: There

text: are

text: however

text: several

text: new

text: points

text: which

text: we

text: expound

text: next.

text: The

text: t

text: channel

text: exchanges

text: can

text: provide

text: a

text: potential

text: in

text: the

text: X¯X

text: channel

text: tuned

text: to

text: produce

text: a

text: near

text: threshold

text: X¯X

text: resonance/

text: bound

text: state,

text: in

text: the

text: partial

text: wave.

text: However

text: we

text: still

text: cannot

text: saturate

text: the

text: upper

text: bound

text: σ

text: in

text: this

text: partial

text: wave

text: and

text: annihilation

text: cross-section

text: enhanced

text: by

text: ann

text: f

text: relative

text: to

text: the

text: nominal

text: are

text: not

text: obtained.

text: We

text: argued

text: above

text: that

text: annihilation

text: requires

text: that

text: a

text: particle

text: of

text: mass

text: M(X)

text: be

text: exchanged

text: in

text: the

text: t

text: channel

text: and

text: the

text: range

text: of

text: the

text: “annihilation

text: potential”

text: r

text: which

text: is

text: deﬁnitely

text: smaller

text: than

text: the

text: eﬀective

text: wavelength

text: size

text: of

text: the

text: resonance

text: (See

text: Fig

text: This

text: suggests

text: a

text: geometric

text: reduction

text: of

text: the

text: cross-section

text: by

text: which

text: certainly

text: would

text: hold

text: at

text: high

text: energies,

text: It

text: is

text: however

text: also

text: the

text: case

text: at

text: low

text: near

text: threshold

text: energies

text: as

text: the

text: following

text: argument

text: indicates.

text: During

text: the

text: lifetime

text: τ(R)

text: f/M(X)

text: of

text: the

text: resonance

text: R,

text: the

text: X¯,

text: say,

text: travels

text: a

text: distance

text: of

text: d

text: vτ

text: relative

text: to

text: X

text: which

text: we

text: assume

text: to

text: be

text: ﬁxed

text: at

text: the

text: origin.

text: Using

text: v

text: k/M(X)

text: f

text: we

text: ﬁnd

text: that

text: d

text: and

text: the

text: eﬀective

text: annihilation

text: volume

text: swept

text: by

text: the

text: X¯

text: is

text: V

text: ol(ann)

text: The

text: full

text: volume

text: of

text: the

text: resonance

text: (or

text: the

text: corresponding

text: bound

text: state

text: obtained

text: by

text: k

text: ik)

text: is

text: V

text: ol(tot)

text: namely

text: f

text: times

text: larger.

text: The

text: probability

text: of

text: actual

text: annihilation

text: rather

text: than

text: scattering

text: back

text: into

text: the

text: initial

text: X¯X

text: state

text: is

text: then

text: the

text: ratio

text: of

text: these

text: volumes

text: namely

text: Thus

text: we

text: are

text: back

text: to

text: the

text: original

text: σ(ann)

text: estimate

text: and

text: upper

text: bounds

text: on

text: M(X)

text: larger

text: by

text: ensue.


text: 12 It is important to note that the large annihilation cross-section of anti-proton and proton near − threshold does not contradict the above arguments for suppression of X¯ X annihilation. There − are several important diﬀerences between the two cases. Indeed p p¯ “annihilation” is a misnomer. − → Unlike in the genuine e+e 2γ annihilation where the electron and positron disappear no quarks and anti-quarks need to be annihilated in p¯p annihilation. Rather the initial q3+q¯3 state rearranges into a three meson (q¯q)3 conﬁguration. As the slow p and p¯ approach each other near threshold, the levels of these two conﬁgurations cross and there is an almost complete adiabatic transition of − p¯ p into the three mesons. Since no heavy t channel exchanges are involved the only suppression of this process comes from the ﬁnite nucleon size. This issue will be important for our subsequent discussion of the relic abundance of heavy X particles which carry the charges of a conﬁning gauge theory and generate stable or long lived baryons. There is however a clear distinction between this and what is rigorously excluded in the ordinary S matrix formalism -namely the cases in which the GK bound is violated. Disregarding any t channel dynamics we can postulate a new heavy gauge particle or a scalar of a mass M(R) tuned to be within a factor of 1/(2f) near the 2M(X) threshold. We then arbitrarily prescribe that it’s coupling to both the XX¯ initial particles and the ﬁnal i¯i particles so as to saturate the GK bound. (see Fig. 2) At this point we depart from the purely formal approach which does allows particles as heavy ≈ as 100T eV and go beyond the original discussion of GK. Utilizing the many experimental bounds on WIMPS which have accumulated in the meantime and also some general comments on the underlying theory we discuss the potential diﬃculties entailed if new particles of the type described above are allowed . The light particles into which the R decays can be new ’dark sector’ x particles. Since the lightest particle in this sector is stable, potential over-closure still looms and one needs detailed scenarios where the new x particles are suﬃciently light to meet CDM or even WDM or HDM ( warm dark matter and hot dark matter) requirements. The other alternative- that the R decays into SM particles is more interesting but more restricted phenomenologically. If the possible i¯i ﬁnal states include the u and d quarks then the maximal couplings of R to the X¯X dark matter particles and to quarks/nuclei can generate excessive X- − ∼ nuclear cross-sections. With target nucleus mass M(A,Z) these cross-sections are σ(X (A, Z)) − − 1/(M(X)4).M(X).M(A, Z) = 4.10 35cm2.(M(X)/T eV ) 3A/100 and may exceed the extremely tight bounds on nuclear cross-sections inferred from direct searches. On the more theoretical side introducing a new strong gauge interaction common to the SM and the dark sectors which would

text: It

text: is

text: important

text: to

text: note

text: that

text: the

text: large

text: annihilation

text: cross-section

text: of

text: anti-proton

text: and

text: proton

text: near

text: threshold

text: does

text: not

text: contradict

text: the

text: above

text: arguments

text: for

text: suppression

text: of

text: X¯

text: X

text: annihilation.

text: There

text: are

text: several

text: important

text: diﬀerences

text: between

text: the

text: two

text: cases.

text: Indeed

text: p

text: p¯

text: “annihilation”

text: is

text: a

text: misnomer.

text: Unlike

text: in

text: the

text: genuine

text: e+e

text: 2γ

text: annihilation

text: where

text: the

text: electron

text: and

text: positron

text: disappear

text: no

text: quarks

text: and

text: anti-quarks

text: need

text: to

text: be

text: annihilated

text: in

text: p¯p

text: annihilation.

text: Rather

text: the

text: initial

text: state

text: rearranges

text: into

text: a

text: three

text: meson

text: conﬁguration.

text: As

text: the

text: slow

text: p

text: and

text: p¯

text: approach

text: each

text: other

text: near

text: threshold,

text: the

text: levels

text: of

text: these

text: two

text: conﬁgurations

text: cross

text: and

text: there

text: is

text: an

text: almost

text: complete

text: adiabatic

text: transition

text: of

text: p¯

text: p

text: into

text: the

text: three

text: mesons.

text: Since

text: no

text: heavy

text: t

text: channel

text: exchanges

text: are

text: involved

text: the

text: only

text: suppression

text: of

text: this

text: process

text: comes

text: from

text: the

text: ﬁnite

text: nucleon

text: size.

text: This

text: issue

text: will

text: be

text: important

text: for

text: our

text: subsequent

text: discussion

text: of

text: the

text: relic

text: abundance

text: of

text: heavy

text: X

text: particles

text: which

text: carry

text: the

text: charges

text: of

text: a

text: conﬁning

text: gauge

text: theory

text: and

text: generate

text: stable

text: or

text: long

text: lived

text: baryons.

text: There

text: is

text: however

text: a

text: clear

text: distinction

text: between

text: this

text: and

text: what

text: is

text: rigorously

text: excluded

text: in

text: the

text: ordinary

text: S

text: matrix

text: formalism

text: -namely

text: the

text: cases

text: in

text: which

text: the

text: GK

text: bound

text: is

text: violated.

text: Disregarding

text: any

text: t

text: channel

text: dynamics

text: we

text: can

text: postulate

text: a

text: new

text: heavy

text: gauge

text: particle

text: or

text: a

text: scalar

text: of

text: a

text: mass

text: M(R)

text: tuned

text: to

text: be

text: within

text: a

text: factor

text: of

text: near

text: the

text: threshold.

text: We

text: then

text: arbitrarily

text: prescribe

text: that

text: it’s

text: coupling

text: to

text: both

text: the

text: XX¯

text: initial

text: particles

text: and

text: the

text: ﬁnal

text: i¯i

text: particles

text: so

text: as

text: to

text: saturate

text: the

text: GK

text: bound.

text: (see

text: Fig.

text: At

text: this

text: point

text: we

text: depart

text: from

text: the

text: purely

text: formal

text: approach

text: which

text: does

text: allows

text: particles

text: as

text: heavy

text: as

text: eV

text: and

text: go

text: beyond

text: the

text: original

text: discussion

text: of

text: GK.

text: Utilizing

text: the

text: many

text: experimental

text: bounds

text: on

text: WIMPS

text: which

text: have

text: accumulated

text: in

text: the

text: meantime

text: and

text: also

text: some

text: general

text: comments

text: on

text: the

text: underlying

text: theory

text: we

text: discuss

text: the

text: potential

text: diﬃculties

text: entailed

text: if

text: new

text: particles

text: of

text: the

text: type

text: described

text: above

text: are

text: allowed

text: The

text: light

text: particles

text: into

text: which

text: the

text: R

text: decays

text: can

text: be

text: new

text: ’dark

text: sector’

text: x

text: particles.

text: Since

text: the

text: lightest

text: particle

text: in

text: this

text: sector

text: is

text: stable,

text: potential

text: over-closure

text: still

text: looms

text: and

text: one

text: needs

text: detailed

text: scenarios

text: where

text: the

text: new

text: x

text: particles

text: are

text: suﬃciently

text: light

text: to

text: meet

text: CDM

text: or

text: even

text: WDM

text: or

text: HDM

text: warm

text: dark

text: matter

text: and

text: hot

text: dark

text: matter)

text: requirements.

text: The

text: other

text: alternative-

text: that

text: the

text: R

text: decays

text: into

text: SM

text: particles

text: is

text: more

text: interesting

text: but

text: more

text: restricted

text: phenomenologically.

text: If

text: the

text: possible

text: i¯i

text: ﬁnal

text: states

text: include

text: the

text: u

text: and

text: d

text: quarks

text: then

text: the

text: maximal

text: couplings

text: of

text: R

text: to

text: the

text: X¯X

text: dark

text: matter

text: particles

text: and

text: to

text: quarks/nuclei

text: can

text: generate

text: excessive

text: X-

text: nuclear

text: cross-sections.

text: With

text: target

text: nucleus

text: mass

text: M(A,Z)

text: these

text: cross-sections

text: are

text: σ(X

text: (A,

text: Z))

text: Z)

text: eV

text: and

text: may

text: exceed

text: the

text: extremely

text: tight

text: bounds

text: on

text: nuclear

text: cross-sections

text: inferred

text: from

text: direct

text: searches.

text: On

text: the

text: more

text: theoretical

text: side

text: introducing

text: a

text: new

text: strong

text: gauge

text: interaction

text: common

text: to

text: the

text: SM

text: and

text: the

text: dark

text: sectors

text: which

text: would


text: 18 (cid:48) (cid:48) of T = Λ where a lot of latent heat is emitted still into the gluonic gas we can expect some ∼ − (cid:48) (cid:48) (cid:48) ≤ − ∼ − signiﬁcant reduction by a factor of exp m(g g )/T exp 7 10 3. The number reducing process we considered has been recently revived in [24] where a new paradigm for O(40MeV ) dark matter has been introduced and where the extra entropy emitted by the dark matter particle reducing interactions, is dumped into the normal SM sector which for a long while is in thermal equilibrium with the dark sector. Generically the interactions producing the x particles are suppressed by some heavy new physics scales Λ and proceed with rates proportional to T n+1/(Λ )n with n = 3, 4 or 5. Still for new new temperatures T which are comparable or even some-what smaller than this new scale the x particles would have been in thermal equilibrium. A well known example is provided by the the ordinary left- handed neutrinos which are coupled to all the remaining SM species and are in thermal chemical equilibrium so long as the temperature exceeds O( MeV). This is because the relevant rate scales ∼ ∼ as G2 T 5 T 5/v4 with v 0.2T eV the scale of ordinary weak interaction falls bellow the Fermi ∼ Hubble expansion rate T 2/M(P lanck) only at this low temperature. At the (at least!) 1010 times higher reheat temperature envisioned any other even supper-weakly coupled species such as sterile neutrinos or ALPS ( axion like particles) would have to be thermally produced. This in turn may exclude a very broad range of scenarios where long lived x particles more massive than say 100eV are required. In particular it would negate the possibility that sterile neutrinos of mass ∼ ≈ 7KeV and lifetimes of 1027 seconds generate the recently observed 3.5KeV X ray line [25]. We should emphasize that this reasoning does not apply to all forms of light long lived particles. An example is provided by the dark photon mentioned above. If there is no dark sector and the lightest particles carrying the new U’(1) are extremely almost Planck scale heavy than its may have never been in thermal equilibrium. Its only coupling is via the mixing (cid:15) with the ordinary (cid:48) ≤ photon which for masses m(A ) MeV decay only via an electron box diagram into three photons and with lifetimes which are far longer than Hubble time. This is so because for such light dark photon considerations related to the cooling of supernovae by photon’ emission strongly bind the ∼ ∼ − strength of its mixing with the ordinary photon (cid:15)2 10 20. The cross-section for producing → (cid:48) ∼ this dark photon e.g. σγ + e γ + e (cid:15)2α2/T 2 yields then in the relevant radiation dominated ∼ ∼ ∼ era where n(e) T 3 a rate of production Γ σn(e) T . unlike the cases considered above Γ ∼ increases more slowly with temperature than the Hubble expansion T 2/M(P lanck) and hence can be out of equilibrium even at the highest possible temperatures. Hence we do not encounter the over-closure diﬃculties and could have the dark photon be warm dark matter.

text: of

text: T

text: Λ

text: where

text: a

text: lot

text: of

text: latent

text: heat

text: is

text: emitted

text: still

text: into

text: the

text: gluonic

text: gas

text: we

text: can

text: expect

text: some

text: signiﬁcant

text: reduction

text: by

text: a

text: factor

text: of

text: exp

text: m(g

text: g

text: )/T

text: exp

text: The

text: number

text: reducing

text: process

text: we

text: considered

text: has

text: been

text: recently

text: revived

text: in

text: where

text: a

text: new

text: paradigm

text: for

text: dark

text: matter

text: has

text: been

text: introduced

text: and

text: where

text: the

text: extra

text: entropy

text: emitted

text: by

text: the

text: dark

text: matter

text: particle

text: reducing

text: interactions,

text: is

text: dumped

text: into

text: the

text: normal

text: SM

text: sector

text: which

text: for

text: a

text: long

text: while

text: is

text: in

text: thermal

text: equilibrium

text: with

text: the

text: dark

text: sector.

text: Generically

text: the

text: interactions

text: producing

text: the

text: x

text: particles

text: are

text: suppressed

text: by

text: some

text: heavy

text: new

text: physics

text: scales

text: Λ

text: and

text: proceed

text: with

text: rates

text: proportional

text: to

text: T

text: )n

text: with

text: n

text: or

text: Still

text: for

text: new

text: new

text: temperatures

text: T

text: which

text: are

text: comparable

text: or

text: even

text: some-what

text: smaller

text: than

text: this

text: new

text: scale

text: the

text: x

text: particles

text: would

text: have

text: been

text: in

text: thermal

text: equilibrium.

text: A

text: well

text: known

text: example

text: is

text: provided

text: by

text: the

text: the

text: ordinary

text: left-

text: handed

text: neutrinos

text: which

text: are

text: coupled

text: to

text: all

text: the

text: remaining

text: SM

text: species

text: and

text: are

text: in

text: thermal

text: chemical

text: equilibrium

text: so

text: long

text: as

text: the

text: temperature

text: exceeds

text: O(

text: MeV).

text: This

text: is

text: because

text: the

text: relevant

text: rate

text: scales

text: as

text: T

text: T

text: with

text: v

text: eV

text: the

text: scale

text: of

text: ordinary

text: weak

text: interaction

text: falls

text: bellow

text: the

text: Fermi

text: Hubble

text: expansion

text: rate

text: T

text: lanck)

text: only

text: at

text: this

text: low

text: temperature.

text: At

text: the

text: (at

text: least!)

text: times

text: higher

text: reheat

text: temperature

text: envisioned

text: any

text: other

text: even

text: supper-weakly

text: coupled

text: species

text: such

text: as

text: sterile

text: neutrinos

text: or

text: ALPS

text: axion

text: like

text: particles)

text: would

text: have

text: to

text: be

text: thermally

text: produced.

text: This

text: in

text: turn

text: may

text: exclude

text: a

text: very

text: broad

text: range

text: of

text: scenarios

text: where

text: long

text: lived

text: x

text: particles

text: more

text: massive

text: than

text: say

text: are

text: required.

text: In

text: particular

text: it

text: would

text: negate

text: the

text: possibility

text: that

text: sterile

text: neutrinos

text: of

text: mass

text: and

text: lifetimes

text: of

text: seconds

text: generate

text: the

text: recently

text: observed

text: X

text: ray

text: line

text: We

text: should

text: emphasize

text: that

text: this

text: reasoning

text: does

text: not

text: apply

text: to

text: all

text: forms

text: of

text: light

text: long

text: lived

text: particles.

text: An

text: example

text: is

text: provided

text: by

text: the

text: dark

text: photon

text: mentioned

text: above.

text: If

text: there

text: is

text: no

text: dark

text: sector

text: and

text: the

text: lightest

text: particles

text: carrying

text: the

text: new

text: are

text: extremely

text: almost

text: Planck

text: scale

text: heavy

text: than

text: its

text: may

text: have

text: never

text: been

text: in

text: thermal

text: equilibrium.

text: Its

text: only

text: coupling

text: is

text: via

text: the

text: mixing

text: with

text: the

text: ordinary

text: photon

text: which

text: for

text: masses

text: m(A

text: MeV

text: decay

text: only

text: via

text: an

text: electron

text: box

text: diagram

text: into

text: three

text: photons

text: and

text: with

text: lifetimes

text: which

text: are

text: far

text: longer

text: than

text: Hubble

text: time.

text: This

text: is

text: so

text: because

text: for

text: such

text: light

text: dark

text: photon

text: considerations

text: related

text: to

text: the

text: cooling

text: of

text: supernovae

text: by

text: photon’

text: emission

text: strongly

text: bind

text: the

text: strength

text: of

text: its

text: mixing

text: with

text: the

text: ordinary

text: photon

text: The

text: cross-section

text: for

text: producing

text: this

text: dark

text: photon

text: e.g.

text: σγ

text: e

text: γ

text: e

text: yields

text: then

text: in

text: the

text: relevant

text: radiation

text: dominated

text: era

text: where

text: n(e)

text: T

text: a

text: rate

text: of

text: production

text: Γ

text: σn(e)

text: T

text: unlike

text: the

text: cases

text: considered

text: above

text: Γ

text: increases

text: more

text: slowly

text: with

text: temperature

text: than

text: the

text: Hubble

text: expansion

text: T

text: lanck)

text: and

text: hence

text: can

text: be

text: out

text: of

text: equilibrium

text: even

text: at

text: the

text: highest

text: possible

text: temperatures.

text: Hence

text: we

text: do

text: not

text: encounter

text: the

text: over-closure

text: diﬃculties

text: and

text: could

text: have

text: the

text: dark

text: photon

text: be

text: warm

text: dark

text: matter.


text: 22 (cid:82) − − ≤ − integrating l times by part and canceling the l! we ﬁnd that I(l) = 2 l −1 (x+1)l(1 x)l 2 l. Finally, 1 ≤ ≤ we have a (k) (k2/(µ2)l and σ (k) (k2/µ2)2l. l l [27] An example from hadron physics is the ρ resonance at 760MeV (playing the role of R) in the ππ channel, the analogs of the X¯X particles here. While the 160MeV broad resonance dominates ππ − − ∼ − scattering for W = 600 900MeV the annihilation into the light e+e pair has a branching of 10 5 due to the weak em coupling and the need to exchange a virtual photon in the s channel which is oﬀ shell by m(ρ)2. ∼ [28] one example among many is an X particle with M(X) 5PeV and lifetime of 1028s the decays of which generate the Ice-Cube PeV neutrinos. [11–13] [29] The conﬂict between conﬁnement and S matrix has been particularly emphasized by the late G. Preparata [30] This intuitive picture was pointed out to me by G. Stermann

text: integrating

text: l

text: times

text: by

text: part

text: and

text: canceling

text: the

text: l!

text: we

text: ﬁnd

text: that

text: I(l)

text: l

text: −1

text: x)l

text: l.

text: Finally,

text: we

text: have

text: a

text: (k)

text: and

text: σ

text: (k)

text: l

text: l

text: An

text: example

text: from

text: hadron

text: physics

text: is

text: the

text: ρ

text: resonance

text: at

text: (playing

text: the

text: role

text: of

text: R)

text: in

text: the

text: ππ

text: channel,

text: the

text: analogs

text: of

text: the

text: X¯X

text: particles

text: here.

text: While

text: the

text: broad

text: resonance

text: dominates

text: ππ

text: scattering

text: for

text: W

text: the

text: annihilation

text: into

text: the

text: light

text: e+e

text: pair

text: has

text: a

text: branching

text: of

text: due

text: to

text: the

text: weak

text: em

text: coupling

text: and

text: the

text: need

text: to

text: exchange

text: a

text: virtual

text: photon

text: in

text: the

text: s

text: channel

text: which

text: is

text: oﬀ

text: shell

text: by

text: one

text: example

text: among

text: many

text: is

text: an

text: X

text: particle

text: with

text: M(X)

text: and

text: lifetime

text: of

text: the

text: decays

text: of

text: which

text: generate

text: the

text: Ice-Cube

text: PeV

text: neutrinos.

text: The

text: conﬂict

text: between

text: conﬁnement

text: and

text: S

text: matrix

text: has

text: been

text: particularly

text: emphasized

text: by

text: the

text: late

text: G.

text: Preparata

text: This

text: intuitive

text: picture

text: was

text: pointed

text: out

text: to

text: me

text: by

text: G.

text: Stermann


text: 23 (a) (b) FIG. 1: (a) A resonance of a bound state in the s-channel (b) Heavy particle exchange in the t-channel FIG. 2: An “elementary” vector boson exchange in the s-channel ##LTFigure## ##LTFigure##

text: (a)

text: (b)

text: FIG.

text: (a)

text: A

text: resonance

text: of

text: a

text: bound

text: state

text: in

text: the

text: s-channel

text: (b)

text: Heavy

text: particle

text: exchange

text: in

text: the

text: t-channel

text: FIG.

text: An

text: “elementary”

text: vector

text: boson

text: exchange

text: in

text: the

text: s-channel

text: ##LTFigure##

label: figure

text: ##LTFigure##


text: 24 FIG. 3: The range for scattering and the smaller range for annihilation (cid:48) (cid:48) FIG. 4: The highly suppressed decay of the g g glueball into two photons via double box diagram ##LTFigure## ##LTFigure##

text: FIG.

text: The

text: range

text: for

text: scattering

text: and

text: the

text: smaller

text: range

text: for

text: annihilation

text: FIG.

text: The

text: highly

text: suppressed

text: decay

text: of

text: the

text: g

text: g

text: glueball

text: into

text: two

text: photons

text: via

text: double

text: box

text: diagram

text: ##LTFigure##

text: ##LTFigure##


text: 4 tivated by an attempt to dilute the density of supper-heavy GUT magnetic monopoles generated → (cid:48) at the ﬁrst stage of GUT breaking in which a U(1) subgroup splits oﬀ: G G XU(1) [6]. GUT Production of monopoles even at energies bigger than their masses is dynamically suppressed − by exp ( 1/α) type factors [7]. The fact that M(monopole) = M(GUT )/α implies that even if ≈ the monopoles were thermally produced at the highest temperatures possible T = T (Reheat) − M(GUT ) there would still be an exp ( 1/α) Boltzmann suppression. However such monopoles may be produced at the phase transition corresponding to the spontaneous breaking of GUTs via the Kibble mechanism [8] at zeros of the GUT breaking Higgs condensate. These points are the junctures of domains where the Higgs VeV points in diﬀerent directions of the gauge group. without subsequent inﬂations the primordial inﬂation at a scale of M(GUT ) = 1016GeV may not achieve − the dilution of n /nγ = 10 25 required to ensure a relic density of the monopoles less than Monopoles that of the allowed dark matter. Demanding that the monopoles will not dissipate the galactic magnetic ﬁeld Yields the much stronger Parker bound on the density of magnetic monopoles. The − capture of m m¯ pairs into “Monopolium” states with ensuing annihilation cannot achieve such a small monopole number density. We discuss related issues in Sec IV. Next in a natural hierarchy are the one dimensional cosmic strings arising in spontaneosly bro- ken U(1) symmetries. We expect that approximately one closed string per horizon of length ct = l(horizon) = l(h) forms at the time of the SSB (spontaneous symmetry breaking). The ∼ ∼ ∗ string tension or energy per unit string length is µ2, where µ T are the SSB scale and the ∗ temperature T of the corresponding phase transition. The energy density due to these strings at the time of SBB is: ≈ ρ(string) µ2l /l3 = µ2/l2. (1) h h h If the closed strings behave as CDM ( cold dark matter) particles then their mass density ρ(string) ≤ scales as T 3 and ρ(string) ρ(CDM) should hold at any temperature T .The CDM density is: ∼ ∼ × − 5ρ(Baryon) = 5η n 1GeV with η(B) 5 10 10 the baryon asymmetry. Thus B γ ≈ n = N(dof)ζ(3)/π2T 3 0.12N(dof)T 3 (2) γ with N(dof) the number of degrees of freedom of relativistic particles at the temperature T. ≥ ∼ For T T eV we take N(dof) = N(dof ) 60 and the comoving dark matter density T standardmodel is :

text: tivated

text: by

text: an

text: attempt

text: to

text: dilute

text: the

text: density

text: of

text: supper-heavy

text: GUT

text: magnetic

text: monopoles

text: generated

text: at

text: the

text: ﬁrst

text: stage

text: of

text: GUT

text: breaking

text: in

text: which

text: a

text: subgroup

text: splits

text: oﬀ:

text: G

text: G

text: GUT

text: Production

text: of

text: monopoles

text: even

text: at

text: energies

text: bigger

text: than

text: their

text: masses

text: is

text: dynamically

text: suppressed

text: by

text: exp

text: type

text: factors

text: The

text: fact

text: that

text: M(monopole)

text: M(GUT

text: )/α

text: implies

text: that

text: even

text: if

text: the

text: monopoles

text: were

text: thermally

text: produced

text: at

text: the

text: highest

text: temperatures

text: possible

text: T

text: T

text: (Reheat)

text: M(GUT

text: there

text: would

text: still

text: be

text: an

text: exp

text: Boltzmann

text: suppression.

text: However

text: such

text: monopoles

text: may

text: be

text: produced

text: at

text: the

text: phase

text: transition

text: corresponding

text: to

text: the

text: spontaneous

text: breaking

text: of

text: GUTs

text: via

text: the

text: Kibble

text: mechanism

text: at

text: zeros

text: of

text: the

text: GUT

text: breaking

text: Higgs

text: condensate.

text: These

text: points

text: are

text: the

text: junctures

text: of

text: domains

text: where

text: the

text: Higgs

text: VeV

text: points

text: in

text: diﬀerent

text: directions

text: of

text: the

text: gauge

text: group.

text: without

text: subsequent

text: inﬂations

text: the

text: primordial

text: inﬂation

text: at

text: a

text: scale

text: of

text: M(GUT

text: may

text: not

text: achieve

text: the

text: dilution

text: of

text: n

text: /nγ

text: required

text: to

text: ensure

text: a

text: relic

text: density

text: of

text: the

text: monopoles

text: less

text: than

text: Monopoles

text: that

text: of

text: the

text: allowed

text: dark

text: matter.

text: Demanding

text: that

text: the

text: monopoles

text: will

text: not

text: dissipate

text: the

text: galactic

text: magnetic

text: ﬁeld

text: Yields

text: the

text: much

text: stronger

text: Parker

text: bound

text: on

text: the

text: density

text: of

text: magnetic

text: monopoles.

text: The

text: capture

text: of

text: m

text: m¯

text: pairs

text: into

text: “Monopolium”

text: states

text: with

text: ensuing

text: annihilation

text: cannot

text: achieve

text: such

text: a

text: small

text: monopole

text: number

text: density.

text: We

text: discuss

text: related

text: issues

text: in

text: Sec

text: IV.

text: Next

text: in

text: a

text: natural

text: hierarchy

text: are

text: the

text: one

text: dimensional

text: cosmic

text: strings

text: arising

text: in

text: spontaneosly

text: bro-

text: ken

text: symmetries.

text: We

text: expect

text: that

text: approximately

text: one

text: closed

text: string

text: per

text: horizon

text: of

text: length

text: ct

text: l(horizon)

text: l(h)

text: forms

text: at

text: the

text: time

text: of

text: the

text: SSB

text: (spontaneous

text: symmetry

text: breaking).

text: The

text: string

text: tension

text: or

text: energy

text: per

text: unit

text: string

text: length

text: is

text: where

text: µ

text: T

text: are

text: the

text: SSB

text: scale

text: and

text: the

text: temperature

text: T

text: of

text: the

text: corresponding

text: phase

text: transition.

text: The

text: energy

text: density

text: due

text: to

text: these

text: strings

text: at

text: the

text: time

text: of

text: SBB

text: is:

text: ρ(string)

text: h

text: h

text: h

text: If

text: the

text: closed

text: strings

text: behave

text: as

text: CDM

text: cold

text: dark

text: matter)

text: particles

text: then

text: their

text: mass

text: density

text: ρ(string)

text: scales

text: as

text: T

text: and

text: ρ(string)

text: ρ(CDM)

text: should

text: hold

text: at

text: any

text: temperature

text: T

text: .The

text: CDM

text: density

text: is:

text: 5η

text: n

text: with

text: η(B)

text: the

text: baryon

text: asymmetry.

text: Thus

text: B

text: γ

text: n

text: γ

text: with

text: N(dof)

text: the

text: number

text: of

text: degrees

text: of

text: freedom

text: of

text: relativistic

text: particles

text: at

text: the

text: temperature

text: T.

text: For

text: T

text: T

text: eV

text: we

text: take

text: N(dof)

text: N(dof

text: and

text: the

text: comoving

text: dark

text: matter

text: density

text: T

text: standardmodel

text: is


text: 7 In the above equation α(W eak) = g2 /4π = 1/30 is the coupling strength of weak interaction W − and v = v(X) v(X¯) = 2v(X) = 2k/(M(X) is the relative velocity of the X¯ and X in their center mass frame where k is the common magnitude of the momenta. Since the scale of SUSY (super- symmetry) breaking and therefor the mass of the LSP’s ( lightest SUSY partners) identiﬁed with X, were supposed be lighter than a TeV to “naturally” explain the mass scale of weak interaction, this gave further impetus to direct and indirect WIMP searches in underground detectors, in satellites, in ICE-CUBE and for their production along with all the other SUSY partners at the LHC (large hadron collider). The lack of positive evidence kept pushing up the masses of the WIMPS, the miracle stopped operating and over closure diﬃculties were encountered. The number density n(X) = n(X¯) = n of X and X¯ particles which are in thermal equilibrium ≥ with the rest of radiation at temperatures T M(X), is exponentially suppressed once T falls ∼ − below M(X): n (exp M(X)/T ). At a temperature T = M(X)/f the annihilation rate: f.o. ∼ Γ(an) = nvσ nα2(W eak)/M(X))2 (6) an falls bellow the Hubble volume expansion rate: v˙ = 3H = 3(8π/3ρ(T ))1/2/M . (7) Planck v The X particles then cease to be in chemical equilibrium and their co-moving density freezes out. The density ρ(T ) is that appropriate for radiation dominated era at temperature T namely: ρ(T ) = N(dof)π2T 4/15. With N(dof) the eﬀective number of degrees of freedom at this temperature ∼ which we again take as N(dof) 60 Using this in Γ(ann) = 3H with T = T = M(X)/f yields f.o n(X) = 2π/51/2m(X)4/m f2α2. Demanding that the relic X density together with the equal f.o Pl density of X¯: ρ(X + X¯) = 2n(X)M(X) does not exceed the total allowed density of cold dark matter ρ(CDM) in Sec II above, yields an upper bound on M(X): ≤ − M(X) 100T eV α(1/f) 1/2N(d.o.f.)1/2(40ζ(3)/6π3.(8π/3)1/2)1/2 (8) − Detailed calculations by many authors yielded f values in the 20 30 range. Taking N to be d.o.f ∼ ≤ 60 and α 1/30 , we ﬁnd that unless M(X) T eV we have excessive relic X density. In general ##LTLine##

text: In

text: the

text: above

text: equation

text: α(W

text: eak)

text: is

text: the

text: coupling

text: strength

text: of

text: weak

text: interaction

text: W

text: and

text: v

text: v(X)

text: v(X¯)

text: is

text: the

text: relative

text: velocity

text: of

text: the

text: X¯

text: and

text: X

text: in

text: their

text: center

text: mass

text: frame

text: where

text: k

text: is

text: the

text: common

text: magnitude

text: of

text: the

text: momenta.

text: Since

text: the

text: scale

text: of

text: SUSY

text: (super-

text: symmetry)

text: breaking

text: and

text: therefor

text: the

text: mass

text: of

text: the

text: LSP’s

text: lightest

text: SUSY

text: partners)

text: identiﬁed

text: with

text: X,

text: were

text: supposed

text: be

text: lighter

text: than

text: a

text: TeV

text: to

text: “naturally”

text: explain

text: the

text: mass

text: scale

text: of

text: weak

text: interaction,

text: this

text: gave

text: further

text: impetus

text: to

text: direct

text: and

text: indirect

text: WIMP

text: searches

text: in

text: underground

text: detectors,

text: in

text: satellites,

text: in

text: ICE-CUBE

text: and

text: for

text: their

text: production

text: along

text: with

text: all

text: the

text: other

text: SUSY

text: partners

text: at

text: the

text: LHC

text: (large

text: hadron

text: collider).

text: The

text: lack

text: of

text: positive

text: evidence

text: kept

text: pushing

text: up

text: the

text: masses

text: of

text: the

text: WIMPS,

text: the

text: miracle

text: stopped

text: operating

text: and

text: over

text: closure

text: diﬃculties

text: were

text: encountered.

text: The

text: number

text: density

text: n(X)

text: n(X¯)

text: n

text: of

text: X

text: and

text: X¯

text: particles

text: which

text: are

text: in

text: thermal

text: equilibrium

text: with

text: the

text: rest

text: of

text: radiation

text: at

text: temperatures

text: T

text: M(X),

text: is

text: exponentially

text: suppressed

text: once

text: T

text: falls

text: below

text: M(X):

text: n

text: (exp

text: M(X)/T

text: At

text: a

text: temperature

text: T

text: M(X)/f

text: the

text: annihilation

text: rate:

text: f.o.

text: Γ(an)

text: nvσ

text: an

text: falls

text: bellow

text: the

text: Hubble

text: volume

text: expansion

text: rate:

text: v˙

text: Planck

text: v

text: The

text: X

text: particles

text: then

text: cease

text: to

text: be

text: in

text: chemical

text: equilibrium

text: and

text: their

text: co-moving

text: density

text: freezes

text: out.

text: The

text: density

text: ρ(T

text: is

text: that

text: appropriate

text: for

text: radiation

text: dominated

text: era

text: at

text: temperature

text: T

text: namely:

text: ρ(T

text: With

text: N(dof)

text: the

text: eﬀective

text: number

text: of

text: degrees

text: of

text: freedom

text: at

text: this

text: temperature

text: which

text: we

text: again

text: take

text: as

text: N(dof)

text: Using

text: this

text: in

text: Γ(ann)

text: with

text: T

text: T

text: M(X)/f

text: yields

text: f.o

text: n(X)

text: Demanding

text: that

text: the

text: relic

text: X

text: density

text: together

text: with

text: the

text: equal

text: f.o

text: Pl

text: density

text: of

text: X¯:

text: ρ(X

text: X¯)

text: does

text: not

text: exceed

text: the

text: total

text: allowed

text: density

text: of

text: cold

text: dark

text: matter

text: ρ(CDM)

text: in

text: Sec

text: II

text: above,

text: yields

text: an

text: upper

text: bound

text: on

text: M(X):

text: M(X)

text: eV

text: Detailed

text: calculations

text: by

text: many

text: authors

text: yielded

text: f

text: values

text: in

text: the

text: range.

text: Taking

text: N

text: to

text: be

text: d.o.f

text: and

text: α

text: we

text: ﬁnd

text: that

text: unless

text: M(X)

text: T

text: eV

text: we

text: have

text: excessive

text: relic

text: X

text: density.

text: In

text: general

text: ##LTLine##


text: 10 with n(X) and n(X¯) the number densities of the X and X¯ particles with momenta k(1) and k(2) and W(1,2), v(1,2) the total energy and relative velocity in their center mass frame. In the thermal environment W (1, 2) is spread over ∆(W ) = T . For the annihilation cross-section σ (W (1, 2)) to be large over the full energy range the resonance R has to be suﬃciently broad: ann ∼ ∼ Γ ∆(W ) (T ). Using T = T = M(X)/f the condition becomes: Γ > M(X)/f. Broad f.o resonances naturally arise if X¯ and X strongly interact with each other though a near threshold bound state pole or resonance requires tuning. The total width is the sum of the width to decay → back to the original X¯X and the decay widths into the annihilation channels R ¯ii. The ﬁrst → − ∼ width γ(R) X¯X = g(R, X¯X)2k/8π is suppressed by k/M(X) = v(X) = f 1/2 1/5. The third ingredient needed to ensure large annihilation cross-section are large decay widths Γ(i) = g2(R;¯ii)M(X). These decay widths are not suppressed by kinematics and jointly can contribute half the the total width maximizing annihilation cross-section. However we claim that in most natural scenarios these are strongly suppressed dynamically a feature best seen from the t channel prospective (ii). (ii) t channel induced large cross-section and their limitation in the case of annihilation. Exchanging mass-less particles in the t channel generates inﬁnite elastic scattering. This follows from the inﬁnite range of the corresponding potential and the Rutherford scattering via photon exchange is a prime example. Indeed while unitarity limits the cross-section con- (cid:80) tributed by each individual partial wave the sum over the partial waves: A(s, t) = A(s; cos(θ) = (2L + 1)a (s)P (cos(θ)) with θ the center of mass scattering angle could diverge. The in- L L variants made up of squares of the sums of four-momenta entering in the diﬀerent channels : − − s = (p(X + p(X¯))2 = W 2, t = (p(X) p(i))2 and u = ((p(X) p(¯i))2 are often used in discussing → 2 2 amplitudes. For ﬁnite mass µ of the exchanged particle the elastic amplitude A(s; cos(θ)) is − analytic in the “Lehman Martin” ellipse in the cos(θ) plane with foci at +1, 1 and a semi-major axis 1 + t /2k2 with t = µ2 the nearest t channel singularity and the sum converges therein. 0 0 ≈ A detailed analysis yields an exponential decrease of the higher partial waves a (s) (k/µ)2l j [? ]. The extra velocity factor suppression of the p annihilation due to the p wave required ∼ − for Majorana relics is well known. Still the elastic cross-section µ 2 are large for small µ. → However, the X¯X ¯ii annihilation cross-sections of interest have no t channel enhancement. This follows from both kinematics and dynamics. We are interested in annihilation near threshold where ∼ ≈ ≈ ∼ W = s1/2 2M(X). At this point k 0 and t u. Recalling that s+t+u = 2M(X)2 +2m(i)2 ≈ ≈ − 2M(X)2 and s 4M(X)2, we ﬁnd that t M(X)2. Thus, the annihilation amplitude is

text: with

text: n(X)

text: and

text: n(X¯)

text: the

text: number

text: densities

text: of

text: the

text: X

text: and

text: X¯

text: particles

text: with

text: momenta

text: and

text: and

text: the

text: total

text: energy

text: and

text: relative

text: velocity

text: in

text: their

text: center

text: mass

text: frame.

text: In

text: the

text: thermal

text: environment

text: W

text: is

text: spread

text: over

text: ∆(W

text: T

text: For

text: the

text: annihilation

text: cross-section

text: σ

text: (W

text: to

text: be

text: large

text: over

text: the

text: full

text: energy

text: range

text: the

text: resonance

text: R

text: has

text: to

text: be

text: suﬃciently

text: broad:

text: ann

text: Γ

text: ∆(W

text: (T

text: Using

text: T

text: T

text: M(X)/f

text: the

text: condition

text: becomes:

text: Γ

text: M(X)/f.

text: Broad

text: f.o

text: resonances

text: naturally

text: arise

text: if

text: X¯

text: and

text: X

text: strongly

text: interact

text: with

text: each

text: other

text: though

text: a

text: near

text: threshold

text: bound

text: state

text: pole

text: or

text: resonance

text: requires

text: tuning.

text: The

text: total

text: width

text: is

text: the

text: sum

text: of

text: the

text: width

text: to

text: decay

text: back

text: to

text: the

text: original

text: X¯X

text: and

text: the

text: decay

text: widths

text: into

text: the

text: annihilation

text: channels

text: R

text: ¯ii.

text: The

text: ﬁrst

text: width

text: γ(R)

text: X¯X

text: g(R,

text: is

text: suppressed

text: by

text: k/M(X)

text: v(X)

text: f

text: The

text: third

text: ingredient

text: needed

text: to

text: ensure

text: large

text: annihilation

text: cross-section

text: are

text: large

text: decay

text: widths

text: Γ(i)

text: These

text: decay

text: widths

text: are

text: not

text: suppressed

text: by

text: kinematics

text: and

text: jointly

text: can

text: contribute

text: half

text: the

text: the

text: total

text: width

text: maximizing

text: annihilation

text: cross-section.

text: However

text: we

text: claim

text: that

text: in

text: most

text: natural

text: scenarios

text: these

text: are

text: strongly

text: suppressed

text: dynamically

text: a

text: feature

text: best

text: seen

text: from

text: the

text: t

text: channel

text: prospective

text: (ii).

text: (ii)

text: t

text: channel

text: induced

text: large

text: cross-section

text: and

text: their

text: limitation

text: in

text: the

text: case

text: of

text: annihilation.

text: Exchanging

text: mass-less

text: particles

text: in

text: the

text: t

text: channel

text: generates

text: inﬁnite

text: elastic

text: scattering.

text: This

text: follows

text: from

text: the

text: inﬁnite

text: range

text: of

text: the

text: corresponding

text: potential

text: and

text: the

text: Rutherford

text: scattering

text: via

text: photon

text: exchange

text: is

text: a

text: prime

text: example.

text: Indeed

text: while

text: unitarity

text: limits

text: the

text: cross-section

text: con-

text: tributed

text: by

text: each

text: individual

text: partial

text: wave

text: the

text: sum

text: over

text: the

text: partial

text: waves:

text: A(s,

text: t)

text: A(s;

text: cos(θ)

text: (s)P

text: (cos(θ))

text: with

text: θ

text: the

text: center

text: of

text: mass

text: scattering

text: angle

text: could

text: diverge.

text: The

text: in-

text: L

text: L

text: variants

text: made

text: up

text: of

text: squares

text: of

text: the

text: sums

text: of

text: four-momenta

text: entering

text: in

text: the

text: diﬀerent

text: channels

text: s

text: (p(X

text: W

text: t

text: (p(X)

text: and

text: u

text: ((p(X)

text: are

text: often

text: used

text: in

text: discussing

text: amplitudes.

text: For

text: ﬁnite

text: mass

text: µ

text: of

text: the

text: exchanged

text: particle

text: the

text: elastic

text: amplitude

text: A(s;

text: cos(θ))

text: is

text: analytic

text: in

text: the

text: “Lehman

text: Martin”

text: ellipse

text: in

text: the

text: cos(θ)

text: plane

text: with

text: foci

text: at

text: and

text: a

text: semi-major

text: axis

text: t

text: with

text: t

text: µ2

text: the

text: nearest

text: t

text: channel

text: singularity

text: and

text: the

text: sum

text: converges

text: therein.

text: A

text: detailed

text: analysis

text: yields

text: an

text: exponential

text: decrease

text: of

text: the

text: higher

text: partial

text: waves

text: a

text: (s)

text: j

text: The

text: extra

text: velocity

text: factor

text: suppression

text: of

text: the

text: p

text: annihilation

text: due

text: to

text: the

text: p

text: wave

text: required

text: for

text: Majorana

text: relics

text: is

text: well

text: known.

text: Still

text: the

text: elastic

text: cross-section

text: µ

text: are

text: large

text: for

text: small

text: µ.

text: However,

text: the

text: X¯X

text: ¯ii

text: annihilation

text: cross-sections

text: of

text: interest

text: have

text: no

text: t

text: channel

text: enhancement.

text: This

text: follows

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text: both

text: kinematics

text: and

text: dynamics.

text: We

text: are

text: interested

text: in

text: annihilation

text: near

text: threshold

text: where

text: W

text: At

text: this

text: point

text: k

text: and

text: t

text: u.

text: Recalling

text: that

text: s+t+u

text: and

text: s

text: we

text: ﬁnd

text: that

text: t

text: Thus,

text: the

text: annihilation

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text: is


text: 1 1 2 1 2 3 4 1 2 3 4 5 6 7 Figure 2: Topology used in experiments { } of D = 4 and sequence length of T = 30, state size of 1, 2, 4, 7 for the four semantic levels, respectively. At the bottom level, 3 observation symbols that are outputted in a generative manner. We generate 50 training sequences for learning and 50 for testing. First, we learn the HSCRF parameters from the training data. Note that the HSCRF is discriminative, the learned model may not correspond to the original HHMM that generates the data. Given the learned HSCRF, we perform various inference tasks with the testing data. We compare the results of Rao-Blackwellised Gibbs sampling introduced in this paper against the exact inference using the Inside-Outside algorithm described in [17]. For sampling, ﬁrst we run the sampler for a ‘burn-in’ period of time, and then discard those samples. The purpose of this practice is to let the sampler ‘forget’ about the starting point to eliminate some possible bias. The burn-in time is set to about 10% of the total iterations. We want to examine the accuracy of the proposed approximation against the exact | methods, along the three dimensions: (i) the state marginals P(xd o) at time t and depth | t level d, (ii) the probability P(e o) that a transition occurs at time t, and (iii) the decoding t | accuracy using xˆd = arg max P(xd o). For the ﬁrst two estimations, two accuracy t xd t t measures are used: a) the average Kullback-Leibler divergence between state marginals and b) the average absolute diﬀerence in state marginals. For decoding evaluation, we use the percentage of matching between the maximal states. Convergence behaviours for a ﬁxed sequence length: Figures 3 and 4 show the three measures using the one-long-run strategy (up to 5000 iterations). It appears that both the KL-divergence and absolute diﬀerence continue to decrease, although it occurs at an extremely slow rate after 500 iterations. The maximal state measure is interesting: it quickly reaches the high accuracy after just 10 iterations. In other words, the mode of | the state marginal P(xd o) is quickly located although the marginal has yet to converge. t Convergence as a function of sequence length: To test how the RBGS scales, three experiments with diﬀerent run times are performed: • The number of iterations is ﬁxed at 100 for all sequence lengths. This gives us a linear run time in term of T. • O The number of iterations scales linearly with T, and thus the total cost is (T2). • O The number of iterations scales quadratically with T. This costs (T3), which means no saving compared to the exact inference. ##LTFigure## ##LTLine## ##LTLine##

text: Figure

text: Topology

text: used

text: in

text: experiments

text: of

text: D

text: and

text: sequence

text: length

text: of

text: T

text: state

text: size

text: of

text: for

text: the

text: four

text: semantic

text: levels,

text: respectively.

text: At

text: the

text: bottom

text: level,

text: observation

text: symbols

text: that

text: are

text: outputted

text: in

text: a

text: generative

text: manner.

text: We

text: generate

text: training

text: sequences

text: for

text: learning

text: and

text: for

text: testing.

text: First,

text: we

text: learn

text: the

text: HSCRF

text: parameters

text: from

text: the

text: training

text: data.

text: Note

text: that

text: the

text: HSCRF

text: is

text: discriminative,

text: the

text: learned

text: model

text: may

text: not

text: correspond

text: to

text: the

text: original

text: HHMM

text: that

text: generates

text: the

text: data.

text: Given

text: the

text: learned

text: HSCRF,

text: we

text: perform

text: various

text: inference

text: tasks

text: with

text: the

text: testing

text: data.

text: We

text: compare

text: the

text: results

text: of

text: Rao-Blackwellised

text: Gibbs

text: sampling

text: introduced

text: in

text: this

text: paper

text: against

text: the

text: exact

text: inference

text: using

text: the

text: Inside-Outside

text: algorithm

text: described

text: in

text: For

text: sampling,

text: ﬁrst

text: we

text: run

text: the

text: sampler

text: for

text: a

text: ‘burn-in’

text: period

text: of

text: time,

text: and

text: then

text: discard

text: those

text: samples.

text: The

text: purpose

text: of

text: this

text: practice

text: is

text: to

text: let

text: the

text: sampler

text: ‘forget’

text: about

text: the

text: starting

text: point

text: to

text: eliminate

text: some

text: possible

text: bias.

text: The

text: burn-in

text: time

text: is

text: set

text: to

text: about

text: of

text: the

text: total

text: iterations.

text: We

text: want

text: to

text: examine

text: the

text: accuracy

text: of

text: the

text: proposed

text: approximation

text: against

text: the

text: exact

text: methods,

text: along

text: the

text: three

text: dimensions:

text: (i)

text: the

text: state

text: marginals

text: P(xd

text: o)

text: at

text: time

text: t

text: and

text: depth

text: t

text: level

text: d,

text: (ii)

text: the

text: probability

text: P(e

text: o)

text: that

text: a

text: transition

text: occurs

text: at

text: time

text: t,

text: and

text: (iii)

text: the

text: decoding

text: t

text: accuracy

text: using

text: xˆd

text: arg

text: max

text: P(xd

text: o).

text: For

text: the

text: ﬁrst

text: two

text: estimations,

text: two

text: accuracy

text: t

text: xd

text: t

text: t

text: measures

text: are

text: used:

text: a)

text: the

text: average

text: Kullback-Leibler

text: divergence

text: between

text: state

text: marginals

text: and

text: b)

text: the

text: average

text: absolute

text: diﬀerence

text: in

text: state

text: marginals.

text: For

text: decoding

text: evaluation,

text: we

text: use

text: the

text: percentage

text: of

text: matching

text: between

text: the

text: maximal

text: states.

text: Convergence

text: behaviours

text: for

text: a

text: ﬁxed

text: sequence

text: length:

text: Figures

text: and

text: show

text: the

text: three

text: measures

text: using

text: the

text: one-long-run

text: strategy

text: (up

text: to

text: iterations).

text: It

text: appears

text: that

text: both

text: the

text: KL-divergence

text: and

text: absolute

text: diﬀerence

text: continue

text: to

text: decrease,

text: although

text: it

text: occurs

text: at

text: an

text: extremely

text: slow

text: rate

text: after

text: iterations.

text: The

text: maximal

text: state

text: measure

text: is

text: interesting:

text: it

text: quickly

text: reaches

text: the

text: high

text: accuracy

text: after

text: just

text: iterations.

text: In

text: other

text: words,

text: the

text: mode

text: of

text: the

text: state

text: marginal

text: P(xd

text: o)

text: is

text: quickly

text: located

text: although

text: the

text: marginal

text: has

text: yet

text: to

text: converge.

text: t

text: Convergence

text: as

text: a

text: function

text: of

text: sequence

text: length:

text: To

text: test

text: how

text: the

text: RBGS

text: scales,

text: three

text: experiments

text: with

text: diﬀerent

text: run

text: times

text: are

text: performed:

text: The

text: number

text: of

text: iterations

text: is

text: ﬁxed

text: at

text: for

text: all

text: sequence

text: lengths.

text: This

text: gives

text: us

text: a

text: linear

text: run

text: time

text: in

text: term

text: of

text: T.

text: O

text: The

text: number

text: of

text: iterations

text: scales

text: linearly

text: with

text: T,

text: and

text: thus

text: the

text: total

text: cost

text: is

text: O

text: The

text: number

text: of

text: iterations

text: scales

text: quadratically

text: with

text: T.

text: This

text: costs

text: which

text: means

text: no

text: saving

text: compared

text: to

text: the

text: exact

text: inference.

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##