text: In this sense the question about if the quantum ﬂuctuations induce forces between the BPS vortices remained unsolved because of the lack of control on the previously mentioned source of errors. The idea to repair this diﬃculty was to include in the heat kernel expansion a (non local) term that takes care of the eﬀect of zero modes surviving in the low temperature range. The new term induced by the zero modes depends of an arbitrary a priori function of the (ﬁctitious) temperature which is chosen by demanding two properties: (1) The known behaviour of the heat kernel not only at high but also at low temperature are reproduced. (2) The solution of the recurrence relations implied by the asymptotic expansion is minimally perturbed by the arbitrary function. This modiﬁcation allowed a much more precise evaluation of the Mellin’s transform of the heat trace to obtain the spectral zeta extending the integration interval to all the temperature range. By this token we are able to ﬁx not only the zeta function near the poles but also the entire part. Because in the kink case many exact evaluations of kink mass quantum corrections are known we were able to check that the improved heat kernel expansion oﬀered much closer approximations to the exact results as compared with the results obtained by using the standard GdW method. Moreover, in Reference [27] we extended the procedure to many component scalar ﬁeld theory. In these type of models there are families of BPS kinks in such a way than other kink zero modes besides the translational zero mode arises. The results also were much more precise than those previously obtained using the standard GdW expansion, see e.g. [28] and [29]. More interesting, in this last paper we do not only consider the problem in (1+1)-dimensional space-time but we analyzed the one- loop ﬂuctuations in a three dimensional perspective where kinks become domain walls. In Reference [30] the same problem was addressed over standard supersymmetric kink domain walls relying on dimensional regularization procedures. Our method consequently also works for extended objects of p-brane type and, in the case of the model we studied, an interesting phenomenon was unveiled: within a family of classically degenerate BPS kinks repulsive forces were induced by the quantum ﬂuctuations that broke the classical degeneracy. We plan to address an identical issue in the moduli space of BPS vortices in the Abelian Higgs model. Jaﬀe and Taubes showed in [31] that the vortex moduli space is the set of N unordered R points in 2. As a consequence vortices with one quantum of magnetic ﬂux move freely without any interaction. The 2N vortex zero modes obey to this freedom in the critical point between Type II and Type I superconductivity phases, in the ﬁrst case surge repulsive whereas in the second case attractive forces between vortices. We shall discuss wether or not this classical picture is maintained at one-loop order after the eﬀect of vortex ﬂuctuations is accounted for. We shall perform the pertinent calculations generalizing the improved GdW heat kernel procedure developed previously for scalar ﬁeld theories to Abelian gauge theories with spontaneous symmetry breaking. The analysis will be ﬁrst focused in the planar AHM where the BPS vortices are topological solitons. After that we shall move to study the same problem in a three dimensional space, where we ﬁnd BPS vortices magnetic tubes or vortex strings. To perform this task, evaluation of the quantum corrections to BPS vortices by using the modiﬁed GdW expansion we need a detailed information of the spectrum of the matrix second-order PDO that governs the vortex ﬂuctuations. All the information needed about vortex zero modes and bound states is collected in our recent papers [32] and [33] where pertinent References can be found. § The organization of the paper is as follows: In Section .2 we thoroughly address the problem de- § scribed above in the planar system. Subsection .2.1 summarize the well known facts about planar BPS § ﬂuctuations, in sub-Section .2.2 the modiﬁed or improved GdW heat kernel expansion is generalized to planar Abelian gauge systems with spontaneous symmetry breaking by one scalar ﬁeld, and ﬁnally, in § sub-Section .2.3 the one-loop mass shifts of rotationally symmetric planar BPS vortices are computed. The new one-loop mass vortex shifts performed in this work, although qualitatively compatible with those obtained in [9], are of much greater precision because the new technique is able to incorporate also the § eﬀect of zero modes in the spectral zeta function. Section .3 is fully devoted to describe the quantum corrections at one loop order of the BPS vortex string tensions in the three dimensional Abelian Higgs model. These results are completely new. Some conclusions about the new approach developed here and § the induction of repulsive forces between BPS due to their quantum ﬂuctuations oﬀered in Section .4, 3

text: In

text: this

text: sense

text: the

text: question

text: about

text: if

text: the

text: quantum

text: ﬂuctuations

text: induce

text: forces

text: between

text: the

text: BPS

text: vortices

text: remained

text: unsolved

text: because

text: of

text: the

text: lack

text: of

text: control

text: on

text: the

text: previously

text: mentioned

text: source

text: of

text: errors.

text: The

text: idea

text: to

text: repair

text: this

text: diﬃculty

text: was

text: to

text: include

text: in

text: the

text: heat

text: kernel

text: expansion

text: a

text: (non

text: local)

text: term

text: that

text: takes

text: care

text: of

text: the

text: eﬀect

text: of

text: zero

text: modes

text: surviving

text: in

text: the

text: low

text: temperature

text: range.

text: The

text: new

text: term

text: induced

text: by

text: the

text: zero

text: modes

text: depends

text: of

text: an

text: arbitrary

text: a

text: priori

text: function

text: of

text: the

text: (ﬁctitious)

text: temperature

text: which

text: is

text: chosen

text: by

text: demanding

text: two

text: properties:

text: The

text: known

text: behaviour

text: of

text: the

text: heat

text: kernel

text: not

text: only

text: at

text: high

text: but

text: also

text: at

text: low

text: temperature

text: are

text: reproduced.

text: The

text: solution

text: of

text: the

text: recurrence

text: relations

text: implied

text: by

text: the

text: asymptotic

text: expansion

text: is

text: minimally

text: perturbed

text: by

text: the

text: arbitrary

text: function.

text: This

text: modiﬁcation

text: allowed

text: a

text: much

text: more

text: precise

text: evaluation

text: of

text: the

text: Mellin’s

text: transform

text: of

text: the

text: heat

text: trace

text: to

text: obtain

text: the

text: spectral

text: zeta

text: extending

text: the

text: integration

text: interval

text: to

text: all

text: the

text: temperature

text: range.

text: By

text: this

text: token

text: we

text: are

text: able

text: to

text: ﬁx

text: not

text: only

text: the

text: zeta

text: function

text: near

text: the

text: poles

text: but

text: also

text: the

text: entire

text: part.

text: Because

text: in

text: the

text: kink

text: case

text: many

text: exact

text: evaluations

text: of

text: kink

text: mass

text: quantum

text: corrections

text: are

text: known

text: we

text: were

text: able

text: to

text: check

text: that

text: the

text: improved

text: heat

text: kernel

text: expansion

text: oﬀered

text: much

text: closer

text: approximations

text: to

text: the

text: exact

text: results

text: as

text: compared

text: with

text: the

text: results

text: obtained

text: by

text: using

text: the

text: standard

text: GdW

text: method.

text: Moreover,

text: in

text: Reference

text: we

text: extended

text: the

text: procedure

text: to

text: many

text: component

text: scalar

text: ﬁeld

text: theory.

text: In

text: these

text: type

text: of

text: models

text: there

text: are

text: families

text: of

text: BPS

text: kinks

text: in

text: such

text: a

text: way

text: than

text: other

text: kink

text: zero

text: modes

text: besides

text: the

text: translational

text: zero

text: mode

text: arises.

text: The

text: results

text: also

text: were

text: much

text: more

text: precise

text: than

text: those

text: previously

text: obtained

text: using

text: the

text: standard

text: GdW

text: expansion,

text: see

text: e.g.

text: and

text: More

text: interesting,

text: in

text: this

text: last

text: paper

text: we

text: do

text: not

text: only

text: consider

text: the

text: problem

text: in

text: space-time

text: but

text: we

text: analyzed

text: the

text: one-

text: loop

text: ﬂuctuations

text: in

text: a

text: three

text: dimensional

text: perspective

text: where

text: kinks

text: become

text: domain

text: walls.

text: In

text: Reference

text: the

text: same

text: problem

text: was

text: addressed

text: over

text: standard

text: supersymmetric

text: kink

text: domain

text: walls

text: relying

text: on

text: dimensional

text: regularization

text: procedures.

text: Our

text: method

text: consequently

text: also

text: works

text: for

text: extended

text: objects

text: of

text: p-brane

text: type

text: and,

text: in

text: the

text: case

text: of

text: the

text: model

text: we

text: studied,

text: an

text: interesting

text: phenomenon

text: was

text: unveiled:

text: within

text: a

text: family

text: of

text: classically

text: degenerate

text: BPS

text: kinks

text: repulsive

text: forces

text: were

text: induced

text: by

text: the

text: quantum

text: ﬂuctuations

text: that

text: broke

text: the

text: classical

text: degeneracy.

text: We

text: plan

text: to

text: address

text: an

text: identical

text: issue

text: in

text: the

text: moduli

text: space

text: of

text: BPS

text: vortices

text: in

text: the

text: Abelian

text: Higgs

text: model.

text: Jaﬀe

text: and

text: Taubes

text: showed

text: in

text: that

text: the

text: vortex

text: moduli

text: space

text: is

text: the

text: set

text: of

text: N

text: unordered

text: R

text: points

text: in

text: As

text: a

text: consequence

text: vortices

text: with

text: one

text: quantum

text: of

text: magnetic

text: ﬂux

text: move

text: freely

text: without

text: any

text: interaction.

text: The

text: vortex

text: zero

text: modes

text: obey

text: to

text: this

text: freedom

text: in

text: the

text: critical

text: point

text: between

text: Type

text: II

text: and

text: Type

text: I

text: superconductivity

text: phases,

text: in

text: the

text: ﬁrst

text: case

text: surge

text: repulsive

text: whereas

text: in

text: the

text: second

text: case

text: attractive

text: forces

text: between

text: vortices.

text: We

text: shall

text: discuss

text: wether

text: or

text: not

text: this

text: classical

text: picture

text: is

text: maintained

text: at

text: one-loop

text: order

text: after

text: the

text: eﬀect

text: of

text: vortex

text: ﬂuctuations

text: is

text: accounted

text: for.

text: We

text: shall

text: perform

text: the

text: pertinent

text: calculations

text: generalizing

text: the

text: improved

text: GdW

text: heat

text: kernel

text: procedure

text: developed

text: previously

text: for

text: scalar

text: ﬁeld

text: theories

text: to

text: Abelian

text: gauge

text: theories

text: with

text: spontaneous

text: symmetry

text: breaking.

text: The

text: analysis

text: will

text: be

text: ﬁrst

text: focused

text: in

text: the

text: planar

text: AHM

text: where

text: the

text: BPS

text: vortices

text: are

text: topological

text: solitons.

text: After

text: that

text: we

text: shall

text: move

text: to

text: study

text: the

text: same

text: problem

text: in

text: a

text: three

text: dimensional

text: space,

text: where

text: we

text: ﬁnd

text: BPS

text: vortices

text: magnetic

text: tubes

text: or

text: vortex

text: strings.

text: To

text: perform

text: this

text: task,

text: evaluation

text: of

text: the

text: quantum

text: corrections

text: to

text: BPS

text: vortices

text: by

text: using

text: the

text: modiﬁed

text: GdW

text: expansion

text: we

text: need

text: a

text: detailed

text: information

text: of

text: the

text: spectrum

text: of

text: the

text: matrix

text: second-order

text: PDO

text: that

text: governs

text: the

text: vortex

text: ﬂuctuations.

text: All

text: the

text: information

text: needed

text: about

text: vortex

text: zero

text: modes

text: and

text: bound

text: states

text: is

text: collected

text: in

text: our

text: recent

text: papers

text: and

text: where

text: pertinent

text: References

text: can

text: be

text: found.

text: The

text: organization

text: of

text: the

text: paper

text: is

text: as

text: follows:

text: In

text: Section

text: we

text: thoroughly

text: address

text: the

text: problem

text: de-

text: scribed

text: above

text: in

text: the

text: planar

text: system.

text: Subsection

text: summarize

text: the

text: well

text: known

text: facts

text: about

text: planar

text: BPS

text: ﬂuctuations,

text: in

text: sub-Section

text: the

text: modiﬁed

text: or

text: improved

text: GdW

text: heat

text: kernel

text: expansion

text: is

text: generalized

text: to

text: planar

text: Abelian

text: gauge

text: systems

text: with

text: spontaneous

text: symmetry

text: breaking

text: by

text: one

text: scalar

text: ﬁeld,

text: and

text: ﬁnally,

text: in

text: sub-Section

text: the

text: one-loop

text: mass

text: shifts

text: of

text: rotationally

text: symmetric

text: planar

text: BPS

text: vortices

text: are

text: computed.

text: The

text: new

text: one-loop

text: mass

text: vortex

text: shifts

text: performed

text: in

text: this

text: work,

text: although

text: qualitatively

text: compatible

text: with

text: those

text: obtained

text: in

text: are

text: of

text: much

text: greater

text: precision

text: because

text: the

text: new

text: technique

text: is

text: able

text: to

text: incorporate

text: also

text: the

text: eﬀect

text: of

text: zero

text: modes

text: in

text: the

text: spectral

text: zeta

text: function.

text: Section

text: is

text: fully

text: devoted

text: to

text: describe

text: the

text: quantum

text: corrections

text: at

text: one

text: loop

text: order

text: of

text: the

text: BPS

text: vortex

text: string

text: tensions

text: in

text: the

text: three

text: dimensional

text: Abelian

text: Higgs

text: model.

text: These

text: results

text: are

text: completely

text: new.

text: Some

text: conclusions

text: about

text: the

text: new

text: approach

text: developed

text: here

text: and

text: the

text: induction

text: of

text: repulsive

text: forces

text: between

text: BPS

text: due

text: to

text: their

text: quantum

text: ﬂuctuations

text: oﬀered

text: in

text: Section


text: (cid:68) (cid:69) − distinguishing between the zero mode contribution 8πN and the contribution of the vortex ﬁelds ex- pressed in terms of the old second Seeley coeﬃcient, that is, derived in the standard GdW procedure, proportional to Σ (ψ, V )(cid:88)where Σ is(cid:88): (cid:88) (cid:88) 2 α 2 −5 | | − 1 2 2 13 − | | − 2 − | | 1 2 Σ (ψ, V ) = ∆ ψ 2 ∆( V V ) + 4 (D ψ )2 + (1 ψ 2)2 2 V V (1 ψ 2) + ( V V )2 . 2 α 6 3 α α α β 4 α α 3 α α α=1 α,β=1 α=1 α=1 All the singular contributions to the vortex Casimir energy can be rearranged in the form: (cid:40) (cid:16) (cid:17) (cid:16) (cid:39) (cid:17) (cid:88) ∆EC(1)(s) + ∆EC(2)(s) + ∆ECZM (s) →− V V V s 1/2 − − − (cid:39) (cid:126) µ2 s m√L √ 1 1 γ √ψ( 1) 4 H →−(cid:16) (cid:17) µ (cid:16) + 2(cid:17)(cid:68)+ o(s + 1)(cid:69) [c ( +)] u2+ s 1/2 m2 16π π 2 π(s + 1) 2 π 2 1 aa 2 a=1 − − (cid:126) µ2 s m√L √ 1 γ + ψ√( 1) − (cid:41) + µ(cid:16) (cid:17) (cid:16) + 2 + o(s + 1) Σ (ψ, V ) (cid:17) m2 16π π 2 π(s + 1) 2 π 2 2 α 2 − − − − −(cid:126) µ2 s m√L √ 1 1 γ √ψ( 1) − √ 1 γ + ψ√( 1) µ + 2 + 2 + o(s + 1) Nu2 m2 2 π 2 π(s + 1) 2 π 2 π(s + 1) 2 π 2 2 2 There appear three types of singularities that need to be cancelled: (1) In the ﬁrst line the diver- gences appear in the contribution to the vortex Casimir energy of the ﬁrst Seeley coeﬃcients. The heat kernel renormalization criterion demands exact cancellation of this divergent term proportional to u2 by subtracting the appropriate contribution to the energy of some mass renormalization counter-terms. In particular a minimal renormalization scheme must be implemented to tame the quadratic divergences of the Higgs tadpole plus the self-energy graphs of all the scalar and vector bosons, as well as the fermionic ghosts. Use of the vacuum spectral zeta function is convenient to regularize the pertinent divergent graphs. We will not develop this delicate procedure here, see [8, 9, 28, 29, 34] to see how this renor- malization works in the superenormalizable, henceforth, easier planar AHM. Simply we shall take equal to zero the contribution written in the ﬁrst line legitimated by the heat kernel renormalization criterion 4. (2) The same situation happens with the divergent contributions in the second line coming from the old second Seeley coeﬃcient because it is proportional to u0 and survives in the inﬁnite mass limit. The divergences, even being smoother, are more involved. One must cope now with the subdominant logarith- mic divergences of the graphs just mentioned plus the logarithmic divergences of one-loop graphs with three Higgs legs plus all the one-loop graphs with four external legs of the ﬁelds working in the AHM. This means that we shall use the energies due to the counter-terms arising in the coupling constant and wave function 5 renormalizations adjusted to exactly cancel the contribution in the second line. (3) In the third line we observe an exact cancelation between the divergences due to the zero modes. There is, however, a ﬁnite remnant that must be kept because the heat kernel renormalization criterion does not apply to massless ﬂuctuations. → − Finally the one-loop vortex mass shift per length unit is obtained by taking the limit s 1 in the 2 sum of the ﬁnite remnant of the whole zero mode contribution plus the partial sum up to n order in T the series of ﬁnite terms ∆EC3(s) taking (cid:88)of co(cid:88)urse the physical value u = 1: V ∆EC − (cid:126)m2 nT 4 H − − (cid:126)m2 V = [c ( +)] Γ[n 2] N . (48) L 32π2 n aa 4π n=3 a=1 H H 4We remark that tr c ( +), like tr c ( +), is infrared divergent, although only as log L2. Mass renormalization takes 1 0 care also of this infrared divergence. 5The terms which are ﬁeld derivatives in Σ are exactly canceled by wave function renormalization of the scalar and 2 vector massive particles. 23 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

text: distinguishing

text: between

text: the

text: zero

text: mode

text: contribution

text: and

text: the

text: contribution

text: of

text: the

text: vortex

text: ﬁelds

text: ex-

text: pressed

text: in

text: terms

text: of

text: the

text: old

text: second

text: Seeley

text: coeﬃcient,

text: that

text: is,

text: derived

text: in

text: the

text: standard

text: GdW

text: procedure,

text: proportional

text: to

text: Σ

text: (ψ,

text: V

text: Σ

text: α

text: −5

text: Σ

text: (ψ,

text: V

text: ψ

text: V

text: V

text: (D

text: ψ

text: ψ

text: V

text: V

text: ψ

text: V

text: V

text: α

text: α

text: α

text: α

text: β

text: α

text: α

text: α

text: α

text: All

text: the

text: singular

text: contributions

text: to

text: the

text: vortex

text: Casimir

text: energy

text: can

text: be

text: rearranged

text: in

text: the

text: form:

text: ∆ECZM

text: (s)

text: V

text: V

text: V

text: s

text: µ2

text: s

text: m√L

text: γ

text: √ψ(

text: H

text: µ

text: o(s

text: [c

text: s

text: π

text: π(s

text: π

text: aa

text: µ2

text: s

text: m√L

text: γ

text: ψ√(

text: o(s

text: Σ

text: (ψ,

text: V

text: π

text: π(s

text: π

text: α

text: µ2

text: s

text: m√L

text: γ

text: √ψ(

text: γ

text: ψ√(

text: µ

text: o(s

text: π

text: π(s

text: π

text: π(s

text: π

text: There

text: appear

text: three

text: types

text: of

text: singularities

text: that

text: need

text: to

text: be

text: cancelled:

text: In

text: the

text: ﬁrst

text: line

text: the

text: diver-

text: gences

text: appear

text: in

text: the

text: contribution

text: to

text: the

text: vortex

text: Casimir

text: energy

text: of

text: the

text: ﬁrst

text: Seeley

text: coeﬃcients.

text: The

text: heat

text: kernel

text: renormalization

text: criterion

text: demands

text: exact

text: cancellation

text: of

text: this

text: divergent

text: term

text: proportional

text: to

text: by

text: subtracting

text: the

text: appropriate

text: contribution

text: to

text: the

text: energy

text: of

text: some

text: mass

text: renormalization

text: counter-terms.

text: In

text: particular

text: a

text: minimal

text: renormalization

text: scheme

text: must

text: be

text: implemented

text: to

text: tame

text: the

text: quadratic

text: divergences

text: of

text: the

text: Higgs

text: tadpole

text: plus

text: the

text: self-energy

text: graphs

text: of

text: all

text: the

text: scalar

text: and

text: vector

text: bosons,

text: as

text: well

text: as

text: the

text: fermionic

text: ghosts.

text: Use

text: of

text: the

text: vacuum

text: spectral

text: zeta

text: function

text: is

text: convenient

text: to

text: regularize

text: the

text: pertinent

text: divergent

text: graphs.

text: We

text: will

text: not

text: develop

text: this

text: delicate

text: procedure

text: here,

text: see

text: to

text: see

text: how

text: this

text: renor-

text: malization

text: works

text: in

text: the

text: superenormalizable,

text: henceforth,

text: easier

text: planar

text: AHM.

text: Simply

text: we

text: shall

text: take

text: equal

text: to

text: zero

text: the

text: contribution

text: written

text: in

text: the

text: ﬁrst

text: line

text: legitimated

text: by

text: the

text: heat

text: kernel

text: renormalization

text: criterion

text: The

text: same

text: situation

text: happens

text: with

text: the

text: divergent

text: contributions

text: in

text: the

text: second

text: line

text: coming

text: from

text: the

text: old

text: second

text: Seeley

text: coeﬃcient

text: because

text: it

text: is

text: proportional

text: to

text: and

text: survives

text: in

text: the

text: inﬁnite

text: mass

text: limit.

text: The

text: divergences,

text: even

text: being

text: smoother,

text: are

text: more

text: involved.

text: One

text: must

text: cope

text: now

text: with

text: the

text: subdominant

text: logarith-

text: mic

text: divergences

text: of

text: the

text: graphs

text: just

text: mentioned

text: plus

text: the

text: logarithmic

text: divergences

text: of

text: one-loop

text: graphs

text: with

text: three

text: Higgs

text: legs

text: plus

text: all

text: the

text: one-loop

text: graphs

text: with

text: four

text: external

text: legs

text: of

text: the

text: ﬁelds

text: working

text: in

text: the

text: AHM.

text: This

text: means

text: that

text: we

text: shall

text: use

text: the

text: energies

text: due

text: to

text: the

text: counter-terms

text: arising

text: in

text: the

text: coupling

text: constant

text: and

text: wave

text: function

text: renormalizations

text: adjusted

text: to

text: exactly

text: cancel

text: the

text: contribution

text: in

text: the

text: second

text: line.

text: In

text: the

text: third

text: line

text: we

text: observe

text: an

text: exact

text: cancelation

text: between

text: the

text: divergences

text: due

text: to

text: the

text: zero

text: modes.

text: There

text: is,

text: however,

text: a

text: ﬁnite

text: remnant

text: that

text: must

text: be

text: kept

text: because

text: the

text: heat

text: kernel

text: renormalization

text: criterion

text: does

text: not

text: apply

text: to

text: massless

text: ﬂuctuations.

text: Finally

text: the

text: one-loop

text: vortex

text: mass

text: shift

text: per

text: length

text: unit

text: is

text: obtained

text: by

text: taking

text: the

text: limit

text: s

text: in

text: the

text: sum

text: of

text: the

text: ﬁnite

text: remnant

text: of

text: the

text: whole

text: zero

text: mode

text: contribution

text: plus

text: the

text: partial

text: sum

text: up

text: to

text: n

text: order

text: in

text: T

text: the

text: series

text: of

text: ﬁnite

text: terms

text: taking

text: the

text: physical

text: value

text: u

text: V

text: ∆EC

text: nT

text: H

text: V

text: [c

text: Γ[n

text: N

text: L

text: n

text: aa

text: 4π

text: H

text: H

text: remark

text: that

text: tr

text: c

text: like

text: tr

text: c

text: is

text: infrared

text: divergent,

text: although

text: only

text: as

text: log

text: Mass

text: renormalization

text: takes

text: care

text: also

text: of

text: this

text: infrared

text: divergence.

text: terms

text: which

text: are

text: ﬁeld

text: derivatives

text: in

text: Σ

text: are

text: exactly

text: canceled

text: by

text: wave

text: function

text: renormalization

text: of

text: the

text: scalar

text: and

text: vector

text: massive

text: particles.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: Plugging this ansatz into the heat equation (15) another equation for A((cid:126)x, (cid:126)y; β) (usually called transfer equation) arises that is solved by expanding A as a power series in β. H This procedure is well behaved if the spectrum of the operator is strictly positive provided that the inﬁnite temperature condition A((cid:126)x, (cid:126)y; 0) = I × is ﬁxed. However, if the operator exhibits zero modes D D the factorization (18) is inconsistent because the left and right members in (18) behave in diﬀerent ways at zero temperature, see (16) and (17(cid:88)): Nzm † lim KH((cid:126)x, (cid:126)y; β) = Ξ ((cid:126)x) Ξ ((cid:126)y) , lim KH ((cid:126)x, (cid:126)y; β) = 0 , (19) β→+∞ 0(cid:96) 0(cid:96) β→+∞ 0 (cid:96)=1 → ∞ due to the fact that A((cid:126)x, (cid:126)y; β) grows as a power of β when β + . In order to amend this discrepancy we replace the factorization (18) by the foll(cid:88)owing one: Nzm −(cid:107)(cid:126)x−y(cid:126)(cid:107)2 † KH((cid:126)x, (cid:126)y; β) = C((cid:126)x, (cid:126)y; β) KH0((cid:126)x, (cid:126)y; β) + e 4β Ξ0(cid:96)((cid:126)x) Ξ0(cid:96)((cid:126)y) G(β) . (20) (cid:96)=1 Good agreement between the zero temperature regime when zero modes are present, together the usual conditions at inﬁnity temperature not aﬀected by zero modes, are guaranteed provided that the matrix function G(β) and the matrix density C((cid:126)x, (cid:126)y; β) satisfy: lim G(β) = I × , lim G(β) = 0 , lim C((cid:126)x, (cid:126)y; β) = I × (21) → ∞ D D → → D D β + β 0 β 0 The matrix density C((cid:126)x, (cid:126)y; β), like A((cid:126)x, (cid:126)y; β) in the standard GdW method, relates the positive part H H H of Spec to Spec in the -heat kernel, whereas the second term in the right hand side of (20) encodes 0 the contribution of zero modes. The power series expansion (cid:88) ∞ C((cid:126)x, (cid:126)y; β) = c ((cid:126)x, (cid:126)y) βn (22) n n=0 together with the factorization (20) is plugged into the heat equation (15) as in the standard GdW procedure. The PDE (15) is converted thereafter into the following relations between the coeﬃcients of the modiﬁed GdW expansion and their derivatives: (cid:88) (cid:104) ∞ − 1 ((cid:126)x − (cid:126)y) · Q(cid:126) ((cid:126)x) c ((cid:126)x, (cid:126)y) + (n + 1)c ((cid:126)x, (cid:126)y) − ∆c ((cid:126)x, (cid:126)y) + ((cid:126)x − (cid:126)y) · ∇(cid:126) c(cid:105) ((cid:126)x, (cid:126)y) + 0 n+1 n n+1 2β n=0 + U(cid:88)((cid:126)x) c(cid:104)((cid:126)x, (cid:126)y) + [u2, c(cid:16)((cid:126)x, (cid:126)y)] + Q(cid:126) ((cid:126)x) · ∇(cid:126)(cid:17)c ((cid:126)x, (cid:126)y) − 1 ((cid:126)x − (cid:126)y) · Q(cid:126) ((cid:126)x) c ((cid:126)x, (cid:126)y) βn + n n n n+1 2 + Nzm 4π Ξ ((cid:126)x) Ξ† ((cid:126)y) β dG(β) + G((cid:105)β) + ((cid:126)x − (cid:126)y) · ∇(cid:126) Ξ ((cid:126)x) Ξ† ((cid:126)y) G(β) − 0(cid:96) 0(cid:96) dβ 0(cid:96) 0(cid:96) (cid:96)=1 − 1 ((cid:126)x − (cid:126)y) · Q(cid:126) ((cid:126)x) Ξ ((cid:126)x) Ξ† ((cid:126)y) G(β) eβu2 = 0 . (23) 2 0(cid:96) 0(cid:96) → Taking into account that eventually we shall take the limit (cid:126)y (cid:126)x we can neglect the contribution of the ﬁrst term in this relation. Before of attempting to solve (23) there is the need of selecting G(β). 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Plugging

text: this

text: ansatz

text: into

text: the

text: heat

text: equation

text: another

text: equation

text: for

text: β)

text: (usually

text: called

text: transfer

text: equation)

text: arises

text: that

text: is

text: solved

text: by

text: expanding

text: A

text: as

text: a

text: power

text: series

text: in

text: β.

text: H

text: This

text: procedure

text: is

text: well

text: behaved

text: if

text: the

text: spectrum

text: of

text: the

text: operator

text: is

text: strictly

text: positive

text: provided

text: that

text: the

text: inﬁnite

text: temperature

text: condition

text: I

text: is

text: ﬁxed.

text: However,

text: if

text: the

text: operator

text: exhibits

text: zero

text: modes

text: D

text: D

text: the

text: factorization

text: is

text: inconsistent

text: because

text: the

text: left

text: and

text: right

text: members

text: in

text: behave

text: in

text: diﬀerent

text: ways

text: at

text: zero

text: temperature,

text: see

text: and

label: equation

text: Nzm

text: lim

text: β)

text: Ξ

text: Ξ

text: lim

text: KH

text: β)

text: β→+∞

text: β→+∞

text: due

text: to

text: the

text: fact

text: that

text: β)

text: grows

text: as

text: a

text: power

text: of

text: β

text: when

text: β

text: In

text: order

text: to

text: amend

text: this

text: discrepancy

text: we

text: replace

text: the

text: factorization

text: by

text: the

text: one:

text: Nzm

text: β)

text: β)

text: β)

text: e

text: 4β

text: G(β)

text: Good

text: agreement

text: between

text: the

text: zero

text: temperature

text: regime

text: when

text: zero

text: modes

text: are

text: present,

text: together

text: the

text: usual

text: conditions

text: at

text: inﬁnity

text: temperature

text: not

text: aﬀected

text: by

text: zero

text: modes,

text: are

text: guaranteed

text: provided

text: that

text: the

text: matrix

text: function

text: G(β)

text: and

text: the

text: matrix

text: density

text: β)

text: satisfy:

text: lim

text: G(β)

text: I

text: lim

text: G(β)

text: lim

text: β)

text: I

text: D

text: D

text: D

text: D

text: β

text: β

text: β

text: The

text: matrix

text: density

text: β),

text: like

text: β)

text: in

text: the

text: standard

text: GdW

text: method,

text: relates

text: the

text: positive

text: part

text: H

text: H

text: H

text: of

text: Spec

text: to

text: Spec

text: in

text: the

text: -heat

text: kernel,

text: whereas

text: the

text: second

text: term

text: in

text: the

text: right

text: hand

text: side

text: of

text: encodes

text: the

text: contribution

text: of

text: zero

text: modes.

text: The

text: power

text: series

text: expansion

text: β)

text: c

text: βn

text: n

text: together

text: with

text: the

text: factorization

text: is

text: plugged

text: into

text: the

text: heat

text: equation

text: as

text: in

text: the

text: standard

text: GdW

text: procedure.

text: The

text: PDE

text: is

text: converted

text: thereafter

text: into

text: the

text: following

text: relations

text: between

text: the

text: coeﬃcients

text: of

text: the

text: modiﬁed

text: GdW

text: expansion

text: and

text: their

text: derivatives:

text: c

text: (n

text: ∆c

text: n

text: 2β

text: c

text: βn

text: n

text: n

text: n

text: Nzm

text: 4π

text: Ξ

text: Ξ†

text: β

text: dG(β)

text: Ξ

text: Ξ†

text: G(β)

text: dβ

text: Ξ

text: Ξ†

text: G(β)

text: Taking

text: into

text: account

text: that

text: eventually

text: we

text: shall

text: take

text: the

text: limit

text: we

text: can

text: neglect

text: the

text: contribution

text: of

text: the

text: ﬁrst

text: term

text: in

text: this

text: relation.

text: Before

text: of

text: attempting

text: to

text: solve

text: there

text: is

text: the

text: need

text: of

text: selecting

text: G(β).

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 ing non-perturbative gluon masses in order to regulate functional form for the uncorrelated part of the proba- the long-range Coulomb tails characteristic of perturba- bility distribution: tive emission kernels [11, 12]. (cid:32) (cid:33) (cid:90) ∞ (cid:32) (cid:33) In this work we shall not delve into those arguments (cid:89)3 − (cid:89)3 √ and simply assume that hot spots are adequate degrees d((cid:126)si; R) → dx xe x d((cid:126)si; xR) . (6) of freedom to discuss inclusive proton-proton scattering i=1 0 i=1 at high energies. Our main assumption about their dy- namical properties is that collisions between hot spots In the absence of non-trivial correlations the latter are fully absorptive over distances smaller than their ra- form yields, after Fourier transforming back to momen- dius, R . Hence, in our effective description, hot spots tum space, a dipole electromagnetic form factor F(t) ∼ hs appear as small black disks of average radius R . We 1/(1 − t4/R4)2, in better agreement with data than a hs also assume implicitly that their ultimate dynamical ori- purely Gaussian one. Nonetheless, the main conclu- gin is correlated to the valence partons, since we model sions of this work are not affected by either choice. Fi- the proton as composed of N =3 hot spots. nally, all the correlation structure is encoded in the func- hs Our goal is to construct the elastic scattering ampli- tion f which, by deﬁnition, is not factorisable in the spa- tial coordinates of the hot spots. We write tude in proton-proton collisions in impact parameter representation. We describe pp interactions as a collision of two systems, each one composed of three hot spots. (cid:89)3 (cid:16) − | − | (cid:17) According to the Glauber model, the natural framework f((cid:126)s1,(cid:126)s2,(cid:126)s3) = δ(2)((cid:126)s1 + (cid:126)s2 + (cid:126)s3) 1 − e µ (cid:126)si (cid:126)sj 2/R2 . to describe high-energy scattering of composite parti- i<j cles, the elastic amplitude for a collision of particles A i,j=1 (7) and B with the hot spots frozen in transverse positions {(cid:126)s } has the form: i The δ-function in Eq. 7 ensures that the hot spots system is described with respect to the proton centre (cid:89)3 (cid:89)3 (cid:104) (cid:105) of mass, thus preventing it from acquiring unphysical Tel((cid:126)b) = 1 − 1 − Θ((cid:126)b + (cid:126)sAi − (cid:126)sBj ) , (2) transverse momentum. Next, we implement repulsive i=1 j=1 short-range correlations between all pairs of hot spots controlled by an effective repulsive core r ≡ R/µ. In the where Θ denotes the scattering amplitude of the i-th limit µ → ∞ we recover the uncorrelatecd case. While and j-th hot spots interaction and (cid:126)b is the impact pa- we have no clear dynamical justiﬁcation for these corre- rameter of the collisions. The physical elastic amplitude lations, their main role in our calculation is to enforce a is obtained after averaging Eq. 2 over the transverse po- larger transverse separation between hot spots with re- sitions of the hot spots as given by their probability dis- spect to the completely uncorrelated case. Indeed, all tributions D((cid:126)s ,(cid:126)s ,(cid:126)s ) in the projectile and target, A and realistic models for the electromagnetic nucleon form 1 2 3 B: factors entail non-trivial spatial correlations between the constituent quarks: diquark models, where the proton is (cid:90) (cid:89) envisaged as a bound diquark state interacting with the T(cid:101) ((cid:126)b)= d2sAd2sBD ({(cid:126)sA})D ({(cid:126)sB})T ((cid:126)b) . (3) el k l A k B l el third quark via gluon ﬂux tubes, depict a rod-like struc- k,l ture of a typical string length l ∼1.5 fm. In turn, baryon s junction models, where the Wilson lines link the three The general structure that we shall consider for the joint valence quarks at a junction, yield a more triangular probability distribution for the transverse positions of structure of the proton. We argue that three-dimensional hot spots inside a proton has the following form: realisations of both diquark and baryon junction mod- els, when projected onto the reaction plane, produce a (cid:32) (cid:33) (cid:89)3 similar correlation structure as the two dimensional re- D({(cid:126)si}) = C d((cid:126)si; R) × f((cid:126)s1,(cid:126)s2,(cid:126)s3). (4) pulsive core correlations in Eq. 7. In Fig. 2 we show i=1 the average transverse distance between two hot spots yielded by the uncorrelated distribution, corresponding The constant C is a normalisation constant to ensure to µ → ∞, a correlated one with r = 0.3 fm and the cor- c that the probability distribution is normalised to unity: responding value for two three-dimensional triangular (cid:82) {d2s }D({s }) = 1. The next term corresponds to the distributions projected onto the reaction plane: equilat- i i product of three uncorrelated probability distributions eral and a highly asymmetric isosceles one, which we for a single hot spot, d((cid:126)s ). In order to facilitate a full an- take as proxies for baryon junction and diquark mod- i alytical calculation of the scattering amplitude, we shall els. The results for the probability distribution given by assume them to be of a gaussian form: Eqs. 4-7 provide a good interpolation between the afore- mentioned, more realistic models of proton substruc- (cid:0) (cid:1) d((cid:126)s ; R) = exp −s2/R2 , (5) ture. i i In the Glauber formulation of the scattering process where R is the average radius of the d distribution. It adopted here, the main effect of correlations in the trans- should not be confused with the proton radius itself R . verse positions is to reweight the contribution of the dif- p Additionally we also consider the following baseline ferent terms of the multiple scattering series spanned by ##LTLine##

font: WPOOWI+URWPalladioL-Roma

text: ing

text: non-perturbative

text: gluon

text: masses

text: in

text: order

text: to

text: regulate

text: functional

text: form

text: for

text: the

text: uncorrelated

text: part

text: of

text: the

text: proba-

text: the

text: long-range

text: Coulomb

text: tails

text: characteristic

text: of

text: perturba-

text: bility

text: distribution:

text: tive

text: emission

text: kernels

text: In

text: this

text: work

text: we

text: shall

text: not

text: delve

text: into

text: those

text: arguments

text: and

text: simply

text: assume

text: that

text: hot

text: spots

text: are

text: adequate

text: degrees

text: R)

text: dx

text: xe

text: x

text: xR)

text: of

text: freedom

text: to

text: discuss

text: inclusive

text: proton-proton

text: scattering

text: at

text: high

text: energies.

text: Our

text: main

text: assumption

text: about

text: their

text: dy-

text: namical

text: properties

text: is

text: that

text: collisions

text: between

text: hot

text: spots

text: In

text: the

text: absence

text: of

text: non-trivial

text: correlations

text: the

text: latter

text: are

text: fully

text: absorptive

text: over

text: distances

text: smaller

text: than

text: their

text: ra-

text: form

text: yields,

text: after

text: Fourier

text: transforming

text: back

text: to

text: momen-

text: dius,

text: R

text: Hence,

text: in

text: our

text: effective

text: description,

text: hot

text: spots

text: tum

text: space,

text: a

text: dipole

text: electromagnetic

text: form

text: factor

text: F(t)

text: hs

text: appear

text: as

text: small

text: black

text: disks

text: of

text: average

text: radius

text: R

text: We

text: in

text: better

text: agreement

text: with

text: data

text: than

text: a

text: hs

text: also

text: assume

text: implicitly

text: that

text: their

text: ultimate

text: dynamical

text: ori-

text: purely

text: Gaussian

text: one.

text: Nonetheless,

text: the

text: main

text: conclu-

text: gin

text: is

text: correlated

text: to

text: the

text: valence

text: partons,

text: since

text: we

text: model

text: sions

text: of

text: this

text: work

text: are

text: not

text: affected

text: by

text: either

text: choice.

text: Fi-

text: the

text: proton

text: as

text: composed

text: of

text: N

text: hot

text: spots.

text: nally,

text: all

text: the

text: correlation

text: structure

text: is

text: encoded

text: in

text: the

text: func-

text: hs

text: Our

text: goal

text: is

text: to

text: construct

text: the

text: elastic

text: scattering

text: ampli-

text: tion

text: f

text: which,

text: by

text: deﬁnition,

text: is

text: not

text: factorisable

text: in

text: the

text: spa-

text: tial

text: coordinates

text: of

text: the

text: hot

text: spots.

text: We

text: write

text: tude

text: in

text: proton-proton

text: collisions

text: in

text: impact

text: parameter

text: representation.

text: We

text: describe

text: pp

text: interactions

text: as

text: a

text: collision

text: of

text: two

text: systems,

text: each

text: one

text: composed

text: of

text: three

text: hot

text: spots.

text: According

text: to

text: the

text: Glauber

text: model,

text: the

text: natural

text: framework

text: e

text: µ

text: to

text: describe

text: high-energy

text: scattering

text: of

text: composite

text: parti-

text: i<j

text: cles,

text: the

text: elastic

text: amplitude

text: for

text: a

text: collision

text: of

text: particles

text: A

text: and

text: B

text: with

text: the

text: hot

text: spots

text: frozen

text: in

text: transverse

text: positions

text: has

text: the

text: form:

text: i

text: The

text: δ-function

text: in

text: Eq.

text: ensures

text: that

text: the

text: hot

text: spots

text: system

text: is

text: described

text: with

text: respect

text: to

text: the

text: proton

text: centre

text: of

text: mass,

text: thus

text: preventing

text: it

text: from

text: acquiring

text: unphysical

text: transverse

text: momentum.

text: Next,

text: we

text: implement

text: repulsive

text: short-range

text: correlations

text: between

text: all

text: pairs

text: of

text: hot

text: spots

text: controlled

text: by

text: an

text: effective

text: repulsive

text: core

text: r

text: R/µ.

text: In

text: the

text: where

text: Θ

text: denotes

text: the

text: scattering

text: amplitude

text: of

text: the

text: i-th

text: limit

text: µ

text: we

text: recover

text: the

text: uncorrelatecd

text: case.

text: While

text: and

text: j-th

text: hot

text: spots

text: interaction

text: and

text: is

text: the

text: impact

text: pa-

text: we

text: have

text: no

text: clear

text: dynamical

text: justiﬁcation

text: for

text: these

text: corre-

text: rameter

text: of

text: the

text: collisions.

text: The

text: physical

text: elastic

text: amplitude

text: lations,

text: their

text: main

text: role

text: in

text: our

text: calculation

text: is

text: to

text: enforce

text: a

text: is

text: obtained

text: after

text: averaging

text: Eq.

text: over

text: the

text: transverse

text: po-

text: larger

text: transverse

text: separation

text: between

text: hot

text: spots

text: with

text: re-

text: sitions

text: of

text: the

text: hot

text: spots

text: as

text: given

text: by

text: their

text: probability

text: dis-

text: spect

text: to

text: the

text: completely

text: uncorrelated

text: case.

text: Indeed,

text: all

text: tributions

text: in

text: the

text: projectile

text: and

text: target,

text: A

text: and

text: realistic

text: models

text: for

text: the

text: electromagnetic

text: nucleon

text: form

text: B:

text: factors

text: entail

text: non-trivial

text: spatial

text: correlations

text: between

text: the

text: constituent

text: quarks:

text: diquark

text: models,

text: where

text: the

text: proton

text: is

text: envisaged

text: as

text: a

text: bound

text: diquark

text: state

text: interacting

text: with

text: the

text: el

text: k

text: l

text: A

text: k

text: B

text: l

text: el

text: third

text: quark

text: via

text: gluon

text: ﬂux

text: tubes,

text: depict

text: a

text: rod-like

text: struc-

text: k,l

text: ture

text: of

text: a

text: typical

text: string

text: length

text: l

text: fm.

text: In

text: turn,

text: baryon

text: s

text: junction

text: models,

text: where

text: the

text: Wilson

text: lines

text: link

text: the

text: three

text: The

text: general

text: structure

text: that

text: we

text: shall

text: consider

text: for

text: the

text: joint

text: valence

text: quarks

text: at

text: a

text: junction,

text: yield

text: a

text: more

text: triangular

text: probability

text: distribution

text: for

text: the

text: transverse

text: positions

text: of

text: structure

text: of

text: the

text: proton.

text: We

text: argue

text: that

text: three-dimensional

text: hot

text: spots

text: inside

text: a

text: proton

text: has

text: the

text: following

text: form:

text: realisations

text: of

text: both

text: diquark

text: and

text: baryon

text: junction

text: mod-

text: els,

text: when

text: projected

text: onto

text: the

text: reaction

text: plane,

text: produce

text: a

text: similar

text: correlation

text: structure

text: as

text: the

text: two

text: dimensional

text: re-

text: C

text: R)

text: pulsive

text: core

text: correlations

text: in

text: Eq.

text: In

text: Fig.

text: we

text: show

text: the

text: average

text: transverse

text: distance

text: between

text: two

text: hot

text: spots

text: yielded

text: by

text: the

text: uncorrelated

text: distribution,

text: corresponding

text: The

text: constant

text: C

text: is

text: a

text: normalisation

text: constant

text: to

text: ensure

text: to

text: µ

text: a

text: correlated

text: one

text: with

text: r

text: fm

text: and

text: the

text: cor-

text: c

text: that

text: the

text: probability

text: distribution

text: is

text: normalised

text: to

text: unity:

text: responding

text: value

text: for

text: two

text: three-dimensional

text: triangular

text: }D({s

text: The

text: next

text: term

text: corresponds

text: to

text: the

text: distributions

text: projected

text: onto

text: the

text: reaction

text: plane:

text: equilat-

text: i

text: i

text: product

text: of

text: three

text: uncorrelated

text: probability

text: distributions

text: eral

text: and

text: a

text: highly

text: asymmetric

text: isosceles

text: one,

text: which

text: we

text: for

text: a

text: single

text: hot

text: spot,

text: In

text: order

text: to

text: facilitate

text: a

text: full

text: an-

text: take

text: as

text: proxies

text: for

text: baryon

text: junction

text: and

text: diquark

text: mod-

text: i

text: alytical

text: calculation

text: of

text: the

text: scattering

text: amplitude,

text: we

text: shall

text: els.

text: The

text: results

text: for

text: the

text: probability

text: distribution

text: given

text: by

text: assume

text: them

text: to

text: be

text: of

text: a

text: gaussian

text: form:

text: Eqs.

text: provide

text: a

text: good

text: interpolation

text: between

text: the

text: afore-

text: mentioned,

text: more

text: realistic

text: models

text: of

text: proton

text: substruc-

text: R)

text: exp

text: ture.

text: i

text: i

text: In

text: the

text: Glauber

text: formulation

text: of

text: the

text: scattering

text: process

text: where

text: R

text: is

text: the

text: average

text: radius

text: of

text: the

text: d

text: distribution.

text: It

text: adopted

text: here,

text: the

text: main

text: effect

text: of

text: correlations

text: in

text: the

text: trans-

text: should

text: not

text: be

text: confused

text: with

text: the

text: proton

text: radius

text: itself

text: R

text: verse

text: positions

text: is

text: to

text: reweight

text: the

text: contribution

text: of

text: the

text: dif-

text: p

text: Additionally

text: we

text: also

text: consider

text: the

text: following

text: baseline

text: ferent

text: terms

text: of

text: the

text: multiple

text: scattering

text: series

text: spanned

text: by

text: ##LTLine##


text: 3 . 1.8 Quark-Diquark GeV respectively. To describe the data on dσel/dt = Our model (1/4π) |T (s, t)|2 we use the following parametrisation: Baryon-junction el 1.6 Uncorrelated ImT (s, t) = a eb1t + a eb2t + a eb3t , el 1 2 3 1.4 ReT (s, t) = c ed1t , (9) el 1 1.2 with the ﬁt parameters a and c subject to the con- i i straints: σ =2Σ a and ρ=Σ (c /a ), where the values tot i i i i i 1 r c = 0.3 f m omfeσnttost gainvdenρinco[r1r,e1s3p]o.nFdortothteheLHexCpecraismeewnetaul smeetahseuerxe-- trapolated ρ value provided by the COMPETE Collab- 0.8 oration [14], same as the TOTEM collaboration in their data analysis. With this set up we obtain a very good de- 0.6 scription of dσel/dt, χ2/d.o.f ∼ 1.1 ÷ 2, and conﬁrm the hollowness effect in LHC experimental data, as shown in Fig. 1. Fits with a larger number of free parameters . . 0.40.2 0.25 0.3 0.35 0.4 0.45 lead to similarly good data description. ⟨s21⟩ [fm2] In order to prove that our model actually accounts for FIG. 2: Mean transverse separation squared between pairs of the onset of hollowness effect at high energies, we scan hot spots as a function of the mean transverse position for dif- the parameter space looking for the presence of a dip of { } ferent D( (cid:126)s ). the inelasticity density at zero impact parameter and a i monotonically decreasing scattering amplitude, as dic- tated by data. I.e, we impose the following conditions: Eqs. 2 and 3 with respect to the uncorrelated case. For in- stance, in baryon junction models, terms where the three (cid:12) hwoitthspcootnssitnitouneentpsrooftotnheuntadregregtoasriemsutrlotanngelyoussuspcpartetesrsiendg, d2T(cid:101)el(s, b)(cid:12)(cid:12)(cid:12) < 0, (10) d2b (cid:12) since the three vertices of an equilateral triangle cannot (cid:12)b=0 overlap in the transverse plane, unlike the uncorrelated d2G (s, b)(cid:12) case. The other parameter that controls the amount of din2b (cid:12)(cid:12) > 0 . (11) effective overlap for different scattering conﬁgurations, b=0 and hence their relative contribution to the scattering se- ries, is the hot spot radius. We shall refer to the region of parameter space that fulﬁlls the two above conditions as hollowness region. To conclude the description of our model we have to specify the elastic scattering amplitude between two hot A ﬁrst, important result is that it is not possible to ob- spots separated a transverse distance sij. Again, in or- tain a growing behaviour of Gin(s, b) at zero impact pa- der to facilitate full analytic calculations, we resort to a rameter in the absence of non-trivial correlations, i.e for Gaussian parametrisation: µ → ∞ in Eq. 7 or, equivalently for zero correlation dis- tances r = 0. Indeed, in this case the full calculation of (cid:0) (cid:1) c Θ(sij) = i exp −s2ij/2Rh2s (1 − iρhs) . (8) tbheeshpohwysnictahlastcGat(cid:48)t(cid:48)e(rbin=g0a)m<p0litiundaellsciamspesli,ﬁreesgaarnddleistsctahne in This amplitude can be thought as resulting from the speciﬁc functional form for the uncorrelated weights in convolution of two gaussian density distributions for a Eq. 5. single hot spot, each of radius R . The absorptive part In turn, for non-zero values of the correlation length hs of Eq. 8 is equal to unity for head-on collisions, in line r < R, we ﬁnd a wide region of the parameter space c with our assumption of hot spot as small black discs. Al- compatible with the hollowness effect. In Fig. 3 we show though hadronic amplitudes are expected to be mostly the hollowness region –represented as a dashed area in imaginary at high energies, we allow for a constant real the plots– in the (R , R )-plane for r = 0.3 (top) and p hs c poradrterρhtso, im.eaitncdhetpheendneonnt-zoenrothveamluoems emnetuasmurterdanesxfepre,riin- 0th.5efemffe(bctoitvteompr)o, wtohnerraedwiueshraevseuldtienﬁgnefrdomRp2t≡heRc2on+vRolh2us-, mentally. It should be noted that all the energy depen- tion of the hot spots distribution with their own den- dence in our model has been left implicit so far. As dis- sity distribution. Results in both plots were obtained cussed below, it is encoded in the transverse growth of with 0.05 < ρ < 0.15. We observe that the hollowness ttohhfeaothu, otrotmgsepotdohteelrr.awdiituhsRR,hrscwanitdh ρinhcsraeraesitnhge cfooullrispioanraemneertegrys, rFseoogmrioeﬁnxﬁeendnitlaRerpgvheaasslnuwdeirothcf, ittnhhceerehhaoostilnlsogpwocntoerrsraesdlaeiutfifsoencRtdkiis.ctkaFnsocirenirnac-t. hs Before discussing the results of our model, we brieﬂy stance, ﬁxing R to the value of the measured proton p present an independent analysis of experimental data charge radius R ≈ 0.88 fm the hollowness region starts p on the differential elastic cross section measured at at R ≥ 0.34 and 0.24 fm for r = 0.3 and 0.5 fm re- hs c the LHC and ISR at collision energies 7 TeV and 62.5 spectively. The results shown in Fig. 3 correspond to ]2mf[ ⟩2|2s⃗− 1s⃗|⟨ ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: Quark-Diquark

text: GeV

text: respectively.

text: To

text: describe

text: the

text: data

text: on

text: dσel/dt

text: Our

text: model

text: |T

text: (s,

text: we

text: use

text: the

text: following

text: parametrisation:

text: Baryon-junction

text: el

text: Uncorrelated

text: ImT

text: (s,

text: t)

text: a

text: a

text: a

text: el

text: ReT

text: (s,

text: t)

text: c

text: el

text: with

text: the

text: ﬁt

text: parameters

text: a

text: and

text: c

text: subject

text: to

text: the

text: con-

text: i

text: i

text: straints:

text: σ

text: a

text: and

text: ρ=Σ

text: (c

text: /a

text: where

text: the

text: values

text: tot

text: i

text: i

text: i

text: i

text: i

text: r

text: c

text: f

text: m

text: omfeσnttost

text: smeetahseuerxe--

text: trapolated

text: ρ

text: value

text: provided

text: by

text: the

text: COMPETE

text: Collab-

text: oration

text: same

text: as

text: the

text: TOTEM

text: collaboration

text: in

text: their

text: data

text: analysis.

text: With

text: this

text: set

text: up

text: we

text: obtain

text: a

text: very

text: good

text: de-

text: scription

text: of

text: dσel/dt,

text: and

text: conﬁrm

text: the

text: hollowness

text: effect

text: in

text: LHC

text: experimental

text: data,

text: as

text: shown

text: in

text: Fig.

text: Fits

text: with

text: a

text: larger

text: number

text: of

text: free

text: parameters

text: lead

text: to

text: similarly

text: good

text: data

text: description.

text: In

text: order

text: to

text: prove

text: that

text: our

text: model

text: actually

text: accounts

text: for

text: FIG.

text: Mean

text: transverse

text: separation

text: squared

text: between

text: pairs

text: of

text: the

text: onset

text: of

text: hollowness

text: effect

text: at

text: high

text: energies,

text: we

text: scan

text: hot

text: spots

text: as

text: a

text: function

text: of

text: the

text: mean

text: transverse

text: position

text: for

text: dif-

text: the

text: parameter

text: space

text: looking

text: for

text: the

text: presence

text: of

text: a

text: dip

text: of

text: ferent

text: D(

text: the

text: inelasticity

text: density

text: at

text: zero

text: impact

text: parameter

text: and

text: a

text: i

text: monotonically

text: decreasing

text: scattering

text: amplitude,

text: as

text: dic-

text: tated

text: by

text: data.

text: I.e,

text: we

text: impose

text: the

text: following

text: conditions:

text: Eqs.

text: and

text: with

text: respect

text: to

text: the

text: uncorrelated

text: case.

text: For

text: in-

text: stance,

text: in

text: baryon

text: junction

text: models,

text: terms

text: where

text: the

text: three

text: since

text: the

text: three

text: vertices

text: of

text: an

text: equilateral

text: triangle

text: cannot

text: overlap

text: in

text: the

text: transverse

text: plane,

text: unlike

text: the

text: uncorrelated

text: (s,

text: case.

text: The

text: other

text: parameter

text: that

text: controls

text: the

text: amount

text: of

text: effective

text: overlap

text: for

text: different

text: scattering

text: conﬁgurations,

text: and

text: hence

text: their

text: relative

text: contribution

text: to

text: the

text: scattering

text: se-

text: ries,

text: is

text: the

text: hot

text: spot

text: radius.

text: We

text: shall

text: refer

text: to

text: the

text: region

text: of

text: parameter

text: space

text: that

text: fulﬁlls

text: the

text: two

text: above

text: conditions

text: as

text: hollowness

text: region.

text: To

text: conclude

text: the

text: description

text: of

text: our

text: model

text: we

text: have

text: to

text: specify

text: the

text: elastic

text: scattering

text: amplitude

text: between

text: two

text: hot

text: A

text: ﬁrst,

text: important

text: result

text: is

text: that

text: it

text: is

text: not

text: possible

text: to

text: ob-

text: spots

text: separated

text: a

text: transverse

text: distance

text: sij.

text: Again,

text: in

text: or-

text: tain

text: a

text: growing

text: behaviour

text: of

text: Gin(s,

text: b)

text: at

text: zero

text: impact

text: pa-

text: der

text: to

text: facilitate

text: full

text: analytic

text: calculations,

text: we

text: resort

text: to

text: a

text: rameter

text: in

text: the

text: absence

text: of

text: non-trivial

text: correlations,

text: i.e

text: for

text: Gaussian

text: parametrisation:

text: µ

text: in

text: Eq.

text: or,

text: equivalently

text: for

text: zero

text: correlation

text: dis-

text: tances

text: r

text: Indeed,

text: in

text: this

text: case

text: the

text: full

text: calculation

text: of

text: c

text: Θ(sij)

text: i

text: exp

text: iρhs)

text: in

text: This

text: amplitude

text: can

text: be

text: thought

text: as

text: resulting

text: from

text: the

text: speciﬁc

text: functional

text: form

text: for

text: the

text: uncorrelated

text: weights

text: in

text: convolution

text: of

text: two

text: gaussian

text: density

text: distributions

text: for

text: a

text: Eq.

text: single

text: hot

text: spot,

text: each

text: of

text: radius

text: R

text: The

text: absorptive

text: part

text: In

text: turn,

text: for

text: non-zero

text: values

text: of

text: the

text: correlation

text: length

text: hs

text: of

text: Eq.

text: is

text: equal

text: to

text: unity

text: for

text: head-on

text: collisions,

text: in

text: line

text: r

text: R,

text: we

text: ﬁnd

text: a

text: wide

text: region

text: of

text: the

text: parameter

text: space

text: c

text: with

text: our

text: assumption

text: of

text: hot

text: spot

text: as

text: small

text: black

text: discs.

text: Al-

text: compatible

text: with

text: the

text: hollowness

text: effect.

text: In

text: Fig.

text: we

text: show

text: though

text: hadronic

text: amplitudes

text: are

text: expected

text: to

text: be

text: mostly

text: the

text: hollowness

text: region

text: –represented

text: as

text: a

text: dashed

text: area

text: in

text: imaginary

text: at

text: high

text: energies,

text: we

text: allow

text: for

text: a

text: constant

text: real

text: the

text: plots–

text: in

text: the

text: (R

text: R

text: )-plane

text: for

text: r

text: (top)

text: and

text: p

text: hs

text: c

text: poradrterρhtso,

text: emnetuasmurterdanesxfepre,riin-

text: mentally.

text: It

text: should

text: be

text: noted

text: that

text: all

text: the

text: energy

text: depen-

text: tion

text: of

text: the

text: hot

text: spots

text: distribution

text: with

text: their

text: own

text: den-

text: dence

text: in

text: our

text: model

text: has

text: been

text: left

text: implicit

text: so

text: far.

text: As

text: dis-

text: sity

text: distribution.

text: Results

text: in

text: both

text: plots

text: were

text: obtained

text: cussed

text: below,

text: it

text: is

text: encoded

text: in

text: the

text: transverse

text: growth

text: of

text: with

text: ρ

text: We

text: observe

text: that

text: the

text: hollowness

text: ttohhfeaothu,

text: ρinhcsraeraesitnhge

text: cfooullrispioanraemneertegrys,

text: hs

text: Before

text: discussing

text: the

text: results

text: of

text: our

text: model,

text: we

text: brieﬂy

text: stance,

text: ﬁxing

text: R

text: to

text: the

text: value

text: of

text: the

text: measured

text: proton

text: p

text: present

text: an

text: independent

text: analysis

text: of

text: experimental

text: data

text: charge

text: radius

text: R

text: fm

text: the

text: hollowness

text: region

text: starts

text: p

text: on

text: the

text: differential

text: elastic

text: cross

text: section

text: measured

text: at

text: at

text: R

text: and

text: fm

text: for

text: r

text: and

text: fm

text: re-

text: hs

text: c

text: the

text: LHC

text: and

text: ISR

text: at

text: collision

text: energies

text: TeV

text: and

text: spectively.

text: The

text: results

text: shown

text: in

text: Fig.

text: correspond

text: to

text: ##LTFigure##

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: 4 between the ground and excited states at each (cid:126)k can be read The eigenvalues λI± (t) of UI(t) provide an expression for off from Eq. (10) as64 the phase bands in reg(cid:126)ikon I and(cid:126)kare given by − | | | | p(cid:126)k = exp[ 2πδ(cid:126)k], δ(cid:126)k = ∆(cid:126)k 2/ 2v(cid:126)k . (11) λI±(cid:126)k(t) = e±iφ(cid:126)Ik(t), cos[φI(t)] = µ (t) cos(ξ (t, 0)). (18) We note that p = 1 for an unavoided level crossing which (cid:126)k (cid:126)k (cid:126)k (cid:126)k − happens for the critical mode (where ∆ = 0 and g(t) b (cid:126)k (cid:54) (cid:126)k crosses zero), while it is zero if ∆(cid:126)k = 0 but g(t) = b(cid:126)k at any An exactly similar method can be used to compute the evo- point of time during the drive. It was shown in Ref. 65 that lution operators in regions II and III. For example, in region within this appr(cid:32)oxi(cid:112)mations we can deﬁne a transfer(cid:33)matrix II, using Eqs. (5), (9), (12), and (13), we obtain (cid:18) (cid:19) (cid:18) (cid:19) (cid:112) √ − − − S(cid:126)k = π 1 (cid:0)√pp(cid:126)k(cid:126)ke −iφ˜s(cid:126)k(cid:1) 1 − p(cid:126)pk(cid:126)keiφ−˜s(cid:126)k , (12) ccI1I2(cid:126)(cid:126)kkII((tt)) = U(cid:126)k(cid:48) (t, t1(cid:126)k)S(cid:126)kU(cid:126)k(cid:48) (t1(cid:126)k, 0) 10 , (19) φ = + δ ln δ 1 + Arg Γ(1 iδ ), (13) s(cid:126)k 4 (cid:126)k (cid:126)k (cid:126)k which leads to − (cid:112) where φ is the Stoke’s phase and φ˜ = φ π. The s(cid:126)k s(cid:126)k s(cid:126)k change of wave functions across the ﬁrst transition point can √ − − be obtain(cid:18)ed using the tra(cid:19)nsfer matr(cid:18)ix S(cid:126)k as (cid:19) cI1(cid:126)kI(t) = 1 p(cid:126)ke iζ1I(cid:126)kI(t), cI2(cid:126)kI(t) = p(cid:126)keiζ2I(cid:126)kI(t), ζII(t) = φ˜ + ξ (t, 0), − 1(cid:126)k s(cid:126)k (cid:126)k − cc1(cid:126)k((tt1(cid:126)k ++ (cid:15)(cid:15))) = S(cid:126)k cc1(cid:126)k((tt1(cid:126)k − (cid:15)(cid:15))) , (14) ζ2I(cid:126)kI(t) = ξ(cid:126)k(t1(cid:126)k, 0) ξ(cid:126)k(t, t1(cid:126)k). (20) 2(cid:126)k 1(cid:126)k 2(cid:126)k 1(cid:126)k where (cid:15) > 0 is inﬁnitesimally small. An analogous condi- Using Eqs. (7), (8) and (20), we then obtain (cid:113) tion with S replaced by ST (where the superscript T denotes transpose) holds for the second transition point. (cid:113) Having obtained these relations, we now obtain explicit an- − (UII(t)) = cII(t)µ (t) + cII(t) 1 µ2(t), alytical expressions for the evolution operator U (t) in each of (cid:126)k 11 1(cid:126)k (cid:126)k 2(cid:126)k (cid:126)k the three regions shown in Fig. 1. In region I, the(cid:126)ksystem starts − − (UII(t)) = cII(t)µ (t) cII(t) (1 µ2(t), (21) from the ground state so that c (0) = 1 and c = 0, Before (cid:126)k 21 2(cid:126)k (cid:126)k 1(cid:126)k (cid:126)k 1(cid:126)k 2(cid:126)k ∗ ∗ − crossing the critical point for the ﬁrst time, the dynamics is (UII(t)) = (UII(t)) , (UII(t)) = (UII(t)) . (cid:126)k 22 (cid:126)k 11 (cid:126)k 12 (cid:126)k 21 purely adiabatic leading to (using Eqs. (5) and (9)) The expression for the phase bands in region II may − cI1(cid:126)k(t) = exp[ iξ(cid:126)k(t, 0)], cI2(cid:126)k(t) = 0. (15) be obtained by diagonalizing the evolution matrix U(cid:126)kII(t). A straightforward calculation yields the expressions for the ± Using Eqs. (5), (7) and(cid:16)(15), we thus(cid:113)obtain (cid:17) eigenvalues of U(cid:126)kII(t) as λI±I(cid:126)k(cid:112)(t) = exp[ iφ(cid:126)IkI(t)], where (cid:113) − |ψ(cid:126)k(t)(cid:105) = e−iξ(cid:126)k(t,0) µ(cid:126)k(t)|ψ(cid:126)kg(cid:105) + 1 − µ(cid:126)k(t)2|ψ(cid:126)ke(cid:105) . cos[φ(cid:126)IkI(t)] = µ(cid:126)k(t) 1− p(cid:126)k cos(ζ1I(cid:126)kI(t)) (16) + p (1 µ2(t)) cos(ζII(t)). (22) (cid:126)k (cid:126)k 2(cid:126)k Using Eqs. (8) and (16) and the fact that U (t) is a unitary A similar calculation can be carried out in region III. Here (cid:126)k matrix, weobtain for region I (cid:113)  one ﬁnds that (cid:18) (cid:19) (cid:18) (cid:19)  (cid:113)  − − − U(cid:126)kI(t) = 1µ−(cid:126)k(µt)(cid:126)2ke(t)ieξ(cid:126)k−(itξ,0(cid:126)k)(t,0) µ1(cid:126)k(tµ)e(cid:126)2ki(ξt(cid:126)k)(et,i0ξ)(cid:126)k(t,0) . ccI1I2(cid:126)(cid:126)kkIIII((tt)) = U(cid:126)k(cid:48) (t, t2(cid:126)k)S(cid:126)kT U(cid:126)k(cid:48) (t2(cid:126)k, 0) 10 , (23) (17) where ST is the transpose of S (Eqs. (12) and (13)). A (cid:126)k (cid:126)k straightforward calculation then leads to ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: between

text: the

text: ground

text: and

text: excited

text: states

text: at

text: each

text: can

text: be

text: read

text: The

text: eigenvalues

text: λI±

text: (t)

text: of

text: UI(t)

text: provide

text: an

text: expression

text: for

text: off

text: from

text: Eq.

text: the

text: phase

text: bands

text: in

text: I

text: given

text: by

text: exp[

text: cos[φI(t)]

text: µ

text: (t)

text: cos(ξ

text: (t,

text: We

text: note

text: that

text: p

text: for

text: an

text: unavoided

text: level

text: crossing

text: which

text: happens

text: for

text: the

text: critical

text: mode

text: (where

text: and

text: g(t)

text: b

text: crosses

text: zero),

text: while

text: it

text: is

text: zero

text: if

text: but

text: g(t)

text: at

text: any

text: An

text: exactly

text: similar

text: method

text: can

text: be

text: used

text: to

text: compute

text: the

text: evo-

text: point

text: of

text: time

text: during

text: the

text: drive.

text: It

text: was

text: shown

text: in

text: Ref.

text: that

text: lution

text: operators

text: in

text: regions

text: II

text: and

text: III.

text: For

text: example,

text: in

text: region

text: within

text: this

text: we

text: can

text: deﬁne

text: a

text: II,

text: using

text: Eqs.

text: and

text: we

text: obtain

text: π

text: (t,

text: φ

text: δ

text: ln

text: δ

text: Arg

text: iδ

text: which

text: leads

text: to

text: where

text: φ

text: is

text: the

text: Stoke’s

text: phase

text: and

text: φ˜

text: φ

text: π.

text: The

text: change

text: of

text: wave

text: functions

text: across

text: the

text: ﬁrst

text: transition

text: point

text: can

text: be

text: using

text: the

text: as

text: ζII(t)

text: φ˜

text: ξ

text: (t,

text: where

text: is

text: inﬁnitesimally

text: small.

text: An

text: analogous

text: condi-

text: Using

text: Eqs.

text: and

text: we

text: then

text: obtain

text: tion

text: with

text: S

text: replaced

text: by

text: ST

text: (where

text: the

text: superscript

text: T

text: denotes

text: transpose)

text: holds

text: for

text: the

text: second

text: transition

text: point.

text: Having

text: obtained

text: these

text: relations,

text: we

text: now

text: obtain

text: explicit

text: an-

text: (UII(t))

text: cII(t)µ

text: (t)

text: cII(t)

text: alytical

text: expressions

text: for

text: the

text: evolution

text: operator

text: U

text: (t)

text: in

text: each

text: of

text: the

text: three

text: regions

text: shown

text: in

text: Fig.

text: In

text: region

text: I,

text: starts

text: (UII(t))

text: cII(t)µ

text: (t)

text: cII(t)

text: from

text: the

text: ground

text: state

text: so

text: that

text: c

text: and

text: c

text: Before

text: crossing

text: the

text: critical

text: point

text: for

text: the

text: ﬁrst

text: time,

text: the

text: dynamics

text: is

text: (UII(t))

text: (UII(t))

text: (UII(t))

text: (UII(t))

text: purely

text: adiabatic

text: leading

text: to

text: (using

text: Eqs.

text: and

text: The

text: expression

text: for

text: the

text: phase

text: bands

text: in

text: region

text: II

text: may

text: exp[

text: be

text: obtained

text: by

text: diagonalizing

text: the

text: evolution

text: matrix

text: A

text: straightforward

text: calculation

text: yields

text: the

text: expressions

text: for

text: the

text: Using

text: Eqs.

text: we

text: eigenvalues

text: of

text: as

text: exp[

text: where

text: 1−

text: p

text: cos(ζII(t)).

text: Using

text: Eqs.

text: and

text: and

text: the

text: fact

text: that

text: U

text: (t)

text: is

text: a

text: unitary

text: A

text: similar

text: calculation

text: can

text: be

text: carried

text: out

text: in

text: region

text: III.

text: Here

text: matrix,

text: weobtain

text: for

text: region

text: I

text: one

text: ﬁnds

text: that

text: (t,

text: where

text: ST

text: is

text: the

text: transpose

text: of

text: S

text: (Eqs.

text: and

text: A

text: straightforward

text: calculation

text: then

text: leads

text: to

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: ∗ Electronic structure of cyanocobalamin: DFT+QMC study Selma Mayda1,2, Zafer Kandemir1, and Nejat Bulut1 (Dated: May 29, 2020) We study the electronic structure and the magnetic correlations of cyanocobalamin (C H CoN O P) by 63 88 14 14 using the framework of the multi-orbital single-impurity Haldane-Anderson model of a transition metal impurity in a semiconductor host. Here, we ﬁrst determine the parameters of the Anderson Hamiltonian by performing density functional theory (DFT) calculations. Then, we use the quantum Monte Carlo (QMC) technique to obtain the electronic structure and the magnetic correlation functions for this effective model. We ﬁnd that new electronic states, which correspond to impurity bound states, form above the lowest unoccupied level of the host semiconductor. These new states derive from the atomic orbitals at the cobalt site and the rest of the molecule. We observe that magnetic moments develop at the Co(3dν) orbitals and over the surrounding sites. We also observe that antiferromagnetic correlations exist between the Co(3dν) orbitals and the surrounding atoms. These antiferromagnetic correlations depend on the ﬁlling of the impurity bound states. I. INTRODUCTION and magnetic correlations of this molecule within the frame- work of the Haldane-Anderson model7. This model was ini- tially introduced to describe the electronic state of Au in a Vitamin B is a very important organometallic molecule 12 for biological systems1–3. In this paper, we study the elec- semiconductor Ge host. Here, we use the Haldane-Anderson model because CNCbl exhibits a semiconductor energy gap tronic structure of cyanocobalamin (CNCbl) which is a form and contains the transition-metal cobalt atom. of vitamin B . In Fig. 1, we illustrate the molecular structure 12 of CNCbl. The cobalt atom neighbours ﬁve nitrogen atoms, of which four are located in the corrin ring. The CN ligand is also In the combined DFT+QMC approach, we ﬁrst determine attached to Co, making the rare cobalt-carbon bonding. The the parameters of the Anderson Hamiltonian8 by the DFT charge neutral CNCbl molecule has 718 electrons. For this calculations carried out with the Gaussian program9. Then, molecule, the HOMO (highest occupied molecular orbital) we study this effective Anderson Hamiltonian by perform- and the LUMO (lowest unoccupied molecular orbital) levels ≈ ing QMC simulations with the Hirsch-Fye10 algorithm. In are separated by an energy gap of 2.2 eV4. The photoab- ≈ our calculations, we use an orbital independent intra-orbital sorption spectrum of CNCbl exhibits distinct peaks at 3.5 Coulomb interaction U at the Co(3d ) orbitals and we neglect eV and 4.5 eV, of which origin remains elusive4. In additon, ν the inter-orbital Coulomb interactions along with the Hund’s it is known that CNCbl has a weak diamagnetic response5,6. coupling. We take the value of U to be 4 eV. In addition, the Co(3d ) energy levels are shifted in order to prevent the dou- ν ble counting of the local Coulomb interactions by both DFT and QMC. The outline of this paper is as follows. In Section 2, we obtain the one-electron parameters of the effective Anderson Hamiltonian. We present the QMC data on the electronic structure and the magnetic correlation functions in Section 3. Here, we ﬁnd that impurity bound states develop above the LUMO level for CNCbl instead of being inside the semicon- ducting energy gap. This is due to the discrete single-particle spectrum of the molecule. We observe that magnetic mo- ments develop at the Co(3d ) orbitals and at the host states ν which have the strongest hybridization. We also observe that there are antiferromagnetic correlations between these host FIG. 1. (Color online) Schematic plot of the molecular structure of magnetic moments and the Co(3d ) moments. These antifer- ν cyanocobalamin (C H CoN O P), which contains 181 atoms. 63 88 14 14 romagnetic correlations disappear when the impurity bound states are ﬁlled. We note that in Ref. [11], we combined Despite many years of research, there remains questions the Hartree-Fock (HF) approximation with the QMC tech- about the electronic structure and the functioning of vitamin nique to study CNCbl. In the HF+QMC calculations, we used B as well as the role of the transition-metal cobalt atom. In only the smaller imidazole part of the CNCbl instead of the 12 this paper, we study the electronic structure of CNCbl from whole molecule. In Section 4, we compare the HF+QMC re- the perspective of many body physics. In particular, we use sults on Im-[CoIII(corrin)]-CN+ with the DFT+QMC results the combined density functional theory (DFT) and quantum on CNCbl. Section 5 gives the summary and conclusions of Monte Carlo (QMC) approach to study the electronic structure the paper. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure##

text: Electronic

label: title

text: structure

text: of

text: cyanocobalamin:

text: DFT+QMC

text: study

text: Selma

label: author

text: Zafer

text: and

text: Nejat

text: (Dated:

text: May

label: date

text: We

label: abstract

text: study

text: the

text: electronic

text: structure

text: and

text: the

text: magnetic

text: correlations

text: of

text: cyanocobalamin

text: (C

text: H

text: CoN

text: O

text: P)

text: by

text: using

text: the

text: framework

text: of

text: the

text: multi-orbital

text: single-impurity

text: Haldane-Anderson

text: model

text: of

text: a

text: transition

text: metal

text: impurity

text: in

text: a

text: semiconductor

text: host.

text: Here,

text: we

text: ﬁrst

text: determine

text: the

text: parameters

text: of

text: the

text: Anderson

text: Hamiltonian

text: by

text: performing

text: density

text: functional

text: theory

text: (DFT)

text: calculations.

text: Then,

text: we

text: use

text: the

text: quantum

text: Monte

text: Carlo

text: (QMC)

text: technique

text: to

text: obtain

text: the

text: electronic

text: structure

text: and

text: the

text: magnetic

text: correlation

text: functions

text: for

text: this

text: effective

text: model.

text: We

text: ﬁnd

text: that

text: new

text: electronic

text: states,

text: which

text: correspond

text: to

text: impurity

text: bound

text: states,

text: form

text: above

text: the

text: lowest

text: unoccupied

text: level

text: of

text: the

text: host

text: semiconductor.

text: These

text: new

text: states

text: derive

text: from

text: the

text: atomic

text: orbitals

text: at

text: the

text: cobalt

text: site

text: and

text: the

text: rest

text: of

text: the

text: molecule.

text: We

text: observe

text: that

text: magnetic

text: moments

text: develop

text: at

text: the

text: orbitals

text: and

text: over

text: the

text: surrounding

text: sites.

text: We

text: also

text: observe

text: that

text: antiferromagnetic

text: correlations

text: exist

text: between

text: the

text: orbitals

text: and

text: the

text: surrounding

text: atoms.

text: These

text: antiferromagnetic

text: correlations

text: depend

text: on

text: the

text: ﬁlling

text: of

text: the

text: impurity

text: bound

text: states.

text: I.

text: INTRODUCTION

text: and

text: magnetic

text: correlations

text: of

text: this

text: molecule

text: within

text: the

text: frame-

text: work

text: of

text: the

text: Haldane-Anderson

text: This

text: model

text: was

text: ini-

text: tially

text: introduced

text: to

text: describe

text: the

text: electronic

text: state

text: of

text: Au

text: in

text: a

text: Vitamin

text: B

text: is

text: a

text: very

text: important

text: organometallic

text: molecule

text: for

text: biological

text: In

text: this

text: paper,

text: we

text: study

text: the

text: elec-

text: semiconductor

text: Ge

text: host.

text: Here,

text: we

text: use

text: the

text: Haldane-Anderson

text: model

text: because

text: CNCbl

text: exhibits

text: a

text: semiconductor

text: energy

text: gap

text: tronic

text: structure

text: of

text: cyanocobalamin

text: (CNCbl)

text: which

text: is

text: a

text: form

text: and

text: contains

text: the

text: transition-metal

text: cobalt

text: atom.

text: of

text: vitamin

text: B

text: In

text: Fig.

text: we

text: illustrate

text: the

text: molecular

text: structure

text: of

text: CNCbl.

text: The

text: cobalt

text: atom

text: neighbours

text: ﬁve

text: nitrogen

text: atoms,

text: of

text: which

text: four

text: are

text: located

text: in

text: the

text: corrin

text: ring.

text: The

text: CN

text: ligand

text: is

text: also

text: In

text: the

text: combined

text: DFT+QMC

text: approach,

text: we

text: ﬁrst

text: determine

text: attached

text: to

text: Co,

text: making

text: the

text: rare

text: cobalt-carbon

text: bonding.

text: The

text: the

text: parameters

text: of

text: the

text: Anderson

text: by

text: the

text: DFT

text: charge

text: neutral

text: CNCbl

text: molecule

text: has

text: electrons.

text: For

text: this

text: calculations

text: carried

text: out

text: with

text: the

text: Gaussian

text: Then,

text: molecule,

text: the

text: HOMO

text: (highest

text: occupied

text: molecular

text: orbital)

text: we

text: study

text: this

text: effective

text: Anderson

text: Hamiltonian

text: by

text: perform-

text: and

text: the

text: LUMO

text: (lowest

text: unoccupied

text: molecular

text: orbital)

text: levels

text: ing

text: QMC

text: simulations

text: with

text: the

text: algorithm.

text: In

text: are

text: separated

text: by

text: an

text: energy

text: gap

text: of

text: The

text: photoab-

text: our

text: calculations,

text: we

text: use

text: an

text: orbital

text: independent

text: intra-orbital

text: sorption

text: spectrum

text: of

text: CNCbl

text: exhibits

text: distinct

text: peaks

text: at

text: Coulomb

text: interaction

text: U

text: at

text: the

text: orbitals

text: and

text: we

text: neglect

text: eV

text: and

text: eV,

text: of

text: which

text: origin

text: remains

text: In

text: additon,

text: ν

text: the

text: inter-orbital

text: Coulomb

text: interactions

text: along

text: with

text: the

text: Hund’s

text: it

text: is

text: known

text: that

text: CNCbl

text: has

text: a

text: weak

text: diamagnetic

text: coupling.

text: We

text: take

text: the

text: value

text: of

text: U

text: to

text: be

text: eV.

text: In

text: addition,

text: the

text: energy

text: levels

text: are

text: shifted

text: in

text: order

text: to

text: prevent

text: the

text: dou-

text: ν

text: ble

text: counting

text: of

text: the

text: local

text: Coulomb

text: interactions

text: by

text: both

text: DFT

text: and

text: QMC.

text: The

text: outline

text: of

text: this

text: paper

text: is

text: as

text: follows.

text: In

text: Section

text: we

text: obtain

text: the

text: one-electron

text: parameters

text: of

text: the

text: effective

text: Anderson

text: Hamiltonian.

text: We

text: present

text: the

text: QMC

text: data

text: on

text: the

text: electronic

text: structure

text: and

text: the

text: magnetic

text: correlation

text: functions

text: in

text: Section

text: Here,

text: we

text: ﬁnd

text: that

text: impurity

text: bound

text: states

text: develop

text: above

text: the

text: LUMO

text: level

text: for

text: CNCbl

text: instead

text: of

text: being

text: inside

text: the

text: semicon-

text: ducting

text: energy

text: gap.

text: This

text: is

text: due

text: to

text: the

text: discrete

text: single-particle

text: spectrum

text: of

text: the

text: molecule.

text: We

text: observe

text: that

text: magnetic

text: mo-

text: ments

text: develop

text: at

text: the

text: orbitals

text: and

text: at

text: the

text: host

text: states

text: ν

text: which

text: have

text: the

text: strongest

text: hybridization.

text: We

text: also

text: observe

text: that

text: there

text: are

text: antiferromagnetic

text: correlations

text: between

text: these

text: host

text: FIG.

text: (Color

text: online)

text: Schematic

text: plot

text: of

text: the

text: molecular

text: structure

text: of

text: magnetic

text: moments

text: and

text: the

text: moments.

text: These

text: antifer-

text: ν

text: cyanocobalamin

text: (C

text: H

text: CoN

text: O

text: P),

text: which

text: contains

text: atoms.

text: romagnetic

text: correlations

text: disappear

text: when

text: the

text: impurity

text: bound

text: states

text: are

text: ﬁlled.

text: We

text: note

text: that

text: in

text: Ref.

text: we

text: combined

text: Despite

text: many

text: years

text: of

text: research,

text: there

text: remains

text: questions

text: the

text: Hartree-Fock

text: (HF)

text: approximation

text: with

text: the

text: QMC

text: tech-

text: about

text: the

text: electronic

text: structure

text: and

text: the

text: functioning

text: of

text: vitamin

text: nique

text: to

text: study

text: CNCbl.

text: In

text: the

text: HF+QMC

text: calculations,

text: we

text: used

text: B

text: as

text: well

text: as

text: the

text: role

text: of

text: the

text: transition-metal

text: cobalt

text: atom.

text: In

text: only

text: the

text: smaller

text: imidazole

text: part

text: of

text: the

text: CNCbl

text: instead

text: of

text: the

text: this

text: paper,

text: we

text: study

text: the

text: electronic

text: structure

text: of

text: CNCbl

text: from

text: whole

text: molecule.

text: In

text: Section

text: we

text: compare

text: the

text: HF+QMC

text: re-

text: the

text: perspective

text: of

text: many

text: body

text: physics.

text: In

text: particular,

text: we

text: use

text: sults

text: on

text: Im-[CoIII(corrin)]-CN+

text: with

text: the

text: DFT+QMC

text: results

text: the

text: combined

text: density

text: functional

text: theory

text: (DFT)

text: and

text: quantum

text: on

text: CNCbl.

text: Section

text: gives

text: the

text: summary

text: and

text: conclusions

text: of

text: Monte

text: Carlo

text: (QMC)

text: approach

text: to

text: study

text: the

text: electronic

text: structure

text: the

text: paper.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##


text: 4 (cid:104) (cid:105) FIG. 5. (Color online) (a) Electron occupation number nν of the Co(3dν) natural atomic orbitals versus the chemical potential µ. (b) Square of magnetic moment (cid:104)(Mνz)2(cid:105) for Co(3dν) natural atomic or- bitals versus the chemical potential µ. Here, the vertical solid and dashed lines denote the values of the HOMO and LUMO levels, re- spectively. These results are for U = 4 eV. m = 336,337 and 340, respectively. Here, we observe that these host states do not become doubly occupied as µ passes FIG. 4. (Color online) Illustration of the m = 336, 337 and 340’th through the εm’s. For example, at the HOMO level, (cid:104)nm(cid:105) = host states in terms of the natural atomic orbitals. 1.72, 1.67 and 1.52 for m = 336,337 and 340 respectively even though they are located deep below the HOMO level. ≈ − Consequently, these host states have ﬁnite magnetic moments hibits a small increase at µ 2.5 eV. Upon further increase (cid:104) (cid:105) when µ is at the HOMO level as seen in Fig. 7(b). The mag- of µ, for all of the Co(3dν) NAO’s, (Mνz)2 vanishes as the netic moments vanish after these host states become doubly orbitals become doubly occupied. occupied for µ >∼ −1 eV. In Fig. 6(a), we present QMC data on the total electron (cid:104) (cid:105) (cid:104) (cid:105) oasccaupfuanticotnioonfotfheµ.CoW(3e≈dsνe−)eNthAaOt (cid:104)’snd(cid:105)ndinc=rea∑s5νe=s1u∑pσto(cid:104)dHν†σ(cid:105)OdMνσO netNicexcot,rrienlaFtiigo.n 8fu(an)c-t(ico)n, (cid:104)wMeνzpMremzs(cid:105)enbtetQwMeeCn tdhaetamoangntheteicmmago-- level, it equals 7.2 at µ 4.8 eV. We observe that nd does ments at the Co(3dν) NAO’s and the m’th host states for m = not change between the HOMO and LUMO levels. The to- 336,337 and 340. These ﬁgures show that the host states tal number of the host electrons (cid:104)nh(cid:105) = ∑Nm=−15 ∑σ(cid:104)c†mσcmσ(cid:105) is wreiltahtiothnes swtriothngtheestmhoymbreindtiszaattiothnehCavoe(3adnt)ifNerAroOm’sa.gTnehteicsecaonr-- shown in Fig. 6(b). Figure 6(c) shows the total electron num- ν ber for CNCbl (cid:104)nT(cid:105) = (cid:104)nd(cid:105)+(cid:104)nh(cid:105) versus µ. Here, we clearly tiferromagnetic correlations vanish as the host states become doubly occupied. see that between the HOMO and LUMO levels the total elec- (cid:104) (cid:105) tron number nT = 718 corresponding to the neutral CNCbl IV. COMPARISON OF THE HF+QMC AND THE DFT+QMC RESULTS molecule. In Fig. 7(a), we present QMC data on the host electron (cid:104) (cid:105) (cid:104) (cid:105) number nm = ∑σ c†mσcmσ versus µ for the m = 336,337 In Ref. [11], numerical results were presented from previ- and 340 host eigenstates. The bare energy levels ε of these ous HF+QMC calculations on Im-[CoIII(corrin)]-CN+, which − − − m states are located at 6.48 eV, 6.43 eV and 6.16 eV for is a smaller piece of the CNCbl molecule. Here, we compare ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: caption

text: FIG.

text: (Color

text: online)

text: (a)

text: Electron

text: occupation

text: number

text: nν

text: of

text: the

text: natural

text: atomic

text: orbitals

text: versus

text: the

text: chemical

text: potential

text: µ.

text: (b)

text: Square

text: of

text: magnetic

text: moment

text: for

text: natural

text: atomic

text: or-

text: bitals

text: versus

text: the

text: chemical

text: potential

text: µ.

text: Here,

text: the

text: vertical

text: solid

text: and

text: dashed

text: lines

text: denote

text: the

text: values

text: of

text: the

text: HOMO

text: and

text: LUMO

text: levels,

text: re-

text: spectively.

text: These

text: results

text: are

text: for

text: U

text: eV.

text: m

text: and

text: respectively.

text: Here,

text: we

text: observe

text: that

text: these

text: host

text: states

text: do

text: not

text: become

text: doubly

text: occupied

text: as

text: µ

text: passes

text: FIG.

text: (Color

text: online)

text: Illustration

text: of

text: the

text: m

text: and

text: through

text: the

text: εm’s.

text: For

text: example,

text: at

text: the

text: HOMO

text: level,

text: host

text: states

text: in

text: terms

text: of

text: the

text: natural

text: atomic

text: orbitals.

text: and

text: for

text: m

text: and

text: respectively

text: even

text: though

text: they

text: are

text: located

text: deep

text: below

text: the

text: HOMO

text: level.

text: Consequently,

text: these

text: host

text: states

text: have

text: ﬁnite

text: magnetic

text: moments

text: hibits

text: a

text: small

text: increase

text: at

text: µ

text: eV.

text: Upon

text: further

text: increase

text: when

text: µ

text: is

text: at

text: the

text: HOMO

text: level

text: as

text: seen

text: in

text: Fig.

text: The

text: mag-

text: of

text: µ,

text: for

text: all

text: of

text: the

text: NAO’s,

text: vanishes

text: as

text: the

text: netic

text: moments

text: vanish

text: after

text: these

text: host

text: states

text: become

text: doubly

text: orbitals

text: become

text: doubly

text: occupied.

text: occupied

text: for

text: µ

text: −1

text: eV.

text: In

text: Fig.

text: we

text: present

text: QMC

text: data

text: on

text: the

text: total

text: electron

text: netNicexcot,rrienlaFtiigo.n

text: tdhaetamoangntheteicmmago--

text: level,

text: it

text: equals

text: at

text: µ

text: eV.

text: We

text: observe

text: that

text: nd

text: does

text: ments

text: at

text: the

text: NAO’s

text: and

text: the

text: m’th

text: host

text: states

text: for

text: m

text: not

text: change

text: between

text: the

text: HOMO

text: and

text: LUMO

text: levels.

text: The

text: to-

text: and

text: These

text: ﬁgures

text: show

text: that

text: the

text: host

text: states

text: tal

text: number

text: of

text: the

text: host

text: electrons

text: is

text: wreiltahtiothnes

text: shown

text: in

text: Fig.

text: Figure

text: shows

text: the

text: total

text: electron

text: num-

text: ν

text: ber

text: for

text: CNCbl

text: versus

text: µ.

text: Here,

text: we

text: clearly

text: tiferromagnetic

text: correlations

text: vanish

text: as

text: the

text: host

text: states

text: become

text: doubly

text: occupied.

text: see

text: that

text: between

text: the

text: HOMO

text: and

text: LUMO

text: levels

text: the

text: total

text: elec-

text: tron

text: number

text: nT

text: corresponding

text: to

text: the

text: neutral

text: CNCbl

text: IV.

text: COMPARISON

text: OF

text: THE

text: HF+QMC

text: AND

text: THE

text: DFT+QMC

text: RESULTS

text: molecule.

text: In

text: Fig.

text: we

text: present

text: QMC

text: data

text: on

text: the

text: host

text: electron

text: number

text: nm

text: ∑σ

text: c†mσcmσ

text: versus

text: µ

text: for

text: the

text: m

text: In

text: Ref.

text: numerical

text: results

text: were

text: presented

text: from

text: previ-

text: and

text: host

text: eigenstates.

text: The

text: bare

text: energy

text: levels

text: ε

text: of

text: these

text: ous

text: HF+QMC

text: calculations

text: on

text: Im-[CoIII(corrin)]-CN+,

text: which

text: m

text: states

text: are

text: located

text: at

text: eV,

text: eV

text: and

text: eV

text: for

text: is

text: a

text: smaller

text: piece

text: of

text: the

text: CNCbl

text: molecule.

text: Here,

text: we

text: compare

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##