text: Note that even when f is a total function, f is always a partial function. See Section for sab more discussion and motivation for this deﬁnition. Now that we have deﬁned f , we can deﬁne sab deterministic and randomized sabotage complexity. Deﬁnition 10. Let f be a partial function. Then DS(f) := D(f ) and RS(f) := R (f ). sab 0 sab We will primarily focus on RS(f) in this work and only discuss DS(f) in Section . To justify deﬁning RS(f) as R (f ) instead of R(f ) (or R(f )), we now show these deﬁnitions are 0 sab sab sab equivalent up to constant factors. ≥ ≥ − Theorem 11. Let f be a partial function. Then R (f ) R (f ) (1 2(cid:15)) R (f ). 0 sab (cid:15) sab 0 sab ∈ Proof. The ﬁrst inequality follows trivially. For the second, let x Q be any valid input to (cid:48) f f . Let x be the input x with asterisks replaced with obelisks and vice versa. Then since sab (cid:54) (cid:48) f (x) = f (x ), by Lemma 8 any (cid:15)-error randomized algorithm that solves f must ﬁnd a sab sab (cid:48) − sab position on which x and x diﬀer with probability at least 1 2(cid:15). The positions at which they diﬀer are either asterisks or obelisks. Since x was an arbitrary input, the algorithm must always − ﬁnd an asterisk or obelisk with probability at least 1 2(cid:15). Since ﬁnding an asterisk or obelisk is a − certiﬁcate for f , by Lemma 7, we get a zero-error algorithm for f that uses R (f )/(1 2(cid:15)) sab ≤ − sab (cid:15) sab expected queries. Thus R (f ) R (f )/(1 2(cid:15)), as desired. 0 sab (cid:15) sab Finally, we prove that RS(f) is indeed a lower bound on R(f), i.e., R(f) = Ω(RS(f)). ≥ ≥ − Theorem 12. Let f be a partial function. Then R (f) R (f) (1 2(cid:15)) RS(f). (cid:15) (cid:15) Proof. Let A be a randomized algorithm for f that uses R (f) randomized queries and outputs the (cid:15) − correct answer on every input in Dom(f) with probability at least 1 (cid:15). Now ﬁx a sabotaged input ∗ † x, and let p be the probability that A ﬁnds a or when run on x. Let q be the probability that ∗ † A outputs 0 if it doesn’t ﬁnd a or when run on x. Let x and x be inputs consistent with x 0 1 − such that f(x ) = 0 and f(x ) = 1. Then A outputs 0 on x with probability at least q(1 p), 0 1 − − 1 and A outputs 1 on x with probability at least (1 q)(1 p). These are both errors, so we have − ≤ − 0 − ≤ − ≤ ≥ − q(1 p) (cid:15) and (1 q)(1 p) (cid:15). Summing them gives 1 p 2(cid:15), or p 1 2(cid:15). ∗ − This means A ﬁnds a entry within R (f) expected queries with probability at least 1 2(cid:15). By ≥ (cid:15) ≥ − Lemma 7, we get −1 R (f) RS(f), or R (f) (1 2(cid:15)) RS(f). 1 2(cid:15) (cid:15) (cid:15) We also deﬁne a variant of RS where the number of asterisks (or obelisks) is limited to one. ⊆ { ∗ †} ∗ † Speciﬁcally, let U 0, 1, , n be the set of all partial assignments with exactly one or . { ∈ { ∗ †} |{ ∈ ∈ { }}| } Formally, U := x 0, 1, , n : i [n] : x / 0, 1 = 1 . i Deﬁnition 13. Let f be an n-bit partial function. We deﬁne f as the restriction of f to usab ∩ sab U, the set of strings with only one asterisk or obelisk. That is, f has domain Q U, but is usab ∩ f equal to f on its domain. We then deﬁne RS (f) := R (f ). If Q U is empty, we deﬁne sab u 0 usab f RS (f) := 0. u The measure RS will play a key role in our lifting result in Section . Since f is a restriction u usab of f to a promise, it is clear that its zero-error randomized query complexity cannot increase, sab ≤ and so RS (f) RS(f). Interestingly, when f is total, RS (f) equals RS(f). In other words, when u u f is total, we may assume without loss of generality that its sabotaged version has only one asterisk or obelisk. Theorem 14. If f is a total function, then RS (f) = RS(f). u 7 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Note

text: that

text: even

text: when

text: f

text: is

text: a

text: total

text: function,

text: f

text: is

text: always

text: a

text: partial

text: function.

text: See

text: Section

text: for

text: sab

text: more

text: discussion

text: and

text: motivation

text: for

text: this

text: deﬁnition.

text: Now

text: that

text: we

text: have

text: deﬁned

text: f

text: we

text: can

text: deﬁne

text: sab

text: deterministic

text: and

text: randomized

text: sabotage

text: complexity.

text: Deﬁnition

text: Let

text: f

text: be

text: a

text: partial

text: function.

text: Then

text: DS(f)

text: D(f

text: and

text: RS(f)

text: R

text: (f

text: sab

text: sab

text: We

text: will

text: primarily

text: focus

text: on

text: RS(f)

text: in

text: this

text: work

text: and

text: only

text: discuss

text: DS(f)

text: in

text: Section

text: To

text: justify

text: deﬁning

text: RS(f)

text: as

text: R

text: (f

text: instead

text: of

text: R(f

text: (or

text: R(f

text: we

text: now

text: show

text: these

text: deﬁnitions

text: are

text: sab

text: sab

text: sab

text: equivalent

text: up

text: to

text: constant

text: factors.

text: Theorem

text: Let

text: f

text: be

text: a

text: partial

text: function.

text: Then

text: R

text: (f

text: R

text: (f

text: R

text: (f

text: sab

text: sab

text: sab

text: Proof.

text: The

text: ﬁrst

text: inequality

text: follows

text: trivially.

text: For

text: the

text: second,

text: let

text: x

text: Q

text: be

text: any

text: valid

text: input

text: to

text: f

text: f

text: Let

text: x

text: be

text: the

text: input

text: x

text: with

text: asterisks

text: replaced

text: with

text: obelisks

text: and

text: vice

text: versa.

text: Then

text: since

text: sab

text: f

text: (x)

text: f

text: (x

text: by

text: Lemma

text: any

text: randomized

text: algorithm

text: that

text: solves

text: f

text: must

text: ﬁnd

text: a

text: sab

text: sab

text: sab

text: position

text: on

text: which

text: x

text: and

text: x

text: diﬀer

text: with

text: probability

text: at

text: least

text: The

text: positions

text: at

text: which

text: they

text: diﬀer

text: are

text: either

text: asterisks

text: or

text: obelisks.

text: Since

text: x

text: was

text: an

text: arbitrary

text: input,

text: the

text: algorithm

text: must

text: always

text: ﬁnd

text: an

text: asterisk

text: or

text: obelisk

text: with

text: probability

text: at

text: least

text: Since

text: ﬁnding

text: an

text: asterisk

text: or

text: obelisk

text: is

text: a

text: certiﬁcate

text: for

text: f

text: by

text: Lemma

text: we

text: get

text: a

text: zero-error

text: algorithm

text: for

text: f

text: that

text: uses

text: R

text: (f

text: sab

text: sab

text: sab

text: expected

text: queries.

text: Thus

text: R

text: (f

text: R

text: (f

text: as

text: desired.

text: sab

text: sab

text: Finally,

text: we

text: prove

text: that

text: RS(f)

text: is

text: indeed

text: a

text: lower

text: bound

text: on

text: R(f),

text: i.e.,

text: R(f)

text: Ω(RS(f)).

text: Theorem

text: Let

text: f

text: be

text: a

text: partial

text: function.

text: Then

text: R

text: (f)

text: R

text: (f)

text: RS(f).

text: Proof.

text: Let

text: A

text: be

text: a

text: randomized

text: algorithm

text: for

text: f

text: that

text: uses

text: R

text: (f)

text: randomized

text: queries

text: and

text: outputs

text: the

text: correct

text: answer

text: on

text: every

text: input

text: in

text: Dom(f)

text: with

text: probability

text: at

text: least

text: Now

text: ﬁx

text: a

text: sabotaged

text: input

text: x,

text: and

text: let

text: p

text: be

text: the

text: probability

text: that

text: A

text: ﬁnds

text: a

text: or

text: when

text: run

text: on

text: x.

text: Let

text: q

text: be

text: the

text: probability

text: that

text: A

text: outputs

text: if

text: it

text: doesn’t

text: ﬁnd

text: a

text: or

text: when

text: run

text: on

text: x.

text: Let

text: x

text: and

text: x

text: be

text: inputs

text: consistent

text: with

text: x

text: such

text: that

text: f(x

text: and

text: f(x

text: Then

text: A

text: outputs

text: on

text: x

text: with

text: probability

text: at

text: least

text: p),

text: and

text: A

text: outputs

text: on

text: x

text: with

text: probability

text: at

text: least

text: p).

text: These

text: are

text: both

text: errors,

text: so

text: we

text: have

text: p)

text: and

text: p)

text: Summing

text: them

text: gives

text: p

text: or

text: p

text: This

text: means

text: A

text: ﬁnds

text: a

text: entry

text: within

text: R

text: (f)

text: expected

text: queries

text: with

text: probability

text: at

text: least

text: By

text: Lemma

text: we

text: get

text: −1

text: R

text: (f)

text: RS(f),

text: or

text: R

text: (f)

text: RS(f).

text: We

text: also

text: deﬁne

text: a

text: variant

text: of

text: RS

text: where

text: the

text: number

text: of

text: asterisks

text: (or

text: obelisks)

text: is

text: limited

text: to

text: one.

text: Speciﬁcally,

text: let

text: U

text: n

text: be

text: the

text: set

text: of

text: all

text: partial

text: assignments

text: with

text: exactly

text: one

text: or

text: Formally,

text: U

text: x

text: n

text: i

text: [n]

text: x

text: i

text: Deﬁnition

text: Let

text: f

text: be

text: an

text: n-bit

text: partial

text: function.

text: We

text: deﬁne

text: f

text: as

text: the

text: restriction

text: of

text: f

text: to

text: usab

text: sab

text: U,

text: the

text: set

text: of

text: strings

text: with

text: only

text: one

text: asterisk

text: or

text: obelisk.

text: That

text: is,

text: f

text: has

text: domain

text: Q

text: U,

text: but

text: is

text: usab

text: f

text: equal

text: to

text: f

text: on

text: its

text: domain.

text: We

text: then

text: deﬁne

text: RS

text: (f)

text: R

text: (f

text: If

text: Q

text: U

text: is

text: empty,

text: we

text: deﬁne

text: sab

text: u

text: usab

text: f

text: RS

text: (f)

text: u

text: The

text: measure

text: RS

text: will

text: play

text: a

text: key

text: role

text: in

text: our

text: lifting

text: result

text: in

text: Section

text: Since

text: f

text: is

text: a

text: restriction

text: u

text: usab

text: of

text: f

text: to

text: a

text: promise,

text: it

text: is

text: clear

text: that

text: its

text: zero-error

text: randomized

text: query

text: complexity

text: cannot

text: increase,

text: sab

text: and

text: so

text: RS

text: (f)

text: RS(f).

text: Interestingly,

text: when

text: f

text: is

text: total,

text: RS

text: (f)

text: equals

text: RS(f).

text: In

text: other

text: words,

text: when

text: u

text: u

text: f

text: is

text: total,

text: we

text: may

text: assume

text: without

text: loss

text: of

text: generality

text: that

text: its

text: sabotaged

text: version

text: has

text: only

text: one

text: asterisk

text: or

text: obelisk.

text: Theorem

text: If

text: f

text: is

text: a

text: total

text: function,

text: then

text: RS

text: (f)

text: RS(f).

text: u

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ Particle creation by peak electric ﬁeld † ‡ § T. C. Adornoa, S. P. Gavrilova,c, and D. M. Gitmana,b,d ¶ ∗∗ (Dated: May 29, 2020) Abstract The particle creation by the so-called peak electric ﬁeld is considered. The latter ﬁeld is a combi- nation of two exponential parts, one exponentially increasing and another exponentially decreasing. We ﬁnd exact solutions of the Dirac equation with the ﬁeld under consideration with appropriate asymptotic conditions and calculate all the characteristics of particle creation eﬀect, in particular, diﬀerential mean numbers of created particle, total number of created particles, and the probability for a vacuum to remain a vacuum. Characteristic asymptotic regimes are discussed in detail and a comparison with the pure asymptotically decaying ﬁeld is considered. PACS numbers: Keywords: ∗ † tg.adorno@mail.tsu.ru, tg.adorno@gmail.com ‡ gavrilovsergeyp@yahoo.com, gavrilovsp@herzen.spb.ru § gitman@if.usp.br ¶ ∗∗ 1 ##LTLine##

label: paragraph

text: Particle

label: title

text: creation

text: by

text: peak

text: electric

text: ﬁeld

text: T.

label: author

text: C.

text: Adornoa,

text: S.

text: P.

text: Gavrilova,c,

text: and

text: D.

text: M.

text: Gitmana,b,d

text: (Dated:

text: May

label: date

text: Abstract

text: The

label: abstract

text: particle

text: creation

text: by

text: the

text: so-called

text: peak

text: electric

text: ﬁeld

text: is

text: considered.

text: The

text: latter

text: ﬁeld

text: is

text: a

text: combi-

text: nation

text: of

text: two

text: exponential

text: parts,

text: one

text: exponentially

text: increasing

text: and

text: another

text: exponentially

text: decreasing.

text: We

text: ﬁnd

text: exact

text: solutions

text: of

text: the

text: Dirac

text: equation

text: with

text: the

text: ﬁeld

text: under

text: consideration

text: with

text: appropriate

text: asymptotic

text: conditions

text: and

text: calculate

text: all

text: the

text: characteristics

text: of

text: particle

text: creation

text: eﬀect,

text: in

text: particular,

text: diﬀerential

text: mean

text: numbers

text: of

text: created

text: particle,

text: total

text: number

text: of

text: created

text: particles,

text: and

text: the

text: probability

text: for

text: a

text: vacuum

text: to

text: remain

text: a

text: vacuum.

text: Characteristic

text: asymptotic

text: regimes

text: are

text: discussed

text: in

text: detail

text: and

text: a

text: comparison

text: with

text: the

text: pure

text: asymptotically

text: decaying

text: ﬁeld

text: is

text: considered.

text: PACS

text: numbers:

text: Keywords:

text: tg.adorno@mail.tsu.ru,

text: tg.adorno@gmail.com

text: gavrilovsergeyp@yahoo.com,

text: gavrilovsp@herzen.spb.ru

text: gitman@if.usp.br

text: ##LTLine##


text: constant electric ﬁeld [additional to the known one-parameter regularizations by the Sauter- − like electric ﬁeld, E cosh 2 (t/T ), and the T -constant electric ﬁeld (an electric ﬁeld which S eﬀectively acts during a suﬃciently large but ﬁnite time interval )]. Let us consider only T this case, supposing that and are suﬃciently small, obeying the conditions k k 1 2 (cid:0) (cid:1) (cid:29) min (h , h ) max 1, m2/eE . (3.1) 1 2 Let us analyze how the numbers Ncr depend on the parameters p and π⊥. It can be (cid:0) (cid:1) n x seen from semiclassical analysis that Ncr is exponentially small in the range of very large n (cid:38) − − π⊥ min eEk 1, eEk 1 . Then the range of ﬁxed π⊥ is of interest, and in what follows, 1 2 we assume that √ π⊥2 λ < K⊥ , λ = , (3.2) eE where K⊥ is any given number satisfying the condition (cid:0) (cid:1) (cid:29) (cid:29) min (h , h ) K⊥2 max 1, m2/eE . (3.3) 1 2 By virtue of symmetry properties of Ncr discussed above, one can only consider p either n x −∞ ≤ positive or negative. Let us, for example, consider the interval . In this case < p 0 x − ≥ is negative and large, , while varies from positive to negative values, π π eE/k π 2 2 2 1 −∞ ≤ − . The case of large negative , , where is any given < π eE/k π 2π /k > K K 1 1 1 1 1 1 1 (cid:29) large number, K K⊥, is quite simple. In this case, using the appropriate asymptotic 1 expressions of the conﬂuent hypergeometric function one can see that Ncr is negligibly small. n (cid:38) − To see this, Eq. (A11) in Appendix A is useful in the range and the h 2π /k > K 1 1 1 1 − expression for large with ﬁxed and and the expression for large with ﬁxed c a h c a c 2 2 2 1 1 1 − (cid:29) and , given in [41], are useful in the range . h 2π /k h 1 1 1 1 We expect a signiﬁcant contribution in the range ≥ − (3.4) h 2π /k > K , 1 1 1 1 that can be divided in four subranges 12 ##LTLine## ##LTLine##

text: constant

text: electric

text: ﬁeld

text: [additional

text: to

text: the

text: known

text: one-parameter

text: regularizations

text: by

text: the

text: Sauter-

text: like

text: electric

text: ﬁeld,

text: E

text: cosh

text: (t/T

text: and

text: the

text: T

text: -constant

text: electric

text: ﬁeld

text: (an

text: electric

text: ﬁeld

text: which

text: S

text: eﬀectively

text: acts

text: during

text: a

text: suﬃciently

text: large

text: but

text: ﬁnite

text: time

text: interval

text: Let

text: us

text: consider

text: only

text: T

text: this

text: case,

text: supposing

text: that

text: and

text: are

text: suﬃciently

text: small,

text: obeying

text: the

text: conditions

text: k

text: k

label: equation

text: min

text: (h

text: h

text: max

text: Let

text: us

text: analyze

text: how

text: the

text: numbers

text: Ncr

text: depend

text: on

text: the

text: parameters

text: p

text: and

text: π⊥.

text: It

text: can

text: be

text: n

text: x

text: seen

text: from

text: semiclassical

text: analysis

text: that

text: Ncr

text: is

text: exponentially

text: small

text: in

text: the

text: range

text: of

text: very

text: large

text: n

text: π⊥

text: min

text: eEk

text: eEk

text: Then

text: the

text: range

text: of

text: ﬁxed

text: π⊥

text: is

text: of

text: interest,

text: and

text: in

text: what

text: follows,

text: we

text: assume

text: that

text: π⊥2

text: λ

text: K⊥

text: λ

text: eE

text: where

text: K⊥

text: is

text: any

text: given

text: number

text: satisfying

text: the

text: condition

text: min

text: (h

text: h

text: max

text: By

text: virtue

text: of

text: symmetry

text: properties

text: of

text: Ncr

text: discussed

text: above,

text: one

text: can

text: only

text: consider

text: p

text: either

text: n

text: x

text: positive

text: or

text: negative.

text: Let

text: us,

text: for

text: example,

text: consider

text: the

text: interval

text: In

text: this

text: case

text: p

text: x

text: is

text: negative

text: and

text: large,

text: while

text: varies

text: from

text: positive

text: to

text: negative

text: values,

text: π

text: π

text: eE/k

text: π

text: The

text: case

text: of

text: large

text: negative

text: where

text: is

text: any

text: given

text: π

text: eE/k

text: π

text: 2π

text: /k

text: K

text: K

text: large

text: number,

text: K

text: K⊥,

text: is

text: quite

text: simple.

text: In

text: this

text: case,

text: using

text: the

text: appropriate

text: asymptotic

text: expressions

text: of

text: the

text: conﬂuent

text: hypergeometric

text: function

text: one

text: can

text: see

text: that

text: Ncr

text: is

text: negligibly

text: small.

text: n

text: To

text: see

text: this,

text: Eq.

text: in

text: Appendix

text: A

text: is

text: useful

text: in

text: the

text: range

text: and

text: the

text: h

text: 2π

text: /k

text: K

text: expression

text: for

text: large

text: with

text: ﬁxed

text: and

text: and

text: the

text: expression

text: for

text: large

text: with

text: ﬁxed

text: c

text: a

text: h

text: c

text: a

text: c

text: and

text: given

text: in

text: are

text: useful

text: in

text: the

text: range

text: h

text: 2π

text: /k

text: h

text: We

text: expect

text: a

text: signiﬁcant

text: contribution

text: in

text: the

text: range

text: h

text: 2π

text: /k

text: K

text: that

text: can

text: be

text: divided

text: in

text: four

text: subranges

text: ##LTLine##

text: ##LTLine##


text: Using the change of the variables 2 − s = (ω π ) , 1 1 k λ 1 and neglecting exponentially small contri(cid:90)butions, we represent the quantity I(1) as p⊥ ∞ ≈ ds − I(1) ω e πλs . (3.17) p⊥ s 1 1 Similarly, using the change of variables 2 − | | s = (ω π ) , 2 2 k λ 2 we represent the quantity I(2) as (cid:90) p⊥ ∞ ≈ ds − I(2) ω e πλs . (3.18) p⊥ s 2 1 The leading contributions for both integrals (3.17) and (3.18) come from the range near → , where and are approximately given by, s 1 ω ω 1 2 ≈ eE ≈ eE ω , ω . 1 2 sk sk 1 2 Consequently the le(cid:18)ading term(cid:19)in(cid:90)I (3.16) takes th(cid:18)e followin(cid:19)g ﬁnal form, p⊥ ∞ ≈ eE eE ds − 1 1 − I + e πλs = eE + e πλG (1, πλ) , (3.19) p⊥ k k s2 k k 1 2 1 1 2 where (cid:90) ∞ ds − − − G (α, x) = e x(s 1) = exxαΓ ( α, x) , (3.20) sα+1 1 − and is the incomplete gamma function. Γ ( α, x) Neglecting the exponentially small contribution, one can represent the integral over p⊥ (cid:90) (cid:90) in Eq. (3.16) (where is given by Eq. (3.19)) as I p⊥ ≈ √ dp⊥Ip⊥ √ dp⊥Ip⊥. ∞ λ<K⊥ λ< Then calculating the Gauss(cid:90)ian integra(cid:18)l, (cid:19) (cid:18) (cid:19) − − p2⊥ eE d/2 1 dp⊥ exp πs = , (3.21) eE s 16 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Using

text: the

text: change

text: of

text: the

text: variables

text: s

text: (ω

text: π

text: k

text: λ

text: and

text: neglecting

text: exponentially

text: small

text: we

text: represent

text: the

text: quantity

text: as

text: p⊥

text: ds

text: ω

text: e

text: πλs

text: p⊥

text: s

text: Similarly,

text: using

text: the

text: change

text: of

text: variables

text: s

text: (ω

text: π

text: k

text: λ

text: we

text: represent

text: the

text: quantity

text: as

text: p⊥

text: ds

text: ω

text: e

text: πλs

text: p⊥

text: s

text: The

text: leading

text: contributions

text: for

text: both

text: integrals

text: and

text: come

text: from

text: the

text: range

text: near

text: where

text: and

text: are

text: approximately

text: given

text: by,

text: s

text: ω

text: ω

text: eE

text: eE

text: ω

text: ω

text: sk

text: sk

text: Consequently

text: the

text: takes

text: ﬁnal

text: form,

text: p⊥

text: eE

text: eE

text: ds

text: I

text: e

text: πλs

text: eE

text: e

text: πλG

text: πλ)

text: p⊥

text: k

text: k

text: k

text: k

text: where

text: ds

text: G

text: (α,

text: x)

text: e

text: x(s

text: exxαΓ

text: α,

text: x)

text: and

text: is

text: the

text: incomplete

text: gamma

text: function.

text: Γ

text: α,

text: x)

text: Neglecting

text: the

text: exponentially

text: small

text: contribution,

text: one

text: can

text: represent

text: the

text: integral

text: over

text: p⊥

text: in

text: Eq.

text: (where

text: is

text: given

text: by

text: Eq.

text: as

text: I

text: p⊥

text: dp⊥Ip⊥

text: dp⊥Ip⊥.

text: λ<K⊥

text: λ<

text: Then

text: calculating

text: the

text: eE

text: dp⊥

text: exp

text: πs

text: eE

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ing and the other exponentially decreasing. This is another new exactly solvable case. We demonstrate that in the ﬁeld under consideration, one can ﬁnd exact solutions with appropri- ate asymptotic conditions and perform nonperturbative calculations of all the characteristics of particle creation process. In some respects, the peak electric ﬁeld shares similar features with the Sauter-like electric ﬁeld, while in other respects it can be treated as a pulse created by laser beams. Switching the peak ﬁeld on and oﬀ, we can imitate electric ﬁelds that are speciﬁc to condensed matter physics, in particular to graphene or Weyl semimetals as was reported, e.g., in Refs. [31–39]. In our calculations, we use the general theory of Ref. [4, 5, 7] and follow in the main the consideration of particle creation eﬀect in a homogeneous electric ﬁeld [24]. To this end we ﬁnd complete sets of exact solutions of the Dirac and Klein-Gordon equations in the peak electric ﬁeld and use them to calculate diﬀerential mean numbers of created particle, total number of created particles, and the probability for a vacuum to remain a vacuum. Characteristic asymptotic regimes (slowly varying peak ﬁeld, short pulse ﬁeld, and the most asymmetric case related to exponentially decaying ﬁeld) are discussed in detail and a com- parison with the pure asymptotically decaying ﬁeld is considered. II. PEAK ELECTRIC FIELD A. General In this section we introduce the so-called peak electric ﬁeld, that is a time-dependent electric ﬁeld directed along an unique direction1 (cid:0) (cid:1) E (t) = Ei (t) = δiE (t) , i = 1, ..., D , (2.1) 1 −∞ ∞ switched on at , and oﬀ at its maximum occurring at a very sharp t = t = + , E > 0 time instant, say at , such that the limit t = 0 (cid:54) ˙ ˙ (2.2) lim E (t) = lim E (t) , →− → t 0 t +0 is not deﬁned. The latter property implies that a peak at is present. Time-dependent t = 0 1 Greek indices refer to the Minkowski spacetime µ = 0, ..., D while Latin indices refer to Euclidean space i = 1, ..., D. Here d = D + 1 is the dimension of the spacetime. Bold letters represent Euclidean vectors (cid:124) (cid:123)(cid:122) (cid:125) − − such as r = x1, x2, ..., xD. The Minkowski metric tensor is diagonal η = diag(+1, 1, ..., 1). µν d 3 ##LTLine##

text: ing

text: and

text: the

text: other

text: exponentially

text: decreasing.

text: This

text: is

text: another

text: new

text: exactly

text: solvable

text: case.

text: We

text: demonstrate

text: that

text: in

text: the

text: ﬁeld

text: under

text: consideration,

text: one

text: can

text: ﬁnd

text: exact

text: solutions

text: with

text: appropri-

text: ate

text: asymptotic

text: conditions

text: and

text: perform

text: nonperturbative

text: calculations

text: of

text: all

text: the

text: characteristics

text: of

text: particle

text: creation

text: process.

text: In

text: some

text: respects,

text: the

text: peak

text: electric

text: ﬁeld

text: shares

text: similar

text: features

text: with

text: the

text: Sauter-like

text: electric

text: ﬁeld,

text: while

text: in

text: other

text: respects

text: it

text: can

text: be

text: treated

text: as

text: a

text: pulse

text: created

text: by

text: laser

text: beams.

text: Switching

text: the

text: peak

text: ﬁeld

text: on

text: and

text: oﬀ,

text: we

text: can

text: imitate

text: electric

text: ﬁelds

text: that

text: are

text: speciﬁc

text: to

text: condensed

text: matter

text: physics,

text: in

text: particular

text: to

text: graphene

text: or

text: Weyl

text: semimetals

text: as

text: was

text: reported,

text: e.g.,

text: in

text: Refs.

text: In

text: our

text: calculations,

text: we

text: use

text: the

text: general

text: theory

text: of

text: Ref.

text: and

text: follow

text: in

text: the

text: main

text: the

text: consideration

text: of

text: particle

text: creation

text: eﬀect

text: in

text: a

text: homogeneous

text: electric

text: ﬁeld

text: To

text: this

text: end

text: we

text: ﬁnd

text: complete

text: sets

text: of

text: exact

text: solutions

text: of

text: the

text: Dirac

text: and

text: Klein-Gordon

text: equations

text: in

text: the

text: peak

text: electric

text: ﬁeld

text: and

text: use

text: them

text: to

text: calculate

text: diﬀerential

text: mean

text: numbers

text: of

text: created

text: particle,

text: total

text: number

text: of

text: created

text: particles,

text: and

text: the

text: probability

text: for

text: a

text: vacuum

text: to

text: remain

text: a

text: vacuum.

text: Characteristic

text: asymptotic

text: regimes

text: (slowly

text: varying

text: peak

text: ﬁeld,

text: short

text: pulse

text: ﬁeld,

text: and

text: the

text: most

text: asymmetric

text: case

text: related

text: to

text: exponentially

text: decaying

text: ﬁeld)

text: are

text: discussed

text: in

text: detail

text: and

text: a

text: com-

text: parison

text: with

text: the

text: pure

text: asymptotically

text: decaying

text: ﬁeld

text: is

text: considered.

text: II.

text: PEAK

text: ELECTRIC

text: FIELD

text: A.

text: General

text: In

text: this

text: section

text: we

text: introduce

text: the

text: so-called

text: peak

text: electric

text: ﬁeld,

text: that

text: is

text: a

text: time-dependent

text: electric

text: ﬁeld

text: directed

text: along

text: an

text: unique

text: E

text: (t)

text: Ei

text: (t)

text: δiE

text: (t)

text: i

text: D

text: switched

text: on

text: at

text: and

text: oﬀ

text: at

text: its

text: maximum

text: occurring

text: at

text: a

text: very

text: sharp

text: t

text: t

text: E

text: time

text: instant,

text: say

text: at

text: such

text: that

text: the

text: limit

text: t

text: lim

text: E

text: (t)

text: lim

text: E

text: (t)

text: t

text: t

text: is

text: not

text: deﬁned.

text: The

text: latter

text: property

text: implies

text: that

text: a

text: peak

text: at

text: is

text: present.

text: Time-dependent

text: t

text: Greek

text: indices

text: refer

text: to

text: the

text: Minkowski

text: spacetime

text: µ

text: D

text: while

text: Latin

text: indices

text: refer

text: to

text: Euclidean

text: space

text: i

text: D.

text: Here

text: d

text: D

text: is

text: the

text: dimension

text: of

text: the

text: spacetime.

text: Bold

text: letters

text: represent

text: Euclidean

text: vectors

text: such

text: as

text: r

text: xD.

text: The

text: Minkowski

text: metric

text: tensor

text: is

text: diagonal

text: η

text: µν

text: d

text: ##LTLine##


text: (cid:29) (cid:28) where and are any given numbers satisfying the conditions and . We g1 √ε g1 1 ε 1 (cid:29) assume that . Note that ε h 1 2 ≈ h k τ = ih /c |2 2| 2 2 2 2 π 2 − in the ranges (a) and (b). Then in the ranges (a) and (b), varies from to . In τ 1 ε g 2 1 the range (b), parameters and are large with ﬁxed and . In this case, using η c a τ > 1 2 2 2 2 the asymptotic expression of the conﬂuent hypergeometric function given by Eq. (A11) in Appendix A, we ﬁnd that (cid:20) (cid:21) (cid:2) (cid:0) (cid:1)(cid:3) −2π |Z |− Ncr = exp (ω + π ) 1 + O 1 (4.15) n k 2 2 2 2 (cid:0) (cid:1) (cid:2) (cid:3) Z both for fermions and bosons, where is given by Eq. (A2) in the Appendix A. We 2 √ √ |Z |− |Z |− ∼ − |Z |− ∼ − − note that modulus 1 varies from 1 ε h 1 to 1 (g 1) h 1. 2 2 2 2 1 2 Approximately, expression (4.15) can be writ(cid:18)ten as (cid:19) ≈ − ππ⊥2 Ncr exp | | . (4.16) n k π 2 2 | | − Note that in the range (b). eE/g < k π < (1 ε) eE 1 2 2 It is clear that the distribution Ncr given by Eq. (4.16) has the following limiting form: n → − | | → − Ncr e πλ as k π (1 ε) eE . n 2 2 In the range (a), we can use the asymptotic expression for the conﬂuent hypergeometric (cid:46) function given by Eq. (A1) in Appendix A to verify that Ncr is ﬁnite and restricted, Ncr n n − e πλ, both for fermions and bosons. Thus, we see that the well-known distribution obtained by Nikishov [2] in a constant uniform electric ﬁeld is reproduced in an exponentially decaying − ∼ electric ﬁeld in the range of a large increment of the longitudinal kinetic momentum, π 2 . eE/k 2 In the range (c), we can use the asymptotic expression of the conﬂuent hypergeometric function for large with ﬁxed and given by Eq. (A12) in Appendix A to get the h a c 2 2 2 following result: (cid:104) (cid:105)   −  − exp π (ω + π ) Ncr ≈ k2 2 2 × sinh [π (ω2 −π2) /k2] for fermions (4.17) n sinh (2πω /k ) 2 2 cosh (π (ω π ) /k ) for bosons 2 2 2 24 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: and

text: are

text: any

text: given

text: numbers

text: satisfying

text: the

text: conditions

text: and

text: We

text: √ε

text: ε

text: assume

text: that

text: Note

text: that

text: ε

text: h

text: h

text: k

text: τ

text: ih

text: /c

text: π

text: in

text: the

text: ranges

text: (a)

text: and

text: (b).

text: Then

text: in

text: the

text: ranges

text: (a)

text: and

text: (b),

text: varies

text: from

text: to

text: In

text: τ

text: ε

text: g

text: the

text: range

text: (b),

text: parameters

text: and

text: are

text: large

text: with

text: ﬁxed

text: and

text: In

text: this

text: case,

text: using

text: η

text: c

text: a

text: τ

text: the

text: asymptotic

text: expression

text: of

text: the

text: conﬂuent

text: hypergeometric

text: function

text: given

text: by

text: Eq.

text: in

text: Appendix

text: A,

text: we

text: ﬁnd

text: that

text: −2π

text: |Z

text: Ncr

text: exp

text: (ω

text: π

text: O

text: n

text: k

text: Z

text: both

text: for

text: fermions

text: and

text: bosons,

text: where

text: is

text: given

text: by

text: Eq.

text: in

text: the

text: Appendix

text: A.

text: We

text: |Z

text: |Z

text: |Z

text: note

text: that

text: modulus

text: varies

text: from

text: ε

text: h

text: to

text: (g

text: h

text: Approximately,

text: expression

text: can

text: be

text: as

text: ππ⊥2

text: Ncr

text: exp

text: n

text: k

text: π

text: Note

text: that

text: in

text: the

text: range

text: (b).

text: eE/g

text: k

text: π

text: ε)

text: eE

text: It

text: is

text: clear

text: that

text: the

text: distribution

text: Ncr

text: given

text: by

text: Eq.

text: has

text: the

text: following

text: limiting

text: form:

text: n

text: Ncr

text: e

text: πλ

text: as

text: k

text: π

text: ε)

text: eE

text: n

text: In

text: the

text: range

text: (a),

text: we

text: can

text: use

text: the

text: asymptotic

text: expression

text: for

text: the

text: conﬂuent

text: hypergeometric

text: function

text: given

text: by

text: Eq.

text: in

text: Appendix

text: A

text: to

text: verify

text: that

text: Ncr

text: is

text: ﬁnite

text: and

text: restricted,

text: Ncr

text: n

text: n

text: e

text: πλ,

text: both

text: for

text: fermions

text: and

text: bosons.

text: Thus,

text: we

text: see

text: that

text: the

text: well-known

text: distribution

text: obtained

text: by

text: Nikishov

text: in

text: a

text: constant

text: uniform

text: electric

text: ﬁeld

text: is

text: reproduced

text: in

text: an

text: exponentially

text: decaying

text: electric

text: ﬁeld

text: in

text: the

text: range

text: of

text: a

text: large

text: increment

text: of

text: the

text: longitudinal

text: kinetic

text: momentum,

text: π

text: eE/k

text: In

text: the

text: range

text: (c),

text: we

text: can

text: use

text: the

text: asymptotic

text: expression

text: of

text: the

text: conﬂuent

text: hypergeometric

text: function

text: for

text: large

text: with

text: ﬁxed

text: and

text: given

text: by

text: Eq.

text: in

text: Appendix

text: A

text: to

text: get

text: the

text: h

text: a

text: c

text: following

text: result:

text: exp

text: π

text: (ω

text: π

text: Ncr

text: sinh

text: [π

text: −π2)

text: for

text: fermions

text: n

text: sinh

text: (2πω

text: /k

text: cosh

text: (π

text: (ω

text: π

text: /k

text: for

text: bosons

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: V. CONCLUDING REMARKS Using the strong-ﬁeld QED, we consider for the ﬁrst time particle creation in the so- called peak electric ﬁeld, which is a combination of two exponential parts, one exponentially increasing and another exponentially decreasing. This is an addition to the few previously known exactly solvable cases, where one can perform nonperturbative calculations of all the characteristics of particle creation process. For a certain choice of parameters, the peak electric ﬁeld produces a particlecreation eﬀect similar to that of the Sauter-like electric ﬁeld, or of the constant electric ﬁeld, or of the electric pulse of laser beams. Besides, by varying the peak ﬁeld parameters we can change the asymmetry rate so that the resulting ﬁeld turns out to be eﬀectively equivalent to the exponentially decaying ﬁeld. All these asymptotic regimes are discussed in detail and a comparison with the pure asymptotically decaying ﬁeld is considered. Moreover, the results obtained allow one to study how the eﬀects of switching on and oﬀ together or separately aﬀect the particle creation processes. Changing the parameters of the peak electric ﬁeld we can adjust its form to a speciﬁc physical situation, in particular, imitate ﬁeld conﬁgurations characteristic for graphene, Weyl semimetals and so on. ACKNOWLEDGEMENTS The work of the authors was supported by a grant from the Russian Science Foundation, Research Project No. 15-12-10009. Appendix A: Some asymptotic expansions The asymptotic expression of the conﬂuent hypergeometric function for large and η c ∼ with ﬁxed and is given by Eq. (13.8.4) in [43] as a τ = η/c 1 √ (cid:39) Z Z − − W Φ (a, c; η) ca/2e 2/4F (a, c; τ) , = (τ 1) (τ) c, (cid:0) (cid:1) (cid:2) (cid:3) W − Z R Z F (a, c; τ) = τ 1 aD− ( ) + D − ( ) , a 1 a R = Wa − τW1−a /Z, W (τ) = 2 (τ − 1 − ln τ) / (τ − 1)2 1/2 (A1) Z where D− ( ) is the Weber parabolic cylinder function (WPCF) [41]. Using Eq. (A1) we a present the functions y2 (η ), y1 (η ) and their derivatives at t = 0 as 1 2 2 1 26 ##LTLine##

text: V.

label: section

text: CONCLUDING

text: REMARKS

text: Using

text: the

text: strong-ﬁeld

text: QED,

text: we

text: consider

text: for

text: the

text: ﬁrst

text: time

text: particle

text: creation

text: in

text: the

text: so-

text: called

text: peak

text: electric

text: ﬁeld,

text: which

text: is

text: a

text: combination

text: of

text: two

text: exponential

text: parts,

text: one

text: exponentially

text: increasing

text: and

text: another

text: exponentially

text: decreasing.

text: This

text: is

text: an

text: addition

text: to

text: the

text: few

text: previously

text: known

text: exactly

text: solvable

text: cases,

text: where

text: one

text: can

text: perform

text: nonperturbative

text: calculations

text: of

text: all

text: the

text: characteristics

text: of

text: particle

text: creation

text: process.

text: For

text: a

text: certain

text: choice

text: of

text: parameters,

text: the

text: peak

text: electric

text: ﬁeld

text: produces

text: a

text: particlecreation

text: eﬀect

text: similar

text: to

text: that

text: of

text: the

text: Sauter-like

text: electric

text: ﬁeld,

text: or

text: of

text: the

text: constant

text: electric

text: ﬁeld,

text: or

text: of

text: the

text: electric

text: pulse

text: of

text: laser

text: beams.

text: Besides,

text: by

text: varying

text: the

text: peak

text: ﬁeld

text: parameters

text: we

text: can

text: change

text: the

text: asymmetry

text: rate

text: so

text: that

text: the

text: resulting

text: ﬁeld

text: turns

text: out

text: to

text: be

text: eﬀectively

text: equivalent

text: to

text: the

text: exponentially

text: decaying

text: ﬁeld.

text: All

text: these

text: asymptotic

text: regimes

text: are

text: discussed

text: in

text: detail

text: and

text: a

text: comparison

text: with

text: the

text: pure

text: asymptotically

text: decaying

text: ﬁeld

text: is

text: considered.

text: Moreover,

text: the

text: results

text: obtained

text: allow

text: one

text: to

text: study

text: how

text: the

text: eﬀects

text: of

text: switching

text: on

text: and

text: oﬀ

text: together

text: or

text: separately

text: aﬀect

text: the

text: particle

text: creation

text: processes.

text: Changing

text: the

text: parameters

text: of

text: the

text: peak

text: electric

text: ﬁeld

text: we

text: can

text: adjust

text: its

text: form

text: to

text: a

text: speciﬁc

text: physical

text: situation,

text: in

text: particular,

text: imitate

text: ﬁeld

text: conﬁgurations

text: characteristic

text: for

text: graphene,

text: Weyl

text: semimetals

text: and

text: so

text: on.

text: ACKNOWLEDGEMENTS

text: The

text: work

text: of

text: the

text: authors

text: was

text: supported

text: by

text: a

text: grant

text: from

text: the

text: Russian

text: Science

text: Foundation,

text: Research

text: Project

text: No.

text: Appendix

text: A:

text: Some

text: asymptotic

text: expansions

text: The

text: asymptotic

text: expression

text: of

text: the

text: conﬂuent

text: hypergeometric

text: function

text: for

text: large

text: and

text: η

text: c

text: with

text: ﬁxed

text: and

text: is

text: given

text: by

text: Eq.

text: in

text: as

text: a

text: τ

text: η/c

text: Z

text: Z

text: W

text: Φ

text: (a,

text: c;

text: η)

text: (a,

text: c;

text: τ)

text: (τ

text: (τ)

text: c,

text: W

text: Z

text: R

text: Z

text: F

text: (a,

text: c;

text: τ)

text: τ

text: aD−

text: D

text: a

text: a

text: R

text: Wa

text: /Z,

text: W

text: (τ)

text: (τ

text: ln

text: τ)

text: (τ

text: Z

text: where

text: D−

text: is

text: the

text: Weber

text: parabolic

text: cylinder

text: function

text: (WPCF)

text: Using

text: Eq.

text: we

text: a

text: present

text: the

text: functions

text: (η

text: (η

text: and

text: their

text: derivatives

text: at

text: t

text: as

text: ##LTLine##


text: electric ﬁelds of this form can, as usual in QED with unstable vacuum [6–9] (see also [40]), be described by -electric potential steps, t (cid:0) (cid:1)   A0 = 0 , A (t) = Ai (t) = δiA (t) , 1 x  − ≥ ˙ dA (t) ≤ → E (t) = A (t) 0 ˙ x x (2.3) Ax (t) = 0 −∞ ∞ , dt A ( ) > A (+ ) x x −∞ ∞ where , are constants (for further discussion and details concerning the A ( ) A (+ ) x x deﬁnition of -electric potential steps; see Ref. [40]). t To study the peak electric ﬁeld we consider an electric ﬁeld that is composed of in- dependent parts, wherein for each one the Dirac equation is exactly solvable. The ﬁeld −∞ in consideration grows exponentially from the inﬁnitely remote past , reaches a t = maximal amplitude at and decreases exponentially to the inﬁnitely remote future E t = 0 ∞ −∞ . We label the exponentially increasing interval by and the exponen- t = + I = ( , 0] ∞ tially decreasing interval by , where the ﬁeld and its -electric potential step II = (0, + ) t are   (cid:0) (cid:1)   (cid:0) (cid:1)  ∈  − − ∈ ek1t , t I , k 1 ek1t + 1 , t I , 1 (2.4) E (t) = E − ∈ , Ax (t) = E − − − ∈ . e k2t , t II k 1 e k2t 1 , t II 2 Here are positive constants. The ﬁeld and its potential are depicted below in Fig. E, k , k 1 2 1. B. Dirac equation with peak electric ﬁeld To describe the problem in the framework of QED with -electric potential steps it is t necessary to solve the Dirac equation for each interval discussed above. In any of them, the Dirac equation in a dimensional Minkowski spacetime is, in its Hamiltonian form, d = D + 1 represented by i∂ ψ (x) = H (t) ψ (x) , H (t) = γ0 (γP + m) , t − − − ∇ − P = i∂ U (t) , P⊥ = i ⊥, U (t) = eA (t) , (2.5) x x x (cid:8) (cid:9) (cid:0) (cid:1) ⊥ where the index stands for spacial components perpendicular to the electric ﬁeld, x⊥ = x2, ..., xD and P⊥ = P 2, . . . , P D . Here ψ(x) is a 2[d/2]-component spinor ([d/2] stands 4 ##LTLine##

text: electric

text: ﬁelds

text: of

text: this

text: form

text: can,

text: as

text: usual

text: in

text: QED

text: with

text: unstable

text: vacuum

text: (see

text: also

text: be

text: described

text: by

text: -electric

text: potential

text: steps,

text: t

text: A

text: (t)

text: Ai

text: (t)

text: δiA

text: (t)

text: x

text: dA

text: (t)

text: E

text: (t)

text: A

text: (t)

text: x

text: x

text: Ax

text: (t)

text: dt

text: A

text: A

text: x

text: x

text: where

text: are

text: constants

text: (for

text: further

text: discussion

text: and

text: details

text: concerning

text: the

text: A

text: A

text: x

text: x

text: deﬁnition

text: of

text: -electric

text: potential

text: steps;

text: see

text: Ref.

text: t

text: To

text: study

text: the

text: peak

text: electric

text: ﬁeld

text: we

text: consider

text: an

text: electric

text: ﬁeld

text: that

text: is

text: composed

text: of

text: in-

text: dependent

text: parts,

text: wherein

text: for

text: each

text: one

text: the

text: Dirac

text: equation

text: is

text: exactly

text: solvable.

text: The

text: ﬁeld

text: in

text: consideration

text: grows

text: exponentially

text: from

text: the

text: inﬁnitely

text: remote

text: past

text: reaches

text: a

text: t

text: maximal

text: amplitude

text: at

text: and

text: decreases

text: exponentially

text: to

text: the

text: inﬁnitely

text: remote

text: future

text: E

text: t

text: We

text: label

text: the

text: exponentially

text: increasing

text: interval

text: by

text: and

text: the

text: exponen-

text: t

text: I

text: tially

text: decreasing

text: interval

text: by

text: where

text: the

text: ﬁeld

text: and

text: its

text: -electric

text: potential

text: step

text: II

text: t

text: are

text: t

text: I

text: k

text: t

text: I

text: E

text: (t)

text: E

text: Ax

text: (t)

text: E

text: e

text: t

text: II

text: k

text: e

text: t

text: II

text: Here

text: are

text: positive

text: constants.

text: The

text: ﬁeld

text: and

text: its

text: potential

text: are

text: depicted

text: below

text: in

text: Fig.

text: E,

text: k

text: k

text: B.

text: Dirac

text: equation

text: with

text: peak

text: electric

text: ﬁeld

text: To

text: describe

text: the

text: problem

text: in

text: the

text: framework

text: of

text: QED

text: with

text: -electric

text: potential

text: steps

text: it

text: is

text: t

text: necessary

text: to

text: solve

text: the

text: Dirac

text: equation

text: for

text: each

text: interval

text: discussed

text: above.

text: In

text: any

text: of

text: them,

text: the

text: Dirac

text: equation

text: in

text: a

text: dimensional

text: Minkowski

text: spacetime

text: is,

text: in

text: its

text: Hamiltonian

text: form,

text: d

text: D

text: represented

text: by

text: i∂

text: ψ

text: (x)

text: H

text: (t)

text: ψ

text: (x)

text: H

text: (t)

text: γ0

text: (γP

text: m)

text: t

text: P

text: i∂

text: U

text: (t)

text: P⊥

text: i

text: U

text: (t)

text: eA

text: (t)

text: x

text: x

text: x

text: where

text: the

text: index

text: stands

text: for

text: spacial

text: components

text: perpendicular

text: to

text: the

text: electric

text: ﬁeld,

text: x⊥

text: xD

text: and

text: P⊥

text: P

text: P

text: D

text: Here

text: ψ(x)

text: is

text: a

text: spinor

text: stands

text: ##LTLine##


text: It is worth noting that the complete set of solutions for the Klein-Gordon equation, (2.18) φ (x) = exp (ipr) ϕ (t) . n n can be obtained from the solutions above by setting in all formulas. In this case χ = 0 . n = p With the help of the exact solutions one may write Dirac spinors throughout the time ∈ −∞ ∞ interval . As can be seen from (2.4), the peak electric ﬁeld is switched on at t ( , + ) → −∞ → ∞ the inﬁnitely remote past and switched oﬀ at the inﬁnitely remote future . t t + At these regions, the exact solutions represent free particles, N − → −∞ N − → ∞ ϕ (t) = e iζω1t if t , ζϕ (t) = ζ e iζω2t if t + , (2.19) ζ n ζ n → −∞ respectively, where denotes energy of initial particles at , denotes energy ω t ω 1 2 → ∞ − of ﬁnal particles at and labels electron and positron states. t + ζ (ζ = +) (ζ = ) N N Here and ζ are normalizat(cid:90)ion constants with respect to the inner product2 ζ (cid:48) † (cid:48) (ψ, ψ ) = ψ (x) ψ (x) dr , dr = dx1...dxD , (2.20) These constants are (cid:16) (cid:17) (cid:16) (cid:17) N N − = CY , ζ = ζCY , Y = V −1/2 , ζ ζ (d 1) − − 1/2 1/2 C = 2ω qζ , ζC = 2ω qζ , ζ 1 1 2 2 − qζ = ω χζπ , (2.21) j j j where V − is the spatial volume. By virtue of these properties, electron (positron) states (d 1) can be selected as follows: N N − ∈ ϕ (t) = exp (iπν /2) y1 (η ) , −ϕ (t) = − exp ( iπν /2) y1 (η ) , t I ; + n + 1 2 1 n 1 1 1 N − − −N ∈ +ϕ (t) = + exp ( iπν /2) y2 (η ) , ϕ (t) = exp (iπν /2) y2 (η ) , t II(2. .22) n 2 1 2 n 2 2 2 2 For a detailed explanation concerning the inner product for t-electric potential steps see, e. g., Ref. [40]. 8 ##LTLine##

text: It

text: is

text: worth

text: noting

text: that

text: the

text: complete

text: set

text: of

text: solutions

text: for

text: the

text: Klein-Gordon

text: equation,

text: φ

text: (x)

text: exp

text: (ipr)

text: ϕ

text: (t)

text: n

text: n

text: can

text: be

text: obtained

text: from

text: the

text: solutions

text: above

text: by

text: setting

text: in

text: all

text: formulas.

text: In

text: this

text: case

text: χ

text: n

text: p

text: With

text: the

text: help

text: of

text: the

text: exact

text: solutions

text: one

text: may

text: write

text: Dirac

text: spinors

text: throughout

text: the

text: time

text: interval

text: As

text: can

text: be

text: seen

text: from

text: the

text: peak

text: electric

text: ﬁeld

text: is

text: switched

text: on

text: at

text: t

text: the

text: inﬁnitely

text: remote

text: past

text: and

text: switched

text: oﬀ

text: at

text: the

text: inﬁnitely

text: remote

text: future

text: t

text: t

text: At

text: these

text: regions,

text: the

text: exact

text: solutions

text: represent

text: free

text: particles,

text: N

text: N

text: ϕ

text: (t)

text: e

text: if

text: t

text: ζϕ

text: (t)

text: ζ

text: e

text: if

text: t

text: ζ

text: n

text: ζ

text: n

text: respectively,

text: where

text: denotes

text: energy

text: of

text: initial

text: particles

text: at

text: denotes

text: energy

text: ω

text: t

text: ω

text: of

text: ﬁnal

text: particles

text: at

text: and

text: labels

text: electron

text: and

text: positron

text: states.

text: t

text: ζ

text: (ζ

text: (ζ

text: N

text: N

text: Here

text: and

text: ζ

text: are

text: constants

text: with

text: respect

text: to

text: the

text: inner

text: ζ

text: (ψ,

text: ψ

text: ψ

text: (x)

text: ψ

text: (x)

text: dr

text: dr

text: These

text: constants

text: are

text: N

text: N

text: CY

text: ζ

text: ζCY

text: Y

text: V

text: ζ

text: ζ

text: (d

text: C

text: 2ω

text: qζ

text: ζC

text: 2ω

text: qζ

text: ζ

text: qζ

text: ω

text: χζπ

text: j

text: j

text: j

text: where

text: V

text: is

text: the

text: spatial

text: volume.

text: By

text: virtue

text: of

text: these

text: properties,

text: electron

text: (positron)

text: states

text: (d

text: can

text: be

text: selected

text: as

text: follows:

text: N

text: N

text: ϕ

text: (t)

text: exp

text: (iπν

text: (η

text: −ϕ

text: (t)

text: exp

text: iπν

text: (η

text: t

text: I

text: n

text: n

text: N

text: −N

text: +ϕ

text: (t)

text: exp

text: iπν

text: (η

text: ϕ

text: (t)

text: exp

text: (iπν

text: (η

text: t

text: n

text: n

text: For

text: a

text: detailed

text: explanation

text: concerning

text: the

text: inner

text: product

text: for

text: t-electric

text: potential

text: steps

text: see,

text: e.

text: g.,

text: Ref.

text: ##LTLine##


text: C. g Coeﬃcients and mean numbers of created particles Taking into account the complete set of exact solutions (2.16), the functions −ϕ (t) and n +ϕ (t) can be presented in the form n    | | ∈ g ( +) ϕ (t) + κg (− +) −ϕ (t) , t I +ϕn (t) =  N+ +−n n ∈ , (2.23) + exp ( iπν /2) y2 (η ) , t II 2 1 2  N − ∈ − exp ( iπν /2) y1 (η ) , t I −ϕn (t) = | 1 1 −|1 − ∈ , (2.24) g (+ −) +ϕ (t) + κg ( −) ϕ (t) , t II n n for the whole axis , where the coeﬃcients are the diagonal matrix elements, t g (cid:16) (cid:17) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) | (cid:48)| | ∗ (cid:48)ψ , ζψ = δ g (cid:48) ζ , g ζ = g (cid:48) ζ . (2.25) ζ l n l,n ζ ζ ζ These coeﬃcients satisfy the unitary relations (cid:88) (cid:0) (cid:1) (cid:16) (cid:17) (cid:88) g ζ|κ g κ|ζ(cid:48) = g ( |κ) g (κ| (cid:48)) = δ (cid:48) . (2.26) ζ ζ ζ,ζ κ κ Here the constant ( above) allows us to cover the Klein-Gordon case, whose details κ κ = +1 are discussed in Eqs. (2.31) and (2.32) below. The functions −ϕ (t) and +ϕ (t) and their derivatives satisfy the following continuity n n conditions: (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:0) (cid:1) (cid:0) (cid:1) +−ϕn(t) t=−0 = +−ϕn(t) t=+0 , ∂t +−ϕn(t) t=−0 = ∂t +−ϕn(t) t=+0 . (2.27) | (cid:48) | Using Eq. (2.27) and the Wronskian (2.17), one can ﬁnd each coeﬃcient g ζ and g ζ (cid:48) ζ ζ in Eqs. (2.23) and (2.24). For example, applying these conditions to the set (2.23), the | coeﬃcient g (− +) takes the form (cid:115) (cid:20) (cid:21) (cid:0) (cid:1) − | (cid:20) C C −1 q iπ − (cid:21)(cid:12) g − + = ∆ , = 1 exp (ν ν ) (cid:12), 2 ω1q2+ω2 2 1 2 (cid:12)(cid:12) d d ∆ = k h y2 (η ) y1 (η ) + k h y1 (η ) y2 (η ) . (2.28) 1 1 1 2 dη 2 1 2 2 2 1 dη 1 2 1 2 t=0 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: C.

text: g

text: Coeﬃcients

text: and

text: mean

text: numbers

text: of

text: created

text: particles

text: Taking

text: into

text: account

text: the

text: complete

text: set

text: of

text: exact

text: solutions

text: the

text: functions

text: −ϕ

text: (t)

text: and

text: n

text: +ϕ

text: (t)

text: can

text: be

text: presented

text: in

text: the

text: form

text: n

text: g

text: ϕ

text: (t)

text: κg

text: −ϕ

text: (t)

text: t

text: I

text: +ϕn

text: (t)

text: N+

text: +−n

text: n

text: exp

text: iπν

text: (η

text: t

text: II

text: N

text: exp

text: iπν

text: (η

text: t

text: I

text: −ϕn

text: (t)

text: g

text: +ϕ

text: (t)

text: κg

text: ϕ

text: (t)

text: t

text: II

text: n

text: n

text: for

text: the

text: whole

text: axis

text: where

text: the

text: coeﬃcients

text: are

text: the

text: diagonal

text: matrix

text: elements,

text: t

text: g

text: ζψ

text: δ

text: g

text: ζ

text: g

text: ζ

text: g

text: ζ

text: ζ

text: l

text: n

text: l,n

text: ζ

text: ζ

text: ζ

text: These

text: coeﬃcients

text: satisfy

text: the

text: unitary

text: relations

text: g

text: ζ|κ

text: g

text: g

text: |κ)

text: g

text: (κ|

text: δ

text: ζ

text: ζ

text: ζ,ζ

text: κ

text: κ

text: Here

text: the

text: constant

text: above)

text: allows

text: us

text: to

text: cover

text: the

text: Klein-Gordon

text: case,

text: whose

text: details

text: κ

text: κ

text: are

text: discussed

text: in

text: Eqs.

text: and

text: below.

text: The

text: functions

text: −ϕ

text: (t)

text: and

text: +ϕ

text: (t)

text: and

text: their

text: derivatives

text: satisfy

text: the

text: following

text: continuity

text: n

text: n

text: conditions:

text: +−ϕn(t)

text: +−ϕn(t)

text: ∂t

text: +−ϕn(t)

text: ∂t

text: +−ϕn(t)

text: Using

text: Eq.

text: and

text: the

text: Wronskian

text: one

text: can

text: ﬁnd

text: each

text: coeﬃcient

text: g

text: ζ

text: and

text: g

text: ζ

text: ζ

text: ζ

text: in

text: Eqs.

text: and

text: For

text: example,

text: applying

text: these

text: conditions

text: to

text: the

text: set

text: the

text: coeﬃcient

text: g

text: takes

text: the

text: form

text: C

text: C

text: −1

text: q

text: iπ

text: g

text: exp

text: (ν

text: ν

text: d

text: d

text: k

text: h

text: (η

text: (η

text: k

text: h

text: (η

text: (η

text: dη

text: dη

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Alternatively, we obtain from the set (2.23) (cid:115) (cid:20) (cid:21) (cid:0) (cid:1) | (cid:26) C(cid:48) (cid:48) C(cid:48) −1 q+ iπ − (cid:27) g + − = ∆ , = 2− exp (ν ν ) , 2 1 2 ω q ω 2 1 1 2 (cid:48) d d ∆ = k h y1 (η ) y2 (η ) + k h y2 (η ) y1 (η ) . (2.29) 2 2 1 1 dη 2 2 1 1 2 2 dη 1 1 2 1 t=0 Comparing Eqs. (2.28) and (2.29) one can easily verify that the symmetry under a simul- (cid:28) (cid:28) − taneous change and holds, k k π π 1 2 1 (cid:0)2 (cid:1) (cid:0) (cid:1) | (cid:28) | g + − g − + . (2.30) A formal transition to the Klein-Gordon case can be performed by setting and χ = 0 − κ = 1 in Eqs. (2.23) and (2.24), and by replacing the normalization factors C, ζC ζ written in (2.21) by − − C = (2ω ) 1/2 , ζC = (2ω ) 1/2 . (2.31) ζ 1 2 | After these substitutions, the coeﬃcient g (− +) for scalar particles reads (cid:0) (cid:1) | C | C − − g − + = ∆ , = (4ω ω ) 1/2 exp [iπ (ν ν ) /2] , (2.32) sc χ=0 sc 1 2 1 2 with given by Eq. (2.28). In this case it is worth noting that the symmetry under the ∆ (cid:28) (cid:28) − simultaneous change and holds as k k π π 1 2 1(cid:0) (cid:1) 2 (cid:0) (cid:1) | (cid:28) − | g + − g − + . (2.33) Note that for scalar particles the coeﬃcients satisfy the unitary relations (cid:88) (cid:0) (cid:1) (cid:16) (cid:17) (cid:88)g κg ζ|κ g κ|ζ(cid:48) = κg ( |κ) g (κ| (cid:48)) = ζδ (cid:48) . (2.34) ζ ζ ζ,ζ κ κ Using a unitary transformation between the initial and ﬁnal Fock spaces, see [7], one V ﬁnds that the diﬀerential mean number of electron-positron pairs created from the vacuum can be expressed via the coeﬃcients as g 10 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Alternatively,

text: we

text: obtain

text: from

text: the

text: set

text: −1

text: q+

text: iπ

text: g

text: 2−

text: exp

text: (ν

text: ν

text: ω

text: q

text: ω

text: d

text: d

text: k

text: h

text: (η

text: (η

text: k

text: h

text: (η

text: (η

text: dη

text: dη

text: Comparing

text: Eqs.

text: and

text: one

text: can

text: easily

text: verify

text: that

text: the

text: symmetry

text: under

text: a

text: simul-

text: taneous

text: change

text: and

text: holds,

text: k

text: k

text: π

text: π

text: g

text: g

text: A

text: formal

text: transition

text: to

text: the

text: Klein-Gordon

text: case

text: can

text: be

text: performed

text: by

text: setting

text: and

text: χ

text: κ

text: in

text: Eqs.

text: and

text: and

text: by

text: replacing

text: the

text: normalization

text: factors

text: C,

text: ζC

text: ζ

text: written

text: in

text: by

text: C

text: ζC

text: ζ

text: After

text: these

text: substitutions,

text: the

text: coeﬃcient

text: g

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text: C

text: C

text: g

text: ω

text: exp

text: [iπ

text: (ν

text: ν

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text: given

text: by

text: Eq.

text: In

text: this

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text: it

text: is

text: worth

text: noting

text: that

text: the

text: symmetry

text: under

text: the

text: simultaneous

text: change

text: and

text: holds

text: as

text: k

text: k

text: π

text: π

text: g

text: g

text: Note

text: that

text: for

text: scalar

text: particles

text: the

text: coeﬃcients

text: satisfy

text: the

text: unitary

text: relations

text: κg

text: ζ|κ

text: g

text: κg

text: |κ)

text: g

text: (κ|

text: ζδ

text: ζ

text: ζ

text: ζ,ζ

text: κ

text: κ

text: Using

text: a

text: unitary

text: transformation

text: between

text: the

text: initial

text: and

text: ﬁnal

text: Fock

text: spaces,

text: see

text: one

text: V

text: ﬁnds

text: that

text: the

text: diﬀerential

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text: electron-positron

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text: the

text: vacuum

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text: netospheric particles results in its external heating. Evi- twisted magnetosphere (Beloborodov & Thompson 2007; dence for high-energy particles is provided by persistent Beloborodov 2013a). noDnrthaeftrmvaerl seiomnisMsiaoyn29fr,o2m020magnetars, which is associ- In this paper we examine the eﬃciencies of the heating atePdrepwrinitthtypceosnettiunsuinaglLAeTlEeXctsrtoynle-pemousliattreoanpj dv.is1c2h/1a6r/g1e1 in the mechanisms (1)-(4) using simple estimates and illustrat- ing with sample numerical models. MAGNETAR HEATING Andrei M. Beloborodov and Xinyu Li 0 Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street New York, NY 10027; amb@phys.columbia.edu Subject headings: dense matter — magnetic ﬁelds — stars: magnetars — stars: neutron 2. COOLING OF A HOT CORE the heat ﬂow to the surface. This is the result of Landau quantization of electron motion in the envelope (elec- The heat capacity of a core with non-superﬂuid neu- trons can only move along B at low densities where the trons determines the maximum thermal energy that (cid:126) electron Fermi energy is below the Landau energy ω ). could be stored in a neutron star (e.g. Page et al. 2004), B In contrast, a horizontal ﬁeld (Θ = π/2) impedes the ∼ −B (cid:28) E 1048T2 erg. (1) heat ﬂow by the factor of (τω ) 2 1, where τ is the th 9 B collisional free path time of electrons. Without heating, most of E is lost to neutrino emission th ∼ The T -T relation also depends on the chemical on a timescale shorter than the typical magnetar age t core s ∼ composition, which must be iron in the lower envelope 1011 s, and the core temperature decreases to Tcore ρ >∼ 109 g cm−3 but may be lighter elements in the sur- 108 K while its surface luminosity L drops well below − s face layers. Potekhin et al. (2003) calculated the T -T 1035 erg s 1 (Yakovlev & Pethick 2004; Page 2009). core s relation and gave its analytical approximation for vari- In this section, we discuss what core temperature ous B and Θ , for both iron and light element envelopes. would be suﬃcient to sustain the observed L of active B s Their calculations assumed that neutrino cooling of the magnetars. Then we estimate the required heating that envelope is negligible and the envelope has a gaseous at- must oﬀset the neutrino cooling to keep the core hot. mosphere. The latter assumption may be invalid, as the Section will address how the high temperature could be magnetar surface is likely condensed (Medin & Lai 2006), sustained by ambipolar diﬀusion. although only approximate calculations are available for 2.1. Core temperature capable of sustaining L s the phase transition to the condensed state. ≈ The surface luminosity of persistent magnetars L Potekhin et al. (2007) included neutrino cooling and − s 1035 erg s 1 approximately corresponds to the average studied heat conduction in stars with condensed surfaces. surface ﬂux They found that replacing the atmosphere with a con- L − − − densed surface weakly aﬀects the Tcore-Ts relation, and F = σ T4 = s = 1022A 1 erg s 1 cm 2, (2) neutrino losses in the envelope become important when s SB s A 13 T >∼ 109 K. The losses eﬀectively impose a ceiling for where σSB ≈ 5.67 × 10−5 erg cm≈−2 s×−1 K−4 is the t∼hcoeresurface −luminosity: L∼s reaches its maximum value Stephan-Boltzmann constant, Ts 4 106 K is the 1035 erg s 1 when Tcore 109 K and does not respond eﬀective surface temperature, and A is the emission to further increase of Tcore, because the heat ﬂux is lost area, which may be smaller than the stellar surface area to neutrino emission on its way through the crust. Be- ≈ × 4πR2 1.5 1013 cm2. low this ceiling the Tcore-Ts relation from Potekhin et al. Such a high T can be sustained if the interior tem- (2003), with neglected neutrino losses, may be used. s perature is comparable to 109 K (Yakovlev & Pethick The dependence of the surface luminosity on Tcore is 2004). The interior region here includes not only the shown in Figure 1 for a neutron star of mass M = 1.4M(cid:12) × − core (ρ > 1.4 1014 g cm 3) but also the lower crust and radius R = 11.7 km. The envelope is assumed to (ρ (cid:29) 1011 g cm−3); this region is nearly isothermal due have an approximately dipole magnetic ﬁeld B; then the angle between B and the radial direction is to its high thermal conductivity. A strong temperature gradient is sustained in the blanketing enve−lope in the sin θ upper crust, especially where ρ < 109 g cm 3, because tan ΘB = 2 cos θ, (3) this region has a lower thermal conductivity. A steady heat ﬂux F is established on the timescale of heat con- where co-latitude θ is measured from the magnetic pole s ∼ − duction across the crust, t 1 10 yr. and general relativistic(cid:90)corrections have been neglected. c The relation between T and T depends on the The surface luminosity of the star is given by core s strength of the magnetic ﬁeld B in the blanketing en- 1 velope and its angle with respect to the radial direc- L = 4πR2 F (Θ , B) d cos θ, (4) s s B tion, Θ , because both aﬀect heat conduction (Potekhin B 0 1999). A strong radial magnetic ﬁeld (Θ = 0) increases ∞ − B and the observed luminosity at inﬁnity L = (1 s 2GM/c2R)L is reduced by the factor of 1.5. One can see s ##LTLine## ##LTLine##

text: netospheric

text: particles

text: results

text: in

text: its

text: external

text: heating.

text: Evi-

text: twisted

text: magnetosphere

text: (Beloborodov

text: Thompson

text: dence

text: for

text: high-energy

text: particles

text: is

text: provided

text: by

text: persistent

text: Beloborodov

text: noDnrthaeftrmvaerl

text: which

text: is

text: associ-

text: In

text: this

text: paper

text: we

text: examine

text: the

text: eﬃciencies

text: of

text: the

text: heating

text: in

text: the

text: mechanisms

text: using

text: simple

text: estimates

text: and

text: illustrat-

text: ing

text: with

text: sample

text: numerical

text: models.

text: MAGNETAR

text: HEATING

text: Andrei

text: M.

text: Beloborodov

text: and

text: Xinyu

text: Li

text: Physics

text: Department

text: and

text: Columbia

text: Astrophysics

text: Laboratory,

text: Columbia

text: University,

text: West

text: Street

text: New

text: York,

text: NY

text: amb@phys.columbia.edu

text: Subject

text: headings:

text: dense

text: matter

text: magnetic

text: ﬁelds

text: stars:

text: magnetars

text: stars:

text: neutron

text: COOLING

text: OF

text: A

text: HOT

text: CORE

text: the

text: heat

text: ﬂow

text: to

text: the

text: surface.

text: This

text: is

text: the

text: result

text: of

text: Landau

text: quantization

text: of

text: electron

text: motion

text: in

text: the

text: envelope

text: (elec-

text: The

text: heat

text: capacity

text: of

text: a

text: core

text: with

text: non-superﬂuid

text: neu-

text: trons

text: can

text: only

text: move

text: along

text: B

text: at

text: low

text: densities

text: where

text: the

text: trons

text: determines

text: the

text: maximum

text: thermal

text: energy

text: that

text: electron

text: Fermi

text: energy

text: is

text: below

text: the

text: Landau

text: energy

text: ω

text: could

text: be

text: stored

text: in

text: a

text: neutron

text: star

text: (e.g.

text: Page

text: et

text: al.

text: B

text: In

text: contrast,

text: a

text: horizontal

text: ﬁeld

text: (Θ

text: impedes

text: the

text: −B

text: E

text: erg.

text: heat

text: ﬂow

text: by

text: the

text: factor

text: of

text: (τω

text: where

text: τ

text: is

text: the

text: th

text: B

text: collisional

text: free

text: path

text: time

text: of

text: electrons.

text: Without

text: heating,

text: most

text: of

text: E

text: is

text: lost

text: to

text: neutrino

text: emission

text: th

text: The

text: T

text: -T

text: relation

text: also

text: depends

text: on

text: the

text: chemical

text: on

text: a

text: timescale

text: shorter

text: than

text: the

text: typical

text: magnetar

text: age

text: t

text: core

text: s

text: composition,

text: which

text: must

text: be

text: iron

text: in

text: the

text: lower

text: envelope

text: s,

text: and

text: the

text: core

text: temperature

text: decreases

text: to

text: Tcore

text: ρ

text: g

text: but

text: may

text: be

text: lighter

text: elements

text: in

text: the

text: sur-

text: K

text: while

text: its

text: surface

text: luminosity

text: L

text: drops

text: well

text: below

text: s

text: face

text: layers.

text: Potekhin

text: et

text: al.

text: calculated

text: the

text: T

text: -T

text: erg

text: s

text: (Yakovlev

text: Pethick

text: Page

text: core

text: s

text: relation

text: and

text: gave

text: its

text: analytical

text: approximation

text: for

text: vari-

text: In

text: this

text: section,

text: we

text: discuss

text: what

text: core

text: temperature

text: ous

text: B

text: and

text: Θ

text: for

text: both

text: iron

text: and

text: light

text: element

text: envelopes.

text: would

text: be

text: suﬃcient

text: to

text: sustain

text: the

text: observed

text: L

text: of

text: active

text: B

text: s

text: Their

text: calculations

text: assumed

text: that

text: neutrino

text: cooling

text: of

text: the

text: magnetars.

text: Then

text: we

text: estimate

text: the

text: required

text: heating

text: that

text: envelope

text: is

text: negligible

text: and

text: the

text: envelope

text: has

text: a

text: gaseous

text: at-

text: must

text: oﬀset

text: the

text: neutrino

text: cooling

text: to

text: keep

text: the

text: core

text: hot.

text: mosphere.

text: The

text: latter

text: assumption

text: may

text: be

text: invalid,

text: as

text: the

text: Section

text: will

text: address

text: how

text: the

text: high

text: temperature

text: could

text: be

text: magnetar

text: surface

text: is

text: likely

text: condensed

text: (Medin

text: Lai

text: sustained

text: by

text: ambipolar

text: diﬀusion.

text: although

text: only

text: approximate

text: calculations

text: are

text: available

text: for

text: Core

text: temperature

text: capable

text: of

text: sustaining

text: L

text: s

text: the

text: phase

text: transition

text: to

text: the

text: condensed

text: state.

text: The

text: surface

text: luminosity

text: of

text: persistent

text: magnetars

text: L

text: Potekhin

text: et

text: al.

text: included

text: neutrino

text: cooling

text: and

text: s

text: erg

text: s

text: approximately

text: corresponds

text: to

text: the

text: average

text: studied

text: heat

text: conduction

text: in

text: stars

text: with

text: condensed

text: surfaces.

text: surface

text: ﬂux

text: They

text: found

text: that

text: replacing

text: the

text: atmosphere

text: with

text: a

text: con-

text: L

text: densed

text: surface

text: weakly

text: aﬀects

text: the

text: Tcore-Ts

text: relation,

text: and

text: F

text: σ

text: s

text: erg

text: s

text: cm

text: neutrino

text: losses

text: in

text: the

text: envelope

text: become

text: important

text: when

text: s

text: SB

text: s

text: A

text: T

text: K.

text: The

text: losses

text: eﬀectively

text: impose

text: a

text: ceiling

text: for

text: where

text: σSB

text: erg

text: s×−1

text: is

text: the

text: t∼hcoeresurface

text: −luminosity:

text: L∼s

text: reaches

text: its

text: maximum

text: value

text: Stephan-Boltzmann

text: constant,

text: Ts

text: K

text: is

text: the

text: erg

text: s

text: when

text: Tcore

text: K

text: and

text: does

text: not

text: respond

text: eﬀective

text: surface

text: temperature,

text: and

text: A

text: is

text: the

text: emission

text: to

text: further

text: increase

text: of

text: Tcore,

text: because

text: the

text: heat

text: ﬂux

text: is

text: lost

text: area,

text: which

text: may

text: be

text: smaller

text: than

text: the

text: stellar

text: surface

text: area

text: to

text: neutrino

text: emission

text: on

text: its

text: way

text: through

text: the

text: crust.

text: Be-

text: low

text: this

text: ceiling

text: the

text: Tcore-Ts

text: relation

text: from

text: Potekhin

text: et

text: al.

text: Such

text: a

text: high

text: T

text: can

text: be

text: sustained

text: if

text: the

text: interior

text: tem-

text: with

text: neglected

text: neutrino

text: losses,

text: may

text: be

text: used.

text: s

text: perature

text: is

text: comparable

text: to

text: K

text: (Yakovlev

text: Pethick

text: The

text: dependence

text: of

text: the

text: surface

text: luminosity

text: on

text: Tcore

text: is

text: The

text: interior

text: region

text: here

text: includes

text: not

text: only

text: the

text: shown

text: in

text: Figure

text: for

text: a

text: neutron

text: star

text: of

text: mass

text: M

text: core

text: (ρ

text: g

text: cm

text: but

text: also

text: the

text: lower

text: crust

text: and

text: radius

text: R

text: km.

text: The

text: envelope

text: is

text: assumed

text: to

text: (ρ

text: g

text: this

text: region

text: is

text: nearly

text: isothermal

text: due

text: have

text: an

text: approximately

text: dipole

text: magnetic

text: ﬁeld

text: B;

text: then

text: the

text: angle

text: between

text: B

text: and

text: the

text: radial

text: direction

text: is

text: to

text: its

text: high

text: thermal

text: conductivity.

text: A

text: strong

text: temperature

text: gradient

text: is

text: sustained

text: in

text: the

text: blanketing

text: enve−lope

text: in

text: the

text: sin

text: θ

text: upper

text: crust,

text: especially

text: where

text: ρ

text: g

text: cm

text: because

text: tan

text: ΘB

text: cos

text: θ,

text: this

text: region

text: has

text: a

text: lower

text: thermal

text: conductivity.

text: A

text: steady

text: heat

text: ﬂux

text: F

text: is

text: established

text: on

text: the

text: timescale

text: of

text: heat

text: con-

text: where

text: co-latitude

text: θ

text: is

text: measured

text: from

text: the

text: magnetic

text: pole

text: s

text: duction

text: across

text: the

text: crust,

text: t

text: yr.

text: and

text: general

text: have

text: been

text: neglected.

text: c

text: The

text: relation

text: between

text: T

text: and

text: T

text: depends

text: on

text: the

text: The

text: surface

text: luminosity

text: of

text: the

text: star

text: is

text: given

text: by

text: core

text: s

text: strength

text: of

text: the

text: magnetic

text: ﬁeld

text: B

text: in

text: the

text: blanketing

text: en-

text: velope

text: and

text: its

text: angle

text: with

text: respect

text: to

text: the

text: radial

text: direc-

text: L

text: F

text: (Θ

text: B)

text: d

text: cos

text: θ,

text: s

text: s

text: B

text: tion,

text: Θ

text: because

text: both

text: aﬀect

text: heat

text: conduction

text: (Potekhin

text: B

text: A

text: strong

text: radial

text: magnetic

text: ﬁeld

text: (Θ

text: increases

text: B

text: and

text: the

text: observed

text: luminosity

text: at

text: inﬁnity

text: L

text: s

text: is

text: reduced

text: by

text: the

text: factor

text: of

text: One

text: can

text: see

text: s

text: ##LTLine##

text: ##LTLine##


text: observations of shrinking hot spots in seven transient magnetars. The observed evolution of the spot area A and luminosity L agrees with the special trend predicted by the untwisting magnetosphere model: A and L de- crease with time. The slope of the L -A relation (con- trolled by the behavior of Φ) varies between 1 and 2, in the theoretically expected region of the L -A plane. The typical timescale of this evolution — months to years — is also consistent with theoretical expectations, although there are outliers that require a more detailed modeling. The predicted and observed localization of external heating in transient magnetars suggests that this mech- spots-eps-converted-to.pdf anism does not dominate Ls in persistent magnetars, as most of their surface emission apparently comes from a large area comparable to 4πR2. Fig. 12.— The evolution of hot spots observed on transient magnetars following their outbursts. The spot shrinks (its area A decreases) and becomes dimmer (its luminosity L decreases) with time, forming tracks on the A-L plane. The theoretica∼l pred×iction is shown −by the strip between the two lines, L 1.3 1033K A211 erg s 1, where K = B14Φ9ψ (Belo∼borod−ov 2009). The value of K de∼pends on the discharge voltage Φ 109 1010 V, the twist angle ψ 1, and the characteristic surface magnetic ﬁeld B. The strip shown in the ﬁgure corresponds to 1 < K < 20, how- ever a broader range is possible, and K may evolve during the out- burst. Data for SGR 1745-2900 are from Coti Zelati et al. (2015); CXOU J1647-45 from Woods et al. (2011) and An et al. (2013); Swift J1822.3-1606 from Rea et al. (2012); SGR 0418+5729 from Esposito et al. (2010); SGR 0501+4516 from Rea et al. (2009); XTE J1810-197 from Gotthelf & Halpern (2007); 1E 1547-5408 froDmrHafatlpverenrseitonal.M(a2y00289),a2n0d20Enoto et al. (2010). The distance toP1Erep1r5i4n7t-t5y4p0e8sewtausscinhganLAgTeEdXtost4ylkepecmfuolllaotweaipnjgvT.i1e2n/g1o6e/t11al. (2010) and Gelfand & Gaensler (2007). MAGNETAR HEATING Andrei M. Beloborodov and Xinyu Li 0 Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street New York, NY 10027; amb@phys.columbia.edu Subject headings: dense matter — magnetic ﬁelds — stars: magnetars — stars: neutron ∼ × 8. DISCUSSION envelope, T > 6 108 K is required (Figure 1). Al- core though ultrastrong magnetic ﬁelds can drive a fast am- The observed surface luminosity of persistent magne- tars L ≈ 1035 erg s−1 is a challenge to magnetar the- bipolar drift that generates a huge heating rate, we ﬁnd s that such hot cores have lifetimes shorter than the typical ory. Energy transport from the core heated by ambipolar magnetar age (Figure 3), as long as the typical wavenum- diﬀusion is an attractive scenario, which lead TD96 to propose an explanation for Ls ≈ 1035 erg s−1: it corre- basesrusmofptthioenv2aπri/aktio<∼n2o0f kBmi.nTthheecloifreetismateisisfyshthoertpblaeucsaiubslee sponds to the highest core temperature that ambipolar the ambipolar drift is fast in the hot core. It is not slowed heating could sustain against neutrino cooling. We ﬁnd, down by the induced pressure gradients in a compressive however, that this scenario faces the following problem. drift and is only limited by the p-n friction, which is Even in the best case of a magnetar with a light-element modest at high temperatures. Assuming stronger mag- netic ﬁelds helps increase the energy reservoir available ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: observations

text: of

text: shrinking

text: hot

text: spots

text: in

text: seven

text: transient

text: magnetars.

text: The

text: observed

text: evolution

text: of

text: the

text: spot

text: area

text: A

text: and

text: luminosity

text: L

text: agrees

text: with

text: the

text: special

text: trend

text: predicted

text: by

text: the

text: untwisting

text: magnetosphere

text: model:

text: A

text: and

text: L

text: de-

text: crease

text: with

text: time.

text: The

text: slope

text: of

text: the

text: L

text: -A

text: relation

text: (con-

text: trolled

text: by

text: the

text: behavior

text: of

text: Φ)

text: varies

text: between

text: and

text: in

text: the

text: theoretically

text: expected

text: region

text: of

text: the

text: L

text: -A

text: plane.

text: The

text: typical

text: timescale

text: of

text: this

text: evolution

text: months

text: to

text: years

text: is

text: also

text: consistent

text: with

text: theoretical

text: expectations,

text: although

text: there

text: are

text: outliers

text: that

text: require

text: a

text: more

text: detailed

text: modeling.

text: The

text: predicted

text: and

text: observed

text: localization

text: of

text: external

text: heating

text: in

text: transient

text: magnetars

text: suggests

text: that

text: this

text: mech-

text: spots-eps-converted-to.pdf

text: anism

text: does

text: not

text: dominate

text: Ls

text: in

text: persistent

text: magnetars,

text: as

text: most

text: of

text: their

text: surface

text: emission

text: apparently

text: comes

text: from

text: a

text: large

text: area

text: comparable

text: to

text: Fig.

text: The

text: evolution

text: of

text: hot

text: spots

text: observed

text: on

text: transient

text: magnetars

text: following

text: their

text: outbursts.

text: The

text: spot

text: shrinks

text: (its

text: area

text: A

text: decreases)

text: and

text: becomes

text: dimmer

text: (its

text: luminosity

text: L

text: decreases)

text: with

text: time,

text: forming

text: tracks

text: on

text: the

text: A-L

text: plane.

text: The

text: theoretica∼l

text: pred×iction

text: is

text: shown

text: −by

text: the

text: strip

text: between

text: the

text: two

text: lines,

text: L

text: erg

text: s

text: where

text: K

text: (Belo∼borod−ov

text: The

text: value

text: of

text: K

text: de∼pends

text: on

text: the

text: discharge

text: voltage

text: Φ

text: V,

text: the

text: twist

text: angle

text: ψ

text: and

text: the

text: characteristic

text: surface

text: magnetic

text: ﬁeld

text: B.

text: The

text: strip

text: shown

text: in

text: the

text: ﬁgure

text: corresponds

text: to

text: K

text: how-

text: ever

text: a

text: broader

text: range

text: is

text: possible,

text: and

text: K

text: may

text: evolve

text: during

text: the

text: out-

text: burst.

text: Data

text: for

text: SGR

text: are

text: from

text: Coti

text: Zelati

text: et

text: al.

text: CXOU

text: from

text: Woods

text: et

text: al.

text: and

text: An

text: et

text: al.

text: Swift

text: from

text: Rea

text: et

text: al.

text: SGR

text: from

text: Esposito

text: et

text: al.

text: SGR

text: from

text: Rea

text: et

text: al.

text: XTE

text: from

text: Gotthelf

text: Halpern

text: et

text: al.

text: The

text: distance

text: and

text: Gelfand

text: Gaensler

text: MAGNETAR

text: HEATING

text: Andrei

text: M.

text: Beloborodov

text: and

text: Xinyu

text: Li

text: Physics

text: Department

text: and

text: Columbia

text: Astrophysics

text: Laboratory,

text: Columbia

text: University,

text: West

text: Street

text: New

text: York,

text: NY

text: amb@phys.columbia.edu

text: Subject

text: headings:

text: dense

text: matter

text: magnetic

text: ﬁelds

text: stars:

text: magnetars

text: stars:

text: neutron

text: DISCUSSION

text: envelope,

text: T

text: K

text: is

text: required

text: (Figure

text: Al-

text: core

text: though

text: ultrastrong

text: magnetic

text: ﬁelds

text: can

text: drive

text: a

text: fast

text: am-

text: The

text: observed

text: surface

text: luminosity

text: of

text: persistent

text: magne-

text: tars

text: L

text: erg

text: is

text: a

text: challenge

text: to

text: magnetar

text: the-

text: bipolar

text: drift

text: that

text: generates

text: a

text: huge

text: heating

text: rate,

text: we

text: ﬁnd

text: s

text: that

text: such

text: hot

text: cores

text: have

text: lifetimes

text: shorter

text: than

text: the

text: typical

text: ory.

text: Energy

text: transport

text: from

text: the

text: core

text: heated

text: by

text: ambipolar

text: magnetar

text: age

text: (Figure

text: as

text: long

text: as

text: the

text: typical

text: wavenum-

text: diﬀusion

text: is

text: an

text: attractive

text: scenario,

text: which

text: lead

text: to

text: propose

text: an

text: explanation

text: for

text: Ls

text: erg

text: it

text: corre-

text: sponds

text: to

text: the

text: highest

text: core

text: temperature

text: that

text: ambipolar

text: the

text: ambipolar

text: drift

text: is

text: fast

text: in

text: the

text: hot

text: core.

text: It

text: is

text: not

text: slowed

text: heating

text: could

text: sustain

text: against

text: neutrino

text: cooling.

text: We

text: ﬁnd,

text: down

text: by

text: the

text: induced

text: pressure

text: gradients

text: in

text: a

text: compressive

text: however,

text: that

text: this

text: scenario

text: faces

text: the

text: following

text: problem.

text: drift

text: and

text: is

text: only

text: limited

text: by

text: the

text: p-n

text: friction,

text: which

text: is

text: Even

text: in

text: the

text: best

text: case

text: of

text: a

text: magnetar

text: with

text: a

text: light-element

text: modest

text: at

text: high

text: temperatures.

text: Assuming

text: stronger

text: mag-

text: netic

text: ﬁelds

text: helps

text: increase

text: the

text: energy

text: reservoir

text: available

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##