text: (cid:2) (cid:3) | A√A | | | | − − − | | | | Ψ) = 1 2 a) V ) a) V ) ei(θ1+φ1 θ2 φ2) b) H) b) H) . (10) 2 S1 S1 S2 S2 S1 S1 S2 S2 (cid:48) After BS the symmetrized state becomes | ⊗ I ⊗ ⊗ I | Ψf) = UˆS1 (cid:48) S1 UˆS2 (cid:48) S2 Ψ) (11) BS | pol | BS po|l | A√A a) √+ b) | a) √+ b) | = 1 2 S1 S1 V ) S2 S2 V ) 2 2 S1 2 S2 | − | | − | − − − A√A a) √ b) | a) √ b) | ei(θ1+φ1 θ2 φ2) 1 2 S1 S1 H) S2 S2 H) . (12) 2 2 S1 2 S2 2.2 New Intensity-Intensity Correlations Let us consider the two observables | | − | | σpol = eiθ1 H) (V + e iθ1 V ) (H , (13) 1θ1 − | S1 S1| | S1 |S1 σpol = e iθ2 H) (V + eiθ2 V ) (H , (14) 2θ2 S2 S2 S2 S2 operating on the polarization Hilbert spaces of the lights from the sources S and S , as well as 1 2 the two observables − | | | | σpath = e iφ1 a) (b + eiφ1 b) (a , (15) 1φ1 | S1 | S1 − | S1 |S1 σpath = eiφ2 a) (b + e iφ2 b) (a (16) 2φ2 S2 S2 S2 S2 operating on the path Hilbert spaces. Let us deﬁne the normalized correlation | ⊗ | (Ψo σpath.σpol σpath.σpol Ψo) C(θ , φ , θ , φ ) = 1φ1| 1|θ1 | 2φ|2 2θ2 (17) 1 1 2 2 ( A 2 + A 2)2 1 2 | for the symmetrized state Ψo) given by Eqn. (9). This can be written as a linear combination of intensity-intensity correlations, as we will now show. First, it can be shown (see Appendix-2 for | details) that this correlation can be expressed in terms of the ﬁnal state Ψf) as (cid:88) 1 − | | C(θ , φ , θ , φ ) = ( 1)k+l+m+n(Ψf Iˆ (kπ, lπ).Iˆ (mπ, nπ) Ψf) 1 1 2 2 S1 S2 (k,l,m,n)=0 | | | | 4 A 2 A 2 − − 1 2 = | | | | cos(θ θ + φ φ ). (18) ( A 2 + A 2)2 1 2 1 2 1 2 5 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: A√A

text: Ψ)

text: a)

text: V

text: a)

text: V

text: θ2

text: b)

text: H)

text: b)

text: H)

text: After

text: BS

text: the

text: symmetrized

text: state

text: becomes

text: I

text: I

text: Ψf)

text: Ψ)

text: BS

text: pol

text: BS

text: po|l

text: A√A

text: a)

text: b)

text: a)

text: b)

text: V

text: V

text: A√A

text: a)

text: b)

text: a)

text: b)

text: θ2

text: H)

text: H)

text: New

text: Intensity-Intensity

text: Correlations

text: Let

text: us

text: consider

text: the

text: two

text: observables

text: σpol

text: H)

text: (V

text: e

text: V

text: (H

text: σpol

text: e

text: H)

text: (V

text: V

text: (H

text: operating

text: on

text: the

text: polarization

text: Hilbert

text: spaces

text: of

text: the

text: lights

text: from

text: the

text: sources

text: S

text: and

text: S

text: as

text: well

text: as

text: the

text: two

text: observables

text: σpath

text: e

text: a)

text: (b

text: b)

text: (a

text: σpath

text: a)

text: (b

text: e

text: b)

text: (a

text: operating

text: on

text: the

text: path

text: Hilbert

text: spaces.

text: Let

text: us

text: deﬁne

text: the

text: normalized

text: correlation

text: (Ψo

text: σpath.σpol

text: σpath.σpol

text: Ψo)

text: C(θ

text: φ

text: θ

text: φ

text: A

text: A

text: for

text: the

text: symmetrized

text: state

text: Ψo)

text: given

text: by

text: Eqn.

text: This

text: can

text: be

text: written

text: as

text: a

text: linear

text: combination

text: of

text: intensity-intensity

text: correlations,

text: as

text: we

text: will

text: now

text: show.

text: First,

text: it

text: can

text: be

text: shown

text: (see

text: for

text: details)

text: that

text: this

text: correlation

text: can

text: be

text: expressed

text: in

text: terms

text: of

text: the

text: ﬁnal

text: state

text: Ψf)

text: as

text: C(θ

text: φ

text: θ

text: φ

text: Iˆ

text: (kπ,

text: lπ).Iˆ

text: (mπ,

text: nπ)

text: Ψf)

text: A

text: A

text: cos(θ

text: θ

text: φ

text: φ

text: A

text: A

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Due to the energy-momentum conservation, the universal AD of the stress tensor T should µν be zero, i.e., γ(j = 2) = 0. (11) It is important, that the AD γ contributing to the DGLAP equation [48]-[52] does not coincide with γ appearing in the BFKL equation. They are related as follows [8, 53, 54] BFKL ω j γ = γ + = + iν , (12) BFKL 2 2 where the additional contribution ω/2 is responsible in particular for the cancelation of the ∼ singular terms 1/γ3 obtained from the NLO corrections (1) to the eigenvalue of the BFKL kernel [8]. Using above relations one obtains ν(j = 2) = i . (13) As a result, from eq. (6) for the Pomeron trajectory we derive the following representation for the correction ∆ (10) to the graviton spin 2 (cid:88) ∞ ∆ = D . (14) m m=1 According to (10) and (14), we have th(cid:88)e follow(cid:16)ing small-ν e(cid:17)xpansion for the eigenvalue of the BFKL kernel ∞ − − − j 2 = D ( ν2)m 1 , (15) m m=1 (cid:18) (cid:19) where ν2 is related to γ according to eq. (12) − j − 2 ν2 = γ . (16) 2 On the other hand, due to the ADS/CFT correspondence the string energies E in dimensionless units are related to the AD γ of the twist-two operators as follows [29, 30]1 − − E2 = (j + Γ)2 4, Γ = 2γ (17) and therefore we can obtain from (16) the relation between the parameter ν for the principal (cid:32) (cid:33) series of unitary representations of the M¨obius group and the string energy E − E2 ν2 = + 1 . (18) 4 This expression for ν2 can be inserted in the r.h.s. of Eq. (15) leading to the following expression (cid:34)(cid:32) (cid:33) (cid:35) for the Regge trajectory of the gravito(cid:88)n in the anti-de-Sitter space ∞ m − E2 − j 2 = D + 1 1 . (19) m 4 m=1 1 Note that our expression (17) for the string energy E diﬀers from a deﬁnition, in which E is equal to the scaling − − dimension ∆ . But eq. (17) is correct, because it can be presented as E2 = (∆ 2)2 4 and coincides with sc sc Eqs. (45) and (3.44) from Refs. [29] and [30], respectively. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Due

label: paragraph

text: to

text: the

text: energy-momentum

text: conservation,

text: the

text: universal

text: AD

text: of

text: the

text: stress

text: tensor

text: T

text: should

text: µν

text: be

text: zero,

text: i.e.,

text: γ(j

text: It

text: is

text: important,

text: that

text: the

text: AD

text: γ

text: contributing

text: to

text: the

text: DGLAP

text: equation

text: does

text: not

text: coincide

text: with

text: γ

text: appearing

text: in

text: the

text: BFKL

text: equation.

text: They

text: are

text: related

text: as

text: follows

text: BFKL

text: ω

text: j

text: γ

text: γ

text: iν

text: BFKL

text: where

text: the

text: additional

text: contribution

text: is

text: responsible

text: in

text: particular

text: for

text: the

text: cancelation

text: of

text: the

text: singular

text: terms

text: obtained

text: from

text: the

text: NLO

text: corrections

text: to

text: the

text: eigenvalue

text: of

text: the

text: BFKL

text: kernel

text: Using

text: above

text: relations

text: one

text: obtains

text: ν(j

text: i

text: As

text: a

text: result,

text: from

text: eq.

text: for

text: the

text: Pomeron

text: trajectory

text: we

text: derive

text: the

text: following

text: representation

text: for

text: the

text: correction

text: to

text: the

text: graviton

text: spin

text: D

text: m

text: According

text: to

text: and

text: we

text: have

text: small-ν

text: for

text: the

text: eigenvalue

text: of

text: the

text: BFKL

text: kernel

text: j

text: D

text: m

text: where

text: ν2

text: is

text: related

text: to

text: γ

text: according

text: to

text: eq.

text: j

text: ν2

text: γ

text: On

text: the

text: other

text: hand,

text: due

text: to

text: the

text: ADS/CFT

text: correspondence

text: the

text: string

text: energies

text: E

text: in

text: dimensionless

text: units

text: are

text: related

text: to

text: the

text: AD

text: γ

text: of

text: the

text: twist-two

text: operators

text: as

text: follows

text: (j

text: Γ

text: 2γ

text: and

text: therefore

text: we

text: can

text: obtain

text: from

text: the

text: relation

text: between

text: the

text: parameter

text: ν

text: for

text: the

text: principal

text: series

text: of

text: unitary

text: representations

text: of

text: the

text: M¨obius

text: group

text: and

text: the

text: string

text: energy

text: E

text: ν2

text: This

text: expression

text: for

text: ν2

text: can

text: be

text: inserted

text: in

text: the

text: r.h.s.

text: of

text: Eq.

text: leading

text: to

text: the

text: following

text: expression

text: for

text: the

text: Regge

text: trajectory

text: of

text: the

text: in

text: the

text: anti-de-Sitter

text: space

text: m

text: j

text: D

text: m

text: Note

text: that

text: our

text: expression

text: for

text: the

text: string

text: energy

text: E

text: diﬀers

text: from

text: a

text: deﬁnition,

text: in

text: which

text: E

text: is

text: equal

text: to

text: the

text: scaling

text: dimension

text: But

text: eq.

text: is

text: correct,

text: because

text: it

text: can

text: be

text: presented

text: as

text: and

text: coincides

text: with

text: sc

text: sc

text: Eqs.

text: and

text: from

text: Refs.

text: and

text: respectively.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: √ 4. Graviton Regge trajectory and Pomeron intercept (cid:28) (cid:28) We assume, that eq. (19) is valid also at large j and large λ in the region 1 j λ, where the strong coupling calculations of energies were performed [43, 47]. These energies can (cid:34) (cid:35) be presented in the form 2 (cid:16) (cid:17) √ E2 S √S S2 = λ h (λ) + h (λ) + h (λ) + O S7/2 , (20) 0 1 2 4 2 λ λ where √a a √a a i1 i2 i3 i2 h (λ) = a + + + + . (21) √ i i0 λ λ λ3 λ2 ∼ The contribution S can be extracted directly from the Basso result [44, 45] taking J = 2 an √ √ according to [46]: I (√λ) √2 I (√λ) − √2 3 1 h (λ) = + = , (22) 0 √ I2( λ) λ I2( λ) λ where I ( λ) is the modiﬁed Bessel functions. It leads to the following values of coeﬃcients a k 0i − 1 15 135 a = 1, a = , a = a = , a = (23) 00 01 02 03 04 2 8 128 The coeﬃcients a and a come from considerations of the classical part of the folded 10 20 spinning string corresponding to the twist-two operators (see, for example, [47]) 3 − 3 a = , a = . (24) 10 20 4 16 The one-loop coeﬃcient a is found recently in the paper [46], considering diﬀerent 11 (cid:16) (cid:17) asymptotical regimes with taking into account the Basso result [44] (ζ is the Euler ζ-function) 3 3 − a = 1 ζ , (25) 11 3 16 Comparing the l.h.s. and r.h.s. of (19) at large j values gives us the coeﬃcients D and ∆ m (see Appendix A in [1]). 5. Conclusion We have shown the intercept of the BFKL pomeron at weak coupling regime and demonstrated an approach to obtain its values at strong couplings (for details, see Ref. [1]). → ∞ (cid:20) (cid:16) (cid:17) (cid:18) − (cid:19) (cid:18) (cid:19)(cid:21) At λ , the correction ∆ for the Pomeron intercept j = 2 ∆ has the form 3 0 2 1 − 1 − 1 − 145 − 9 1 1 ∆ = 1 + 1 + 3ζ + 2a ζ + O . (26) λ1/2 2λ1/2 8λ 3 λ3/2 12 128 2 3 λ2 λ5/2 The fourth corrections in (26) contain unknown coeﬃcient a , which will be obtained after 12 the evaluation of spinning folded string on the two-loop level. Some estimations were given in Section 6 of [1] . Acknowledgments A.V.K. was supported in part by RFBR grant No. 13-02-01060-a. He thanks Eugene Levin and the Organizing Committee of International Moscow Phenomenology Workshop for invitation. − 2 Here we put S = j 2, which in particular is related to the use of the angular momentum J = 2 in calculations an of Refs [43, 47]. ∼ − ∼ − 3 Using a similar approach, the coeﬃcients λ 1 and λ 3/2 were calculated also in the paper [55]. After correction of some errors, the results in [55] coincide with ours. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##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: section

text: Graviton

text: Regge

text: trajectory

text: and

text: Pomeron

text: intercept

text: We

text: assume,

text: that

text: eq.

text: is

text: valid

text: also

text: at

text: large

text: j

text: and

text: large

text: λ

text: in

text: the

text: region

text: j

text: λ,

text: where

text: the

text: strong

text: coupling

text: calculations

text: of

text: energies

text: were

text: performed

text: These

text: energies

text: can

text: be

text: presented

text: in

text: the

text: form

text: S

text: √S

text: λ

text: h

text: (λ)

text: h

text: (λ)

text: h

text: (λ)

text: O

text: λ

text: λ

text: where

text: √a

text: a

text: √a

text: a

text: h

text: (λ)

text: a

text: i

text: λ

text: λ

text: λ3

text: λ2

text: The

text: contribution

text: S

text: can

text: be

text: extracted

text: directly

text: from

text: the

text: Basso

text: result

text: taking

text: J

text: an

text: according

text: to

text: I

text: (√λ)

text: √2

text: I

text: (√λ)

text: √2

text: h

text: (λ)

text: λ)

text: λ

text: λ)

text: λ

text: where

text: I

text: λ)

text: is

text: the

text: modiﬁed

text: Bessel

text: functions.

text: It

text: leads

text: to

text: the

text: following

text: values

text: of

text: coeﬃcients

text: a

text: k

text: a

text: a

text: a

text: a

text: a

text: The

text: coeﬃcients

text: a

text: and

text: a

text: come

text: from

text: considerations

text: of

text: the

text: classical

text: part

text: of

text: the

text: folded

text: spinning

text: string

text: corresponding

text: to

text: the

text: twist-two

text: operators

text: (see,

text: for

text: example,

text: a

text: a

text: The

text: one-loop

text: coeﬃcient

text: a

text: is

text: found

text: recently

text: in

text: the

text: paper

text: considering

text: diﬀerent

text: asymptotical

text: regimes

text: with

text: taking

text: into

text: account

text: the

text: Basso

text: result

text: (ζ

text: is

text: the

text: Euler

text: ζ-function)

text: a

text: ζ

text: Comparing

text: the

text: l.h.s.

text: and

text: r.h.s.

text: of

text: at

text: large

text: j

text: values

text: gives

text: us

text: the

text: coeﬃcients

text: D

text: and

text: m

text: (see

text: Appendix

text: A

text: in

text: Conclusion

text: We

text: have

text: shown

text: the

text: intercept

text: of

text: the

text: BFKL

text: pomeron

text: at

text: weak

text: coupling

text: regime

text: and

text: demonstrated

text: an

text: approach

text: to

text: obtain

text: its

text: values

text: at

text: strong

text: couplings

text: (for

text: details,

text: see

text: Ref.

text: At

text: λ

text: the

text: correction

text: for

text: the

text: Pomeron

text: intercept

text: j

text: has

text: the

text: form

text: 3ζ

text: ζ

text: O

text: 8λ

text: λ2

text: The

text: fourth

text: corrections

text: in

text: contain

text: unknown

text: coeﬃcient

text: a

text: which

text: will

text: be

text: obtained

text: after

text: the

text: evaluation

text: of

text: spinning

text: folded

text: string

text: on

text: the

text: two-loop

text: level.

text: Some

text: estimations

text: were

text: given

text: in

text: Section

text: of

text: Acknowledgments

text: A.V.K.

text: was

text: supported

text: in

text: part

text: by

text: RFBR

text: grant

text: No.

text: He

text: thanks

text: Eugene

text: Levin

text: and

text: the

text: Organizing

text: Committee

text: of

text: International

text: Moscow

text: Phenomenology

text: Workshop

text: for

text: invitation.

text: Here

text: we

text: put

text: S

text: j

text: which

text: in

text: particular

text: is

text: related

text: to

text: the

text: use

text: of

text: the

text: angular

text: momentum

text: J

text: in

text: calculations

text: an

text: of

text: Refs

text: Using

text: a

text: similar

text: approach,

text: the

text: coeﬃcients

text: λ

text: and

text: λ

text: were

text: calculated

text: also

text: in

text: the

text: paper

text: After

text: correction

text: of

text: some

text: errors,

text: the

text: results

text: in

text: coincide

text: with

text: ours.

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

text: ##LTLine##


text: References [1] Kotikov A V and Lipatov L N 2013 Nucl. Phys. B 874 889 [2] Kotikov A V and Lipatov L N 2000 Nucl. Phys. B 582 19 [3] Kotikov A V and Lipatov L N 2003 Nucl. Phys. B 661 19 [4] Kotikov A V and Lipatov L N 2001 DGLAP and BFKL evolution equations in the N=4 supersymmetric gauge theory Preprint hep-ph/0112346. [5] Fadin V F and Fiore R 2008 Phys. Lett. B 661 139 [6] Fadin V F, Fiore R and Grabovsky A V 2010 Nucl. Phys. B 831 248 [7] Balitsky I and Chirilli G A 2010 Phys. Lett. B 687 204 [8] Fadin V F and Lipatov L N 1998 Phys. Lett. B 429 127 [9] Camici G and M. Ciafaloni M Phys. Lett. B 430 349 [10] Lipatov L N 1976 Sov. J. Nucl. Phys. 23 338 [11] Fadin V F, Kuraev E A and Lipatov L N 1975 Phys. Lett. B 60 50 [12] Fadin V F, Kuraev E A and Lipatov L N 1975 Sov. Phys. JETP 44 443 [13] Fadin V F, Kuraev E A and Lipatov L N 1977 Sov. Phys. JETP 45 199 [14] Balitsky I I and Lipatov L N 1978 Sov. J. Nucl. Phys. 28 822 [15] Lipatov L N 1993 Phys. Lett. B 309 394 [16] Lipatov L N 1993 High energy asymptotics of mult-color QCD and exactly solvable lattice models it Preprin hep-th/9311037 [17] Lipatov L N 1997 Proc, Int. Conf. ”Perspectives in Hadronic Physics” (Triest) (Singapore: World Scientiﬁc) p. 562 [18] Lipatov L N 2001 Nucl. Phys. Proc. Suppl. A 99 175 [19] Dolan F A and Osborn H 2002 Nucl. Phys. B 629 3 [20] Kotikov A V, Lipatov L N and Velizhanin V N 2003 Phys. Lett. B 557 114 [21] Kotikov A V, Lipatov L N, Onishchenko A I and Velizhanin V N 2004 Phys. Lett. B 595 521 [22] Kotikov A V, Lipatov L N, Rej A, Staudacher M and Velizhanin V N 2007 J. Stat. Mech. 0710 P10003 [23] Bajnok Z, Janik R A and Lukowski T 2009 Nucl. Phys. B 816 376 [24] Lukowski T, Rej A and Velizhanin V N 2010 Nucl. Phys. B 831 105 [25] Moch S, Vermaseren J A M and Vogt A 2004 Nucl. Phys. B 688 101 [26] Beisert N and Staudacher M 2005 Nucl. Phys. B 727 1 [27] Staudacher M 2005 JHEP 0505 054 [28] Maldacena J 1998 Adv. Theor. Math. Phys. 2 231 [29] Gubser S S, Klebanov I R and Polyakov A M 1998 Phys. Lett. B 428 105 [30] Witten E 1998 Adv. Theor. Math. Phys. 2 253 [31] Beisert N, Eden B and Staudacher M 2007 J. Stat. Mech. 0701 P021 [32] Benna M K, Benvenuti S, Klebanov I R and Scardicchio A 2007 Phys. Rev. Lett. 98 131603 [33] Kotikov A V and Lipatov L N 2007 Nucl. Phys. B 769 217 [34] Basso B, Korchemsky G P and Kotanski J 2008 Phys. Rev. Lett. 100 091601 [35] Gubser S S , Klebanov I R and Polyakov A M 2002 Nucl. Phys. B 636 99 [36] Frolov S and Tseytlin A A 2002 JHEP 0206 007 [37] Roiban R , Tirziu A and Tseytlin A A 2007 JHEP 0707 056 [38] Polchinski J and Strassler M J 2002 Phys. Rev. Lett. 88 031601 [39] Polchinski J and Strassler M J 2003 JHEP 0305 012 [40] Kotikov A V, Lipatov L N, Onishchenko A I and Velizhanin V N 2006 Phys. Lett. B 632 754 [41] Brower R C, Polchinski J, Strassler M J and Tan C I 2007 JHEP 0712 005 [42] Lipatov L N 2013 Phys. Part. Nucl. 44 391 [43] Gromov N, Serban D, Shenderovich I and Volin D 2011 JHEP 1108 046 [44] Basso B 2011 An exact slope for AdS/CFT Preprint arXiv:1109.3154 [hep-th] [45] Basso B 2912 Scaling dimensions at small spin in N=4 SYM theory Preprint arXiv:1205.0054 [hep-th] [46] Gromov N and Valatka S 2012 JHEP 1203 058 [47] Roiban R and Tseytlin A A 2011 Nucl. Phys. B 848 251 [48] Gribov V N and Lipatov L N 1972 Sov. J. Nucl. Phys. 15 438 [49] Gribov V N and Lipatov L N 1972 Sov. J. Nucl. Phys. 15 675 [50] Lipatov L N 1975 Sov. J. Nucl. Phys. 20 94 [51] Altarelli G and Parisi G 1977 Nucl. Phys. B 126 298 [52] Dokshitzer Yu L 1977 Sov. Phys. JETP 46 641 [53] Salam G P 1998 JHEP 9807 019 [54] Salam G P 1999 Acta Phys. Polon. B 30 3679 [55] Costa M S, Goncalves V and Penedones J 2012 JHEP 1212 091

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text: Math.

text: Phys.

text: Gubser

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text: S,

text: Klebanov

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text: R

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text: Polyakov

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text: Witten

text: E

text: Adv.

text: Theor.

text: Math.

text: Phys.

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text: Eden

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text: M

text: K,

text: Benvenuti

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text: Klebanov

text: I

text: R

text: and

text: Scardicchio

text: A

text: Phys.

text: Rev.

text: Lett.

text: Kotikov

text: A

text: V

text: and

text: Lipatov

text: L

text: N

text: Nucl.

text: Phys.

text: B

text: Basso

text: B,

text: Korchemsky

text: G

text: P

text: and

text: Kotanski

text: J

text: Phys.

text: Rev.

text: Lett.

text: Gubser

text: S

text: S

text: Klebanov

text: I

text: R

text: and

text: Polyakov

text: A

text: M

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text: Phys.

text: B

text: Frolov

text: S

text: and

text: Tseytlin

text: A

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text: Roiban

text: R

text: Tirziu

text: A

text: and

text: Tseytlin

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text: A

text: JHEP

text: Polchinski

text: J

text: and

text: Strassler

text: M

text: J

text: Phys.

text: Rev.

text: Lett.

text: Polchinski

text: J

text: and

text: Strassler

text: M

text: J

text: JHEP

text: Kotikov

text: A

text: V,

text: Lipatov

text: L

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text: Onishchenko

text: A

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text: Velizhanin

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text: N

text: Phys.

text: Lett.

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text: R

text: C,

text: Polchinski

text: J,

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text: Scaling

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text: at

text: small

text: spin

text: in

text: SYM

text: theory

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text: [hep-th]

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text: N

text: and

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text: Tseytlin

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text: A

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text: Phys.

text: B

text: Gribov

text: V

text: N

text: and

text: Lipatov

text: L

text: N

text: Sov.

text: J.

text: Nucl.

text: Phys.

text: Gribov

text: V

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text: and

text: Lipatov

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text: Phys.

text: Lipatov

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text: Nucl.

text: Phys.

text: Altarelli

text: G

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text: Parisi

text: G

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text: Phys.

text: B

text: Dokshitzer

text: Yu

text: L

text: Sov.

text: Phys.

text: JETP

text: Salam

text: G

text: P

text: JHEP

text: Salam

text: G

text: P

text: Acta

text: Phys.

text: Polon.

text: B

text: Costa

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text: For simplicity in (McCallum, 1999) the concept is ﬁrst deﬁned for the case when A is an irreducible basis. We emulate this approach, generalising for other cases by considering (cid:83) (cid:83) contents and irreducible factors of positive degree when verifying the algorithm in Section A { } E { } 1. So let = A t be a list of irreducible bases A and let = E t be a list of ⊆ i i=1 i i i=1 subsets E A . Put A = t A and E = t E . Note that we use the convention of i i i=1 i i=1 i uppercase Roman letters for sets of polynomials and calligraphic letters for lists of these. (cid:83) A E Deﬁnition 15. With the notation above the reduced projection of with respect to is A ∪ × E PE( ) := t P (A ) RES ( ) (15) i=1 Ei i × E where RES ( ) is the cross resultant set × E { | ∃ ∈ ∈ (cid:54) } RES ( ) = res (f, fˆ) i, j such that f E , fˆ E , i < j, f = fˆ (16) xn i j and ∪ { | ∈ ∈ ∈ } P (A) = P(E) res (f, g) f E, g A, g / E , E { xn | ∈ (cid:54) } P(A) = coeﬀs(f), disc(f), res (f, g) f, g A, f = g . xn R − A Theorem 16. Let S be a connected submanifold of n 1. Suppose each element of PE( ) ∈ is order invariant in S. Then each f E either vanishes identically on S or is analytically ∈ delineable on S; the sections over S of the f E which do not vanish identically are ∈ pairwise disjoint; and each element f E which does not vanish identically is order- invariant in such sections. ≤ ≤ ∈ \ Moreover, for each i, in 1 i t every g A E is sign-invariant in each section ∈ i i over S of every f E which does not vanish identically. i ⊆ A Proof. The crucial observation for the(cid:83)ﬁrst part is that P(E) PE( ). To see this, recall equation (15) and note that we can write ∪ × E P(E) = P(E ) RES ( ). i i We can therefore apply Theorem 4 to the set E and obtain the ﬁrst three conclusions immediately, leaving only the ﬁnal conclusion to prove. ≤ ≤ ∈ \ ∈ Let i be in the range 1 i t, let g A E and let f E . Suppose that f does i∈ i A i not vanish identically on S. Now res (f, g) PE( ), and so is order-invariant in S by xn hypothesis. Further, we already concluded that f is delineable. Therefore by Theorem 5, g is sign-invariant in each section of f over S. (cid:50) Theorem 16 is the key tool for the veriﬁcation of our TTICAD algorithm in Section 1. ∈ It allows us to conclude the output is correct so long as no f E vanishes identically on the lower dimensional manifold, S. A polynomial f in r variables that vanishes identically ∈ R − at a point α r 1 is said to be nulliﬁed at α. The theory of this subsection appears identical to the work in (Bradford et al., 2013a). The diﬀerence is in the application of the theory in Section 1. We suppose that the input { } is a list of QFFs, φ , with each A deﬁned from the polynomials in each φ . In (Bradford i i i et al., 2013a) there was an assumption (no longer made) that each of these formulae had a designated EC f = 0 from which the subsets E are deﬁned. Instead, we deﬁne E to i i i 13

text: For

text: simplicity

text: in

text: (McCallum,

text: the

text: concept

text: is

text: ﬁrst

text: deﬁned

text: for

text: the

text: case

text: when

text: A

text: is

text: an

text: irreducible

text: basis.

text: We

text: emulate

text: this

text: approach,

text: generalising

text: for

text: other

text: cases

text: by

text: considering

text: contents

text: and

text: irreducible

text: factors

text: of

text: positive

text: degree

text: when

text: verifying

text: the

text: algorithm

text: in

text: Section

text: A

text: E

text: So

text: let

text: A

text: t

text: be

text: a

text: list

text: of

text: irreducible

text: bases

text: A

text: and

text: let

text: E

text: t

text: be

text: a

text: list

text: of

text: i

text: i

text: i

text: subsets

text: E

text: A

text: Put

text: A

text: t

text: A

text: and

text: E

text: t

text: E

text: Note

text: that

text: we

text: use

text: the

text: convention

text: of

text: i

text: i

text: i

text: i

text: uppercase

text: Roman

text: letters

text: for

text: sets

text: of

text: polynomials

text: and

text: calligraphic

text: letters

text: for

text: lists

text: of

text: these.

text: A

text: E

text: Deﬁnition

text: With

text: the

text: notation

text: above

text: the

text: reduced

text: projection

text: of

text: with

text: respect

text: to

text: is

text: A

text: E

text: PE(

text: t

text: P

text: (A

text: RES

text: Ei

text: i

text: E

text: where

text: RES

text: is

text: the

text: cross

text: resultant

text: set

text: E

text: RES

text: res

text: (f,

text: fˆ)

text: i,

text: j

text: such

text: that

text: f

text: E

text: fˆ

text: E

text: i

text: j,

text: f

text: fˆ

text: xn

text: i

text: j

text: and

text: P

text: (A)

text: P(E)

text: res

text: (f,

text: g)

text: f

text: E,

text: g

text: A,

text: g

text: E

text: E

text: xn

text: P(A)

text: coeﬀs(f),

text: disc(f),

text: res

text: (f,

text: g)

text: f,

text: g

text: A,

text: f

text: g

text: xn

text: R

text: A

text: Theorem

text: Let

text: S

text: be

text: a

text: connected

text: submanifold

text: of

text: n

text: Suppose

text: each

text: element

text: of

text: PE(

text: is

text: order

text: invariant

text: in

text: S.

text: Then

text: each

text: f

text: E

text: either

text: vanishes

text: identically

text: on

text: S

text: or

text: is

text: analytically

text: delineable

text: on

text: S;

text: the

text: sections

text: over

text: S

text: of

text: the

text: f

text: E

text: which

text: do

text: not

text: vanish

text: identically

text: are

text: pairwise

text: disjoint;

text: and

text: each

text: element

text: f

text: E

text: which

text: does

text: not

text: vanish

text: identically

text: is

text: order-

text: invariant

text: in

text: such

text: sections.

text: Moreover,

text: for

text: each

text: i,

text: in

text: i

text: t

text: every

text: g

text: A

text: E

text: is

text: sign-invariant

text: in

text: each

text: section

text: i

text: i

text: over

text: S

text: of

text: every

text: f

text: E

text: which

text: does

text: not

text: vanish

text: identically.

text: i

text: A

text: Proof.

text: The

text: crucial

text: observation

text: for

text: part

text: is

text: that

text: P(E)

text: PE(

text: To

text: see

text: this,

text: recall

text: equation

text: and

text: note

text: that

text: we

text: can

text: write

text: E

text: P(E)

text: P(E

text: RES

text: i

text: i

text: We

text: can

text: therefore

text: apply

text: Theorem

text: to

text: the

text: set

text: E

text: and

text: obtain

text: the

text: ﬁrst

text: three

text: conclusions

text: immediately,

text: leaving

text: only

text: the

text: ﬁnal

text: conclusion

text: to

text: prove.

text: Let

text: i

text: be

text: in

text: the

text: range

text: i

text: t,

text: let

text: g

text: A

text: E

text: and

text: let

text: f

text: E

text: Suppose

text: that

text: f

text: does

text: i∈

text: i

text: A

text: i

text: not

text: vanish

text: identically

text: on

text: S.

text: Now

text: res

text: (f,

text: g)

text: PE(

text: and

text: so

text: is

text: order-invariant

text: in

text: S

text: by

text: xn

text: hypothesis.

text: Further,

text: we

text: already

text: concluded

text: that

text: f

text: is

text: delineable.

text: Therefore

text: by

text: Theorem

text: g

text: is

text: sign-invariant

text: in

text: each

text: section

text: of

text: f

text: over

text: S.

text: Theorem

text: is

text: the

text: key

text: tool

text: for

text: the

text: veriﬁcation

text: of

text: our

text: TTICAD

text: algorithm

text: in

text: Section

text: It

text: allows

text: us

text: to

text: conclude

text: the

text: output

text: is

text: correct

text: so

text: long

text: as

text: no

text: f

text: E

text: vanishes

text: identically

text: on

text: the

text: lower

text: dimensional

text: manifold,

text: S.

text: A

text: polynomial

text: f

text: in

text: r

text: variables

text: that

text: vanishes

text: identically

text: R

text: at

text: a

text: point

text: α

text: r

text: is

text: said

text: to

text: be

text: nulliﬁed

text: at

text: α.

text: The

text: theory

text: of

text: this

text: subsection

text: appears

text: identical

text: to

text: the

text: work

text: in

text: (Bradford

text: et

text: al.,

text: The

text: diﬀerence

text: is

text: in

text: the

text: application

text: of

text: the

text: theory

text: in

text: Section

text: We

text: suppose

text: that

text: the

text: input

text: is

text: a

text: list

text: of

text: QFFs,

text: φ

text: with

text: each

text: A

text: deﬁned

text: from

text: the

text: polynomials

text: in

text: each

text: φ

text: In

text: (Bradford

text: i

text: i

text: i

text: et

text: al.,

text: there

text: was

text: an

text: assumption

text: (no

text: longer

text: made)

text: that

text: each

text: of

text: these

text: formulae

text: had

text: a

text: designated

text: EC

text: f

text: from

text: which

text: the

text: subsets

text: E

text: are

text: deﬁned.

text: Instead,

text: we

text: deﬁne

text: E

text: to

text: i

text: i

text: i


text: { } be a basis for f if there is such a designated EC and deﬁne E = A otherwise. That i i i is, we need to treat all the polynomials in QFFs with no EC with the importance usually reserved for ECs. 3.2. Comparison with using a single implicit equational constraint A It is clear that in general the reduced projection PE( ) will lead to fewer projection polynomials than using the full projection P. However, a comparison with the existing theory of equational constraints requires a little more care. First, we note that the TTICAD theory is applicable to a sequence of formulae while the theory of McCallum (1999) is applicable only to a single formula. Hence if the truth value of each QFF is needed then TTICAD is the only option; a truth-invariant CAD for a parent formula will not necessarily suﬃce. Second we note that even if the sequence do form a parent formula then this must have an overall EC to use (McCallum, 1999) while the TTICAD theory is applicable even if this is not the case. Let us consider the situation where both theories are applicable, i.e. we have a sequence (cid:81) of formulae (forming a parent formula) for which each has an EC and thus the parent formula an implicit EC (their product). In the context of Section this corresponds to using f as the EC. The implicit EC approach would correspond to using the i i ∪ ∪ reduced projection P (A) of (McCallum, 1999), with E = E and A = A . We E i i iAi make the simplifying assumption that A is an irreducible basis. In general PE( ) will still contain fewer polynomials than P (A) since P (A) contains all resultants res(f, g) ∈ ∈ ∈ E A E where f E , g A (and g / E), while PE( ) contains only those with i = j (and ∈ i j g / E ). Thus even in situations where the previous theory applies there is an advantage i in using the new TTICAD theory. These savings are highlighted by the worked examples in the next subsection and the complexity analysis later. 3.3. Worked examples In Section 1 we deﬁne an algorithm for producing TTICADs. First we illustrate the savings with our worked examples from Section , which satisfy the simplifying assump- tions from Section 1. We start by considering Φ from equation (2). In the notation above we have: { } { } A := f , g , E := f ; 1 { 1 1} 1 { 1} A := f , g , E := f . (cid:8) 2 2 2 (cid:9) 2 2 We construct the reduced(cid:8)projection sets for each φi, (cid:9) − − P (A ) = x2 1, x4 x2 + 1 , E1 1 − −16 − P (A ) = x2 8x + 15, x4 16x3 + 95x2 248x + 3841 , E2 2 16 and the cross-resultant set × E { } { − } Res ( ) = res (f , f ) = 68x2 272x + 285 . y 1 2 A PE( ) is then the union of these three sets. In Figure 3 we plot the polynomials (solid A curves) and identify the 12 real solutions of PE( ) (solid vertical lines). We can see the solutions align with the asymptotes of the f ’s and the important intersections (those of i f with g and f with g ). 1 1 2 2 If we were to instead use a projection operator based on an implicit EC f f = 0 { 1 2} then in the notation above we would construct P (A) from A = f , f , g , g and E 1 2 1 2 14 ##LTLine## ##LTLine##

text: be

text: a

text: basis

text: for

text: f

text: if

text: there

text: is

text: such

text: a

text: designated

text: EC

text: and

text: deﬁne

text: E

text: A

text: otherwise.

text: That

text: i

text: i

text: i

text: is,

text: we

text: need

text: to

text: treat

text: all

text: the

text: polynomials

text: in

text: QFFs

text: with

text: no

text: EC

text: with

text: the

text: importance

text: usually

text: reserved

text: for

text: ECs.

text: Comparison

text: with

text: using

text: a

text: single

text: implicit

text: equational

text: constraint

text: A

text: It

text: is

text: clear

text: that

text: in

text: general

text: the

text: reduced

text: projection

text: PE(

text: will

text: lead

text: to

text: fewer

text: projection

text: polynomials

text: than

text: using

text: the

text: full

text: projection

text: P.

text: However,

text: a

text: comparison

text: with

text: the

text: existing

text: theory

text: of

text: equational

text: constraints

text: requires

text: a

text: little

text: more

text: care.

text: First,

text: we

text: note

text: that

text: the

text: TTICAD

text: theory

text: is

text: applicable

text: to

text: a

text: sequence

text: of

text: formulae

text: while

text: the

text: theory

text: of

text: McCallum

text: is

text: applicable

text: only

text: to

text: a

text: single

text: formula.

text: Hence

text: if

text: the

text: truth

text: value

text: of

text: each

text: QFF

text: is

text: needed

text: then

text: TTICAD

text: is

text: the

text: only

text: option;

text: a

text: truth-invariant

text: CAD

text: for

text: a

text: parent

text: formula

text: will

text: not

text: necessarily

text: suﬃce.

text: Second

text: we

text: note

text: that

text: even

text: if

text: the

text: sequence

text: do

text: form

text: a

text: parent

text: formula

text: then

text: this

text: must

text: have

text: an

text: overall

text: EC

text: to

text: use

text: (McCallum,

text: while

text: the

text: TTICAD

text: theory

text: is

text: applicable

text: even

text: if

text: this

text: is

text: not

text: the

text: case.

text: Let

text: us

text: consider

text: the

text: situation

text: where

text: both

text: theories

text: are

text: applicable,

text: i.e.

text: we

text: have

text: a

text: sequence

text: of

text: formulae

text: (forming

text: a

text: parent

text: formula)

text: for

text: which

text: each

text: has

text: an

text: EC

text: and

text: thus

text: the

text: parent

text: formula

text: an

text: implicit

text: EC

text: (their

text: product).

text: In

text: the

text: context

text: of

text: Section

text: this

text: corresponds

text: to

text: using

text: f

text: as

text: the

text: EC.

text: The

text: implicit

text: EC

text: approach

text: would

text: correspond

text: to

text: using

text: the

text: i

text: i

text: reduced

text: projection

text: P

text: (A)

text: of

text: (McCallum,

text: with

text: E

text: E

text: and

text: A

text: A

text: We

text: E

text: i

text: i

text: iAi

text: make

text: the

text: simplifying

text: assumption

text: that

text: A

text: is

text: an

text: irreducible

text: basis.

text: In

text: general

text: PE(

text: will

text: still

text: contain

text: fewer

text: polynomials

text: than

text: P

text: (A)

text: since

text: P

text: (A)

text: contains

text: all

text: resultants

text: res(f,

text: g)

text: E

text: A

text: E

text: where

text: f

text: E

text: g

text: A

text: (and

text: g

text: E),

text: while

text: PE(

text: contains

text: only

text: those

text: with

text: i

text: j

text: (and

text: i

text: j

text: g

text: E

text: Thus

text: even

text: in

text: situations

text: where

text: the

text: previous

text: theory

text: applies

text: there

text: is

text: an

text: advantage

text: i

text: in

text: using

text: the

text: new

text: TTICAD

text: theory.

text: These

text: savings

text: are

text: highlighted

text: by

text: the

text: worked

text: examples

text: in

text: the

text: next

text: subsection

text: and

text: the

text: complexity

text: analysis

text: later.

text: Worked

text: examples

text: In

text: Section

text: we

text: deﬁne

text: an

text: algorithm

text: for

text: producing

text: TTICADs.

text: First

text: we

text: illustrate

text: the

text: savings

text: with

text: our

text: worked

text: examples

text: from

text: Section

text: which

text: satisfy

text: the

text: simplifying

text: assump-

text: tions

text: from

text: Section

text: We

text: start

text: by

text: considering

text: Φ

text: from

text: equation

text: In

text: the

text: notation

text: above

text: we

text: have:

text: A

text: f

text: g

text: E

text: f

text: A

text: f

text: g

text: E

text: f

text: We

text: construct

text: the

text: sets

text: for

text: each

text: φi,

text: P

text: (A

text: P

text: (A

text: and

text: the

text: cross-resultant

text: set

text: E

text: Res

text: res

text: (f

text: f

text: y

text: A

text: PE(

text: is

text: then

text: the

text: union

text: of

text: these

text: three

text: sets.

text: In

text: Figure

text: we

text: plot

text: the

text: polynomials

text: (solid

text: A

text: curves)

text: and

text: identify

text: the

text: real

text: solutions

text: of

text: PE(

text: (solid

text: vertical

text: lines).

text: We

text: can

text: see

text: the

text: solutions

text: align

text: with

text: the

text: asymptotes

text: of

text: the

text: f

text: ’s

text: and

text: the

text: important

text: intersections

text: (those

text: of

text: i

text: f

text: with

text: g

text: and

text: f

text: with

text: g

text: If

text: we

text: were

text: to

text: instead

text: use

text: a

text: projection

text: operator

text: based

text: on

text: an

text: implicit

text: EC

text: f

text: f

text: then

text: in

text: the

text: notation

text: above

text: we

text: would

text: construct

text: P

text: (A)

text: from

text: A

text: f

text: f

text: g

text: g

text: and

text: E

text: ##LTLine##

text: ##LTLine##


text: regions between) which together form a stack. Taking the union of these stacks gives the R R CAD of 2. This is repeated until a CAD of n is produced. At each stage the cells are represented by (at least) a sample point and an index: a list of integers corresponding to the ordered roots of the projection polynomials which locates the cell in the CAD. To conclude that a CAD produced in this way is sign-invariant we need delineability. A polynomial is delineable in a cell if the portion of its zero set in the cell consists of disjoint sections. A set of polynomials are delineable in a cell if each is delineable and the sections of diﬀerent polynomials in the cell are either identical or disjoint. The projection operator used must be deﬁned so that over each cell of a sign-invariant CAD for projection polynomials in r variables (the word over meaning we are now talking about an (r + 1)-dim space) the polynomials in r + 1 variables are delineable. The output of this and subsequent CAD algorithms (including the one presented in this paper) depends heavily on the variable ordering. We usually work with polynomials Z Z in [x] = [x , . . . , x ] with the variables, x, in ascending order (so we ﬁrst project with 1 n respect to x and continue to reach univariate polynomials in x ). The main variable of n 1 a polynomial (mvar) is the greatest variable present with respect to the ordering. CAD has doubly exponential complexity in the number of variables (Brown and Dav- enport, 2007; Davenport and Heintz, 1988). There now exist algorithms with better complexity for some CAD applications (see for example Basu et al. (1996)) but CAD implementations often remain the best general purpose approach. There have been many developments to the theory since Collin’s treatment, including the following: • Improvements to the projection operator (Hong, 1990; McCallum, 1988, 1998; Brown, 2001; Han et al., 2014), reducing the number of projection polynomials computed. • Algorithms to identify the adjacency of cells in a CAD (Arnon et al., 1984b, 1988) and following from this the idea of clustering (Arnon, 1988) to minimise the lifting. • Partial CAD, introduced by Collins and Hong (1991), where the structure of F is used R R to lift less of the decomposition of k to n, if it is suﬃcient to deduce ψ. • The theory of equational constraints, (McCallum, 1999, 2001; Brown and McCallum, 2005), also aiming to deduce ψ itself, this time using more eﬃcient projections. • The use of certiﬁed numerics in the lifting phase to minimise the amount of symbolic computation required (Strzebon´ski, 2006; Iwane et al., 2009). • New approaches which break with the normal projection and lifting model: local pro- jection (Strzebon´ski, 2014), the building of single CAD cells (Brown, 2013; Jovanovic and de Moura, 2012) and CAD via Triangular Decomposition (Chen et al., 2009b). The latter is now used for the CAD command built into Maple, and works by ﬁrst creating a cylindrical decomposition of complex space. 1.2. TTICAD Brown (1998) deﬁned a truth-invariant CAD as one for which a formula had invariant truth value on each cell. Given a QFF, a sign-invariant CAD for the deﬁning polynomi- als is trivially truth-invariant. Brown considered the reﬁnement of sign-invariant CADs whilst maintaining truth-invariance, while some of the developments listed above can be viewed as methods to produce truth-invariant CADs directly. We deﬁne a new but related type of CAD, the topic of this paper. { } Deﬁnition 1. Let φ t refer to a list of QFFs. We say a cylindrical algebraic decom- D i i=1 position is a Truth Table Invariant CAD for the QFFs (TTICAD) if the Boolean value D of each φ is constant (either true or false) on each cell of . i 3

text: regions

text: between)

text: which

text: together

text: form

text: a

text: stack.

text: Taking

text: the

text: union

text: of

text: these

text: stacks

text: gives

text: the

text: R

text: R

text: CAD

text: of

text: This

text: is

text: repeated

text: until

text: a

text: CAD

text: of

text: n

text: is

text: produced.

text: At

text: each

text: stage

text: the

text: cells

text: are

text: represented

text: by

text: (at

text: least)

text: a

text: sample

text: point

text: and

text: an

text: index:

text: a

text: list

text: of

text: integers

text: corresponding

text: to

text: the

text: ordered

text: roots

text: of

text: the

text: projection

text: polynomials

text: which

text: locates

text: the

text: cell

text: in

text: the

text: CAD.

text: To

text: conclude

text: that

text: a

text: CAD

text: produced

text: in

text: this

text: way

text: is

text: sign-invariant

text: we

text: need

text: delineability.

text: A

text: polynomial

text: is

text: delineable

text: in

text: a

text: cell

text: if

text: the

text: portion

text: of

text: its

text: zero

text: set

text: in

text: the

text: cell

text: consists

text: of

text: disjoint

text: sections.

text: A

text: set

text: of

text: polynomials

text: are

text: delineable

text: in

text: a

text: cell

text: if

text: each

text: is

text: delineable

text: and

text: the

text: sections

text: of

text: diﬀerent

text: polynomials

text: in

text: the

text: cell

text: are

text: either

text: identical

text: or

text: disjoint.

text: The

text: projection

text: operator

text: used

text: must

text: be

text: deﬁned

text: so

text: that

text: over

text: each

text: cell

text: of

text: a

text: sign-invariant

text: CAD

text: for

text: projection

text: polynomials

text: in

text: r

text: variables

text: (the

text: word

text: over

text: meaning

text: we

text: are

text: now

text: talking

text: about

text: an

text: (r

text: space)

text: the

text: polynomials

text: in

text: r

text: variables

text: are

text: delineable.

text: The

text: output

text: of

text: this

text: and

text: subsequent

text: CAD

text: algorithms

text: (including

text: the

text: one

text: presented

text: in

text: this

text: paper)

text: depends

text: heavily

text: on

text: the

text: variable

text: ordering.

text: We

text: usually

text: work

text: with

text: polynomials

text: Z

text: Z

text: in

text: [x]

text: [x

text: x

text: with

text: the

text: variables,

text: x,

text: in

text: ascending

text: order

text: (so

text: we

text: ﬁrst

text: project

text: with

text: n

text: respect

text: to

text: x

text: and

text: continue

text: to

text: reach

text: univariate

text: polynomials

text: in

text: x

text: The

text: main

text: variable

text: of

text: n

text: a

text: polynomial

text: (mvar)

text: is

text: the

text: greatest

text: variable

text: present

text: with

text: respect

text: to

text: the

text: ordering.

text: CAD

text: has

text: doubly

text: exponential

text: complexity

text: in

text: the

text: number

text: of

text: variables

text: (Brown

text: and

text: Dav-

text: enport,

text: Davenport

text: and

text: Heintz,

text: There

text: now

text: exist

text: algorithms

text: with

text: better

text: complexity

text: for

text: some

text: CAD

text: applications

text: (see

text: for

text: example

text: Basu

text: et

text: al.

text: but

text: CAD

text: implementations

text: often

text: remain

text: the

text: best

text: general

text: purpose

text: approach.

text: There

text: have

text: been

text: many

text: developments

text: to

text: the

text: theory

text: since

text: Collin’s

text: treatment,

text: including

text: the

text: following:

text: Improvements

text: to

text: the

text: projection

text: operator

text: (Hong,

text: McCallum,

text: Brown,

text: Han

text: et

text: al.,

text: reducing

text: the

text: number

text: of

text: projection

text: polynomials

text: computed.

text: Algorithms

text: to

text: identify

text: the

text: adjacency

text: of

text: cells

text: in

text: a

text: CAD

text: (Arnon

text: et

text: al.,

text: and

text: following

text: from

text: this

text: the

text: idea

text: of

text: clustering

text: (Arnon,

text: to

text: minimise

text: the

text: lifting.

text: Partial

text: CAD,

text: introduced

text: by

text: Collins

text: and

text: Hong

text: where

text: the

text: structure

text: of

text: F

text: is

text: used

text: R

text: R

text: to

text: lift

text: less

text: of

text: the

text: decomposition

text: of

text: k

text: to

text: n,

text: if

text: it

text: is

text: suﬃcient

text: to

text: deduce

text: ψ.

text: The

text: theory

text: of

text: equational

text: constraints,

text: (McCallum,

text: Brown

text: and

text: McCallum,

text: also

text: aiming

text: to

text: deduce

text: ψ

text: itself,

text: this

text: time

text: using

text: more

text: eﬃcient

text: projections.

text: The

text: use

text: of

text: certiﬁed

text: numerics

text: in

text: the

text: lifting

text: phase

text: to

text: minimise

text: the

text: amount

text: of

text: symbolic

text: computation

text: required

text: (Strzebon´ski,

text: Iwane

text: et

text: al.,

text: New

text: approaches

text: which

text: break

text: with

text: the

text: normal

text: projection

text: and

text: lifting

text: model:

text: local

text: pro-

text: jection

text: (Strzebon´ski,

text: the

text: building

text: of

text: single

text: CAD

text: cells

text: (Brown,

text: Jovanovic

text: and

text: de

text: Moura,

text: and

text: CAD

text: via

text: Triangular

text: Decomposition

text: (Chen

text: et

text: al.,

text: The

text: latter

text: is

text: now

text: used

text: for

text: the

text: CAD

text: command

text: built

text: into

text: Maple,

text: and

text: works

text: by

text: ﬁrst

text: creating

text: a

text: cylindrical

text: decomposition

text: of

text: complex

text: space.

text: TTICAD

text: Brown

text: deﬁned

text: a

text: truth-invariant

text: CAD

text: as

text: one

text: for

text: which

text: a

text: formula

text: had

text: invariant

text: truth

text: value

text: on

text: each

text: cell.

text: Given

text: a

text: QFF,

text: a

text: sign-invariant

text: CAD

text: for

text: the

text: deﬁning

text: polynomi-

text: als

text: is

text: trivially

text: truth-invariant.

text: Brown

text: considered

text: the

text: reﬁnement

text: of

text: sign-invariant

text: CADs

text: whilst

text: maintaining

text: truth-invariance,

text: while

text: some

text: of

text: the

text: developments

text: listed

text: above

text: can

text: be

text: viewed

text: as

text: methods

text: to

text: produce

text: truth-invariant

text: CADs

text: directly.

text: We

text: deﬁne

text: a

text: new

text: but

text: related

text: type

text: of

text: CAD,

text: the

text: topic

text: of

text: this

text: paper.

text: Deﬁnition

text: Let

text: φ

text: t

text: refer

text: to

text: a

text: list

text: of

text: QFFs.

text: We

text: say

text: a

text: cylindrical

text: algebraic

text: decom-

text: D

text: i

text: position

text: is

text: a

text: Truth

text: Table

text: Invariant

text: CAD

text: for

text: the

text: QFFs

text: (TTICAD)

text: if

text: the

text: Boolean

text: value

text: D

text: of

text: each

text: φ

text: is

text: constant

text: (either

text: true

text: or

text: false)

text: on

text: each

text: cell

text: of

text: i


text: A sign-invariant CAD for all polynomials occurring in a list of formulae would clearly be a TTICAD for the list. However, we aim to produce smaller TTICADs for many such lists. We will achieve this by utilising the presence of equational constraints, a technique ﬁrst suggested by Collins (1998) with key theory developed by McCallum (1999). Deﬁnition 2. Suppose some quantiﬁed formula is given: ∗ · · · φ = (Q x ) (Q x )φ(x) k+1 k+1 n n where the Q are quantiﬁers and φ is quantiﬁer free. An equation f = 0 is an equational i ∗ ∗ constraint (EC) of φ if f = 0 is logically implied by φ (the quantiﬁer-free part of φ ). Such a constraint may be either explicit (an atom of the formula) or otherwise implicit. In Sections 1 and 1 we will describe how TTICADs can be produced eﬃciently when there are ECs present in the list of formulae. There are two reasons to use this theory. ∗ (1) As a tool to build a truth-invariant CAD eﬃciently: If a parent formula φ is built { } { } ∗ from the formulae φ then any TTICAD for φ is also truth-invariant for φ . i i We note that for such a formula a TTICAD may need to contain more cells than a truth-invariant CAD. For example, consider a cell in a truth-invariant CAD for ∗ ∨ φ = φ φ within which φ is always true. If φ changed truth value in such 1 2 1 2 a cell then it would need to be split in order to achieve a TTICAD, but this is ∗ unnecessary for a truth-invariant CAD of φ . Nevertheless, we ﬁnd that our TTICAD theory is often able to produce smaller truth-invariant CADs than any other available approach. We demonstrate the sav- ings oﬀered via worked examples introduced in the next subsection. (2) When given a problem for which truth table invariance is required: That is, a prob- lem for which the list of formulae are not derived from a larger parent formula and thus a truth-invariant CAD for their disjunction may not suﬃce. For example, decomposing complex space according to a set of branch cuts for the purpose of algebraic simpliﬁcation (Bradford and Davenport, 2002; Phisanbut et al., 2010). Here the idea is to represent each branch cut as a semi-algebraic set to give input admissible to CAD, (recent progress on this has been described by England et al. (2013)). Then a TTICAD for the list of formulae these sets deﬁne provides the necessary decomposition. Example 33 is from this class. 1.3. Worked examples To demonstrate our ideas we will provide details for two worked examples. Assume ≺ we have the variable ordering x y (meaning 1-dimensional CADs are with respect to x) and consider the following polynomials, graphed in Figure 1. − − f := x2 + y2 1 g := xy 1 1 − − − 1 − 4 − − f := (x 4)2 + (y 1)2 1 g := (x 4)(y 1) 1 2 2 4 R Suppose we wish to ﬁnd the regions of 2 where the following formula is true: ∧ ∨ ∧ Φ := (f = 0 g < 0) (f = 0 g < 0) . (2) 1 1 2 2 4 ##LTLine## ##LTLine##

text: A

text: sign-invariant

text: CAD

text: for

text: all

text: polynomials

text: occurring

text: in

text: a

text: list

text: of

text: formulae

text: would

text: clearly

text: be

text: a

text: TTICAD

text: for

text: the

text: list.

text: However,

text: we

text: aim

text: to

text: produce

text: smaller

text: TTICADs

text: for

text: many

text: such

text: lists.

text: We

text: will

text: achieve

text: this

text: by

text: utilising

text: the

text: presence

text: of

text: equational

text: constraints,

text: a

text: technique

text: ﬁrst

text: suggested

text: by

text: Collins

text: with

text: key

text: theory

text: developed

text: by

text: McCallum

text: Deﬁnition

text: Suppose

text: some

text: quantiﬁed

text: formula

text: is

text: given:

text: φ

text: (Q

text: x

text: (Q

text: x

text: )φ(x)

text: n

text: n

text: where

text: the

text: Q

text: are

text: quantiﬁers

text: and

text: φ

text: is

text: quantiﬁer

text: free.

text: An

text: equation

text: f

text: is

text: an

text: equational

text: i

text: constraint

text: (EC)

text: of

text: φ

text: if

text: f

text: is

text: logically

text: implied

text: by

text: φ

text: (the

text: quantiﬁer-free

text: part

text: of

text: φ

text: Such

text: a

text: constraint

text: may

text: be

text: either

text: explicit

text: (an

text: atom

text: of

text: the

text: formula)

text: or

text: otherwise

text: implicit.

text: In

text: Sections

text: and

text: we

text: will

text: describe

text: how

text: TTICADs

text: can

text: be

text: produced

text: eﬃciently

text: when

text: there

text: are

text: ECs

text: present

text: in

text: the

text: list

text: of

text: formulae.

text: There

text: are

text: two

text: reasons

text: to

text: use

text: this

text: theory.

text: As

text: a

text: tool

text: to

text: build

text: a

text: truth-invariant

text: CAD

text: eﬃciently:

text: If

text: a

text: parent

text: formula

text: φ

text: is

text: built

text: from

text: the

text: formulae

text: φ

text: then

text: any

text: TTICAD

text: for

text: φ

text: is

text: also

text: truth-invariant

text: for

text: φ

text: i

text: i

text: We

text: note

text: that

text: for

text: such

text: a

text: formula

text: a

text: TTICAD

text: may

text: need

text: to

text: contain

text: more

text: cells

text: than

text: a

text: truth-invariant

text: CAD.

text: For

text: example,

text: consider

text: a

text: cell

text: in

text: a

text: truth-invariant

text: CAD

text: for

text: φ

text: φ

text: φ

text: within

text: which

text: φ

text: is

text: always

text: true.

text: If

text: φ

text: changed

text: truth

text: value

text: in

text: such

text: a

text: cell

text: then

text: it

text: would

text: need

text: to

text: be

text: split

text: in

text: order

text: to

text: achieve

text: a

text: TTICAD,

text: but

text: this

text: is

text: unnecessary

text: for

text: a

text: truth-invariant

text: CAD

text: of

text: φ

text: Nevertheless,

text: we

text: ﬁnd

text: that

text: our

text: TTICAD

text: theory

text: is

text: often

text: able

text: to

text: produce

text: smaller

text: truth-invariant

text: CADs

text: than

text: any

text: other

text: available

text: approach.

text: We

text: demonstrate

text: the

text: sav-

text: ings

text: oﬀered

text: via

text: worked

text: examples

text: introduced

text: in

text: the

text: next

text: subsection.

text: When

text: given

text: a

text: problem

text: for

text: which

text: truth

text: table

text: invariance

text: is

text: required:

text: That

text: is,

text: a

text: prob-

text: lem

text: for

text: which

text: the

text: list

text: of

text: formulae

text: are

text: not

text: derived

text: from

text: a

text: larger

text: parent

text: formula

text: and

text: thus

text: a

text: truth-invariant

text: CAD

text: for

text: their

text: disjunction

text: may

text: not

text: suﬃce.

text: For

text: example,

text: decomposing

text: complex

text: space

text: according

text: to

text: a

text: set

text: of

text: branch

text: cuts

text: for

text: the

text: purpose

text: of

text: algebraic

text: simpliﬁcation

text: (Bradford

text: and

text: Davenport,

text: Phisanbut

text: et

text: al.,

text: Here

text: the

text: idea

text: is

text: to

text: represent

text: each

text: branch

text: cut

text: as

text: a

text: semi-algebraic

text: set

text: to

text: give

text: input

text: admissible

text: to

text: CAD,

text: (recent

text: progress

text: on

text: this

text: has

text: been

text: described

text: by

text: England

text: et

text: al.

text: Then

text: a

text: TTICAD

text: for

text: the

text: list

text: of

text: formulae

text: these

text: sets

text: deﬁne

text: provides

text: the

text: necessary

text: decomposition.

text: Example

text: is

text: from

text: this

text: class.

text: Worked

text: examples

text: To

text: demonstrate

text: our

text: ideas

text: we

text: will

text: provide

text: details

text: for

text: two

text: worked

text: examples.

text: Assume

text: we

text: have

text: the

text: variable

text: ordering

text: x

text: y

text: (meaning

text: CADs

text: are

text: with

text: respect

text: to

text: x)

text: and

text: consider

text: the

text: following

text: polynomials,

text: graphed

text: in

text: Figure

text: f

text: g

text: xy

text: f

text: (x

text: (y

text: g

text: (x

text: R

text: Suppose

text: we

text: wish

text: to

text: ﬁnd

text: the

text: regions

text: of

text: where

text: the

text: following

text: formula

text: is

text: true:

text: Φ

text: (f

text: g

text: (f

text: g

text: ##LTLine##

text: ##LTLine##


text: England, M., Bradford, R., Davenport, J.H., Wilson, D., 2013. Understanding branch cuts of expressions. In: Intelligent Computer Mathematics (LNAI 7961). Springer Berlin Heidelberg, pp. 136–151. England, M., Bradford, R., Davenport, J.H., Wilson, D., 2014b. Choosing a variable ordering for truth-table invariant cylindrical algebraic decomposition by incremen- tal triangular decomposition. In: Mathematical Software – ICMS 2014 (LNCS 8592). Springer Heidelberg, pp. 450–457. England, M., Cheb-Terrab, E., Bradford, R., Davenport, J.H., Wilson, D., 2014c. Branch cuts in maple 17. ACM CCA 187 (1), 24–27. England, M., Wilson, D., Bradford, R., Davenport, J.H., 2014d. Using the Regular Chains Library to build cylindrical algebraic decompositions by projecting and lifting. In: Mathematical Software – ICMS 2014 (LNCS 8592). Springer Heidelberg, pp. 458–465. Erascu, M., Hong, H., 2014. Synthesis of optimal numerical algorithms using real quan- tiﬁer elimination (Case Study: Square root computation). In: ISSAC ’14. ACM, pp. 162–169. Fotiou, I., Parrilo, P., Morari, M., 2005. Nonlinear parametric optimization using cylin- drical algebraic decomposition. In: CDC-ECC ’05. pp. 3735–3740. Han, J., Dai, L., Xia, B., 2014. Constructing fewer open cells by gcd computation in CAD projection. In: ISSAC ’14. ACM, pp. 240–247. Hong, H., 1990. An improvement of the projection operator in cylindrical algebraic de- composition. In: ISSAC ’90. ACM, pp. 261–264. Huang, Z., England, M., Wilson, D., Davenport, J.H., Paulson, L., Bridge, J., 2014. Applying machine learning to the problem of choosing a heuristic to select the variable ordering for cylindrical algebraic decomposition. In: Intelligent Computer Mathematics (LNAI 8543). Springer International, pp. 92–107. Iwane, H., Yanami, H., Anai, H., Yokoyama, K., 2009. An eﬀective implementation of a symbolic-numeric cylindrical algebraic decomposition for quantiﬁer elimination. In: SNC ’09. pp. 55–64. Jovanovic, D., de Moura, L., 2012. Solving non-linear arithmetic. In: IJCAR (LNCS 7364). Springer, pp. 339–354. Kahan, W., 1987. Branch cuts for complex elementary functions. In: Iserles, A., Powell, M. (Eds.), Proc. State of Art in Numerical Analysis. Clarendon Press, pp. 165–211. Lazard, D., 1994. An improved projection for cylindrical algebraic decomposition. In: Algebraic Geometry and its Applications: Collections of Papers from Abhyankar’s 60th Birthday Conference. Springer Berlin, pp. 467–476. McCallum, S., 1985. An improved projection operation for cylindrical algebraic decom- position. PhD Thesis (Computer Sciences Tech. Rep. 578), Uni. Wisconsin-Madison. McCallum, S., 1988. An improved projection operation for cylindrical algebraic decom- position of three-dimensional space. J. Symbolic Comp. 5 (1-2), 141–161. McCallum, S., 1998. An improved projection operation for cylindrical algebraic decom- position. In: Caviness, B., Johnson, J. (Eds.), Quantiﬁer Elimination and Cylindrical Algebraic Decomposition. Texts & Monographs in Symbolic Computation. Springer- Verlag, pp. 242–268. McCallum, S., 1999. On projection in CAD-based quantiﬁer elimination with equational constraint. In: ISSAC ’99. ACM, pp. 145–149. McCallum, S., 2001. On propagation of equational constraints in CAD-based quantiﬁer elimination. In: ISSAC ’01. ACM, pp. 223–231. 40

text: England,

text: M.,

text: Bradford,

text: R.,

text: Davenport,

text: J.H.,

text: Wilson,

text: D.,

text: Understanding

text: branch

text: cuts

text: of

text: expressions.

text: In:

text: Intelligent

text: Computer

text: Mathematics

text: (LNAI

text: Springer

text: Berlin

text: Heidelberg,

text: pp.

text: England,

text: M.,

text: Bradford,

text: R.,

text: Davenport,

text: J.H.,

text: Wilson,

text: D.,

text: Choosing

text: a

text: variable

text: ordering

text: for

text: truth-table

text: invariant

text: cylindrical

text: algebraic

text: decomposition

text: by

text: incremen-

text: tal

text: triangular

text: decomposition.

text: In:

text: Mathematical

text: Software

text: ICMS

text: (LNCS

text: Springer

text: Heidelberg,

text: pp.

text: England,

text: M.,

text: Cheb-Terrab,

text: E.,

text: Bradford,

text: R.,

text: Davenport,

text: J.H.,

text: Wilson,

text: D.,

text: Branch

text: cuts

text: in

text: maple

text: ACM

text: CCA

text: England,

text: M.,

text: Wilson,

text: D.,

text: Bradford,

text: R.,

text: Davenport,

text: J.H.,

text: Using

text: the

text: Regular

text: Chains

text: Library

text: to

text: build

text: cylindrical

text: algebraic

text: decompositions

text: by

text: projecting

text: and

text: lifting.

text: In:

text: Mathematical

text: Software

text: ICMS

text: (LNCS

text: Springer

text: Heidelberg,

text: pp.

text: Erascu,

text: M.,

text: Hong,

text: H.,

text: Synthesis

text: of

text: optimal

text: numerical

text: algorithms

text: using

text: real

text: quan-

text: tiﬁer

text: elimination

text: (Case

text: Study:

text: Square

text: root

text: computation).

text: In:

text: ISSAC

text: ACM,

text: pp.

text: Fotiou,

text: I.,

text: Parrilo,

text: P.,

text: Morari,

text: M.,

text: Nonlinear

text: parametric

text: optimization

text: using

text: cylin-

text: drical

text: algebraic

text: decomposition.

text: In:

text: CDC-ECC

text: pp.

text: Han,

text: J.,

text: Dai,

text: L.,

text: Xia,

text: B.,

text: Constructing

text: fewer

text: open

text: cells

text: by

text: gcd

text: computation

text: in

text: CAD

text: projection.

text: In:

text: ISSAC

text: ACM,

text: pp.

text: Hong,

text: H.,

text: An

text: improvement

text: of

text: the

text: projection

text: operator

text: in

text: cylindrical

text: algebraic

text: de-

text: composition.

text: In:

text: ISSAC

text: ACM,

text: pp.

text: Huang,

text: Z.,

text: England,

text: M.,

text: Wilson,

text: D.,

text: Davenport,

text: J.H.,

text: Paulson,

text: L.,

text: Bridge,

text: J.,

text: Applying

text: machine

text: learning

text: to

text: the

text: problem

text: of

text: choosing

text: a

text: heuristic

text: to

text: select

text: the

text: variable

text: ordering

text: for

text: cylindrical

text: algebraic

text: decomposition.

text: In:

text: Intelligent

text: Computer

text: Mathematics

text: (LNAI

text: Springer

text: International,

text: pp.

text: Iwane,

text: H.,

text: Yanami,

text: H.,

text: Anai,

text: H.,

text: Yokoyama,

text: K.,

text: An

text: eﬀective

text: implementation

text: of

text: a

text: symbolic-numeric

text: cylindrical

text: algebraic

text: decomposition

text: for

text: quantiﬁer

text: elimination.

text: In:

text: SNC

text: pp.

text: Jovanovic,

text: D.,

text: de

text: Moura,

text: L.,

text: Solving

text: non-linear

text: arithmetic.

text: In:

text: IJCAR

text: (LNCS

text: Springer,

text: pp.

text: Kahan,

text: W.,

text: Branch

text: cuts

text: for

text: complex

text: elementary

text: functions.

text: In:

text: Iserles,

text: A.,

text: Powell,

text: M.

text: (Eds.),

text: Proc.

text: State

text: of

text: Art

text: in

text: Numerical

text: Analysis.

text: Clarendon

text: Press,

text: pp.

text: Lazard,

text: D.,

text: An

text: improved

text: projection

text: for

text: cylindrical

text: algebraic

text: decomposition.

text: In:

text: Algebraic

text: Geometry

text: and

text: its

text: Applications:

text: Collections

text: of

text: Papers

text: from

text: Abhyankar’s

text: Birthday

text: Conference.

text: Springer

text: Berlin,

text: pp.

text: McCallum,

text: S.,

text: An

text: improved

text: projection

text: operation

text: for

text: cylindrical

text: algebraic

text: decom-

text: position.

text: PhD

text: Thesis

text: (Computer

text: Sciences

text: Tech.

text: Rep.

text: Uni.

text: Wisconsin-Madison.

text: McCallum,

text: S.,

text: An

text: improved

text: projection

text: operation

text: for

text: cylindrical

text: algebraic

text: decom-

text: position

text: of

text: three-dimensional

text: space.

text: J.

text: Symbolic

text: Comp.

text: McCallum,

text: S.,

text: An

text: improved

text: projection

text: operation

text: for

text: cylindrical

text: algebraic

text: decom-

text: position.

text: In:

text: Caviness,

text: B.,

text: Johnson,

text: J.

text: (Eds.),

text: Quantiﬁer

text: Elimination

text: and

text: Cylindrical

text: Algebraic

text: Decomposition.

text: Texts

text: Monographs

text: in

text: Symbolic

text: Computation.

text: Springer-

text: Verlag,

text: pp.

text: McCallum,

text: S.,

text: On

text: projection

text: in

text: CAD-based

text: quantiﬁer

text: elimination

text: with

text: equational

text: constraint.

text: In:

text: ISSAC

text: ACM,

text: pp.

text: McCallum,

text: S.,

text: On

text: propagation

text: of

text: equational

text: constraints

text: in

text: CAD-based

text: quantiﬁer

text: elimination.

text: In:

text: ISSAC

text: ACM,

text: pp.


text: Correlations and neutrinoless ββ decay nuclear matrix elements of pf -shell nuclei Javier Men´endeza,b, Tom´as R. Rodr´ıgueza, Gabriel Mart´ınez-Pinedoa,c, Alfredo Povesd aInstitut fu¨r Kernphysik, Technische Universita¨t Darmstadt, Schlossgartenstr. 2, D-64289 Darmstadt, Germany bExtreMe Matter Institute EMMI, GSI Helmholtzzentrum fu¨r Schwerionenforschung GmbH, D-64291 Darmstadt, Germany cGSI Helmholtzzentrum fu¨r Schwerionenforschung, Plankstr. 1, D-64291 Darmstadt, Germany dDepartamento de F´ısica Te´orica and IFT UAM/CSIC, Universidad Aut´onoma de Madrid, Cantoblanco, E-28049, Madrid, Spain Abstract We calculate the nuclear matrix elements (NMEs) for neutrinoless double-beta decays (0νββ) of pf-shell nuclei using the shell model (SM) and energy density functional (EDF) methods. The systematic study of non-physical decays (except → → → for 48Ca) of Ca Ti, Ti Cr and Cr Fe allows for a detailed comparison between the two nuclear structure approaches. We observe that while the dominant Gamow-Teller part of the NME diﬀers roughly by a factor of two between SM and EDF, when we restrict the calculations to spherical EDF states and seniority-zero SM conﬁgurations, the NMEs obtained by both methods are strikingly similar. This points out to the important role of nuclear structure correlations for 0νββ decay NMEs. We identify correlations associated with high-seniority components in the initial and ﬁnal states of the decay as one of the reasons for the discrepancies between SM and EDF results. We also explore exact projection to good isospin, and conclude that it has only a moderate eﬀect in the Gamow-Teller part of the NMEs but strongly aﬀects the Fermi contribution. This work opens up the door for NME benchmarks between diﬀerent theoretical approaches, and constitutes a step forward towards more reliable estimations of the NMEs. 1. Introduction model (IBM) [22]. However, state-of-the-art NME pre- dictions by these approaches diﬀer up to factor two [23], Experimental searches for the lepton-number violating strongly limiting the precision to which information on weak process neutrinoless double beta decay (0νββ de- neutrino masses would be known in case of a 0νββ decay cay) are the most promising approach to determine some measurement. In addition, the half-lives of experimentally of the fundamental properties of neutrinos. The detec- relevant nuclei could be signiﬁcantly under/overestimated (cid:104) (cid:105) tion of 0νββ decay would establish the Majorana character for a given m . ββ of neutrinos and provide information about their absolute A better understanding of the NMEs is therefore cru- mass and hierarchy [1]. Ongoing experiments EXO [2], cial. Recently, experimental observables relevant for 0νββ KamLAND-Zen [3] and GERDA [4] have recently set im- decay have been measured [24–26], allowing checks for pressive lower-limits, well over 1025 years, on the half-lives the nuclear structure methods [27], that in some cases of 136Xe and 76Ge, and the dozen most favorable nuclei resulted in closer NMEs between diﬀerent theoretical ap- are being explored worldwide [5–13]. However, an even- proaches [28–30]. tual 0νββ decay detection does not guarantee the precise In this article we follow a complementary approach, determination of absolute neutrino masses, because the studying non-physical 0νββ decays along isotopic chains in half-life depends on the transition nuclear matrix elements the pf-shell, with SM and EDF methods. Although within (NME(cid:104)), M0ν [1]: (cid:105) (cid:18) (cid:19) this region only 48Ca is an actual 0νββ decay candidate, a comparison between methods is better established with → −1 | | (cid:104)m (cid:105) 2 systematic calculations. For instance, systematic EDF cal- T10/ν2(0+i 0+f ) = G0ν M0ν 2 mββ , (1) culations for the cadmium isotopes provided a better un- e derstanding of the role of deformation, pairing and shell (cid:80) eﬀects in 0νββ decay [31]. Systematic studies allow to an- with m the electron mass and G a well-known phase- e 0ν alyze not only numerical values, but also trends, which are space factor [14, 15]. The combination of neutrino masses (cid:104) (cid:105) | | useful to identify similarities and diﬀerences between SM that appears in 0νββ decay is m = U2 m , with ββ i ei i and EDF. Understanding these is essential to reduce the U the neutrino mixing matrix. ij theoretical uncertainty in the 0νββ decay NMEs. NMEs have been predicted by diﬀerent theoretical nu- 2. Theoretical framework clear structure approaches. These comprise large-scale shell model (SM) [16, 17], energy density functional meth- Here we brieﬂy describe the EDF and SM calculations ods (EDF) [18, 19], the quasiparticle random phase ap- performed in this work, as well as the 0νββ decay transi- proximation (QRPA) [20, 21], and the interacting boson tion operator employed. A more extensive description of Preprint submitted to Elsevier May 26, 2020 ##LTLine## ##LTLine## ##LTLine##

text: Correlations

label: title

text: and

text: neutrinoless

text: ββ

text: decay

text: nuclear

text: matrix

text: elements

text: of

text: pf

text: -shell

text: nuclei

text: Javier

label: author

text: Men´endeza,b,

text: Tom´as

text: R.

text: Rodr´ıgueza,

text: Gabriel

text: Mart´ınez-Pinedoa,c,

text: Alfredo

text: Povesd

text: aInstitut

text: fu¨r

text: Kernphysik,

text: Technische

text: Universita¨t

text: Darmstadt,

text: Schlossgartenstr.

text: Darmstadt,

text: Germany

text: bExtreMe

text: Matter

text: Institute

text: EMMI,

text: GSI

text: Helmholtzzentrum

text: fu¨r

text: Schwerionenforschung

text: GmbH,

text: Darmstadt,

text: Germany

text: cGSI

text: Helmholtzzentrum

text: fu¨r

text: Schwerionenforschung,

text: Plankstr.

text: Darmstadt,

text: Germany

text: dDepartamento

text: de

text: F´ısica

text: Te´orica

text: and

text: IFT

text: UAM/CSIC,

text: Universidad

text: Aut´onoma

text: de

text: Madrid,

text: Cantoblanco,

text: Madrid,

text: Spain

text: Abstract

text: We

label: abstract

text: calculate

text: the

text: nuclear

text: matrix

text: elements

text: (NMEs)

text: for

text: neutrinoless

text: double-beta

text: decays

text: (0νββ)

text: of

text: pf-shell

text: nuclei

text: using

text: the

text: shell

text: model

text: (SM)

text: and

text: energy

text: density

text: functional

text: (EDF)

text: methods.

text: The

text: systematic

text: study

text: of

text: non-physical

text: decays

text: (except

text: for

text: of

text: Ca

text: Ti,

text: Ti

text: Cr

text: and

text: Cr

text: Fe

text: allows

text: for

text: a

text: detailed

text: comparison

text: between

text: the

text: two

text: nuclear

text: structure

text: approaches.

text: We

text: observe

text: that

text: while

text: the

text: dominant

text: Gamow-Teller

text: part

text: of

text: the

text: NME

text: diﬀers

text: roughly

text: by

text: a

text: factor

text: of

text: two

text: between

text: SM

text: and

text: EDF,

text: when

text: we

text: restrict

text: the

text: calculations

text: to

text: spherical

text: EDF

text: states

text: and

text: seniority-zero

text: SM

text: conﬁgurations,

text: the

text: NMEs

text: obtained

text: by

text: both

text: methods

text: are

text: strikingly

text: similar.

text: This

text: points

text: out

text: to

text: the

text: important

text: role

text: of

text: nuclear

text: structure

text: correlations

text: for

text: 0νββ

text: decay

text: NMEs.

text: We

text: identify

text: correlations

text: associated

text: with

text: high-seniority

text: components

text: in

text: the

text: initial

text: and

text: ﬁnal

text: states

text: of

text: the

text: decay

text: as

text: one

text: of

text: the

text: reasons

text: for

text: the

text: discrepancies

text: between

text: SM

text: and

text: EDF

text: results.

text: We

text: also

text: explore

text: exact

text: projection

text: to

text: good

text: isospin,

text: and

text: conclude

text: that

text: it

text: has

text: only

text: a

text: moderate

text: eﬀect

text: in

text: the

text: Gamow-Teller

text: part

text: of

text: the

text: NMEs

text: but

text: strongly

text: aﬀects

text: the

text: Fermi

text: contribution.

text: This

text: work

text: opens

text: up

text: the

text: door

text: for

text: NME

text: benchmarks

text: between

text: diﬀerent

text: theoretical

text: approaches,

text: and

text: constitutes

text: a

text: step

text: forward

text: towards

text: more

text: reliable

text: estimations

text: of

text: the

text: NMEs.

text: Introduction

text: model

text: (IBM)

text: However,

text: state-of-the-art

text: NME

text: pre-

text: dictions

text: by

text: these

text: approaches

text: diﬀer

text: up

text: to

text: factor

text: two

text: Experimental

text: searches

text: for

text: the

text: lepton-number

text: violating

text: strongly

text: limiting

text: the

text: precision

text: to

text: which

text: information

text: on

text: weak

text: process

text: neutrinoless

text: double

text: beta

text: decay

text: (0νββ

text: de-

text: neutrino

text: masses

text: would

text: be

text: known

text: in

text: case

text: of

text: a

text: 0νββ

text: decay

text: cay)

text: are

text: the

text: most

text: promising

text: approach

text: to

text: determine

text: some

text: measurement.

text: In

text: addition,

text: the

text: half-lives

text: of

text: experimentally

text: of

text: the

text: fundamental

text: properties

text: of

text: neutrinos.

text: The

text: detec-

text: relevant

text: nuclei

text: could

text: be

text: signiﬁcantly

text: under/overestimated

text: tion

text: of

text: 0νββ

text: decay

text: would

text: establish

text: the

text: Majorana

text: character

text: for

text: a

text: given

text: m

text: ββ

text: of

text: neutrinos

text: and

text: provide

text: information

text: about

text: their

text: absolute

text: A

text: better

text: understanding

text: of

text: the

text: NMEs

text: is

text: therefore

text: cru-

text: mass

text: and

text: hierarchy

text: Ongoing

text: experiments

text: EXO

text: cial.

text: Recently,

text: experimental

text: observables

text: relevant

text: for

text: 0νββ

text: KamLAND-Zen

text: and

text: GERDA

text: have

text: recently

text: set

text: im-

text: decay

text: have

text: been

text: measured

text: allowing

text: checks

text: for

text: pressive

text: lower-limits,

text: well

text: over

text: years,

text: on

text: the

text: half-lives

text: the

text: nuclear

text: structure

text: methods

text: that

text: in

text: some

text: cases

text: of

text: and

text: and

text: the

text: dozen

text: most

text: favorable

text: nuclei

text: resulted

text: in

text: closer

text: NMEs

text: between

text: diﬀerent

text: theoretical

text: ap-

text: are

text: being

text: explored

text: worldwide

text: However,

text: an

text: even-

text: proaches

text: tual

text: 0νββ

text: decay

text: detection

text: does

text: not

text: guarantee

text: the

text: precise

text: In

text: this

text: article

text: we

text: follow

text: a

text: complementary

text: approach,

text: determination

text: of

text: absolute

text: neutrino

text: masses,

text: because

text: the

text: studying

text: non-physical

text: 0νββ

text: decays

text: along

text: isotopic

text: chains

text: in

text: half-life

text: depends

text: on

text: the

text: transition

text: nuclear

text: matrix

text: elements

text: the

text: pf-shell,

text: with

text: SM

text: and

text: EDF

text: methods.

text: Although

text: within

text: this

text: region

text: only

text: is

text: an

text: actual

text: 0νββ

text: decay

text: candidate,

text: a

text: comparison

text: between

text: methods

text: is

text: better

text: established

text: with

text: −1

text: systematic

text: calculations.

text: For

text: instance,

text: systematic

text: EDF

text: cal-

text: mββ

text: culations

text: for

text: the

text: cadmium

text: isotopes

text: provided

text: a

text: better

text: un-

text: e

text: derstanding

text: of

text: the

text: role

text: of

text: deformation,

text: pairing

text: and

text: shell

text: eﬀects

text: in

text: 0νββ

text: decay

text: Systematic

text: studies

text: allow

text: to

text: an-

text: with

text: m

text: the

text: electron

text: mass

text: and

text: G

text: a

text: well-known

text: phase-

text: e

text: 0ν

text: alyze

text: not

text: only

text: numerical

text: values,

text: but

text: also

text: trends,

text: which

text: are

text: space

text: factor

text: The

text: combination

text: of

text: neutrino

text: masses

text: useful

text: to

text: identify

text: similarities

text: and

text: diﬀerences

text: between

text: SM

text: that

text: appears

text: in

text: 0νββ

text: decay

text: is

text: m

text: m

text: with

text: ββ

text: i

text: ei

text: i

text: and

text: EDF.

text: Understanding

text: these

text: is

text: essential

text: to

text: reduce

text: the

text: U

text: the

text: neutrino

text: mixing

text: matrix.

text: ij

text: theoretical

text: uncertainty

text: in

text: the

text: 0νββ

text: decay

text: NMEs.

text: NMEs

text: have

text: been

text: predicted

text: by

text: diﬀerent

text: theoretical

text: nu-

text: Theoretical

text: framework

text: clear

text: structure

text: approaches.

text: These

text: comprise

text: large-scale

text: shell

text: model

text: (SM)

text: energy

text: density

text: functional

text: meth-

text: Here

text: we

text: brieﬂy

text: describe

text: the

text: EDF

text: and

text: SM

text: calculations

text: ods

text: (EDF)

text: the

text: quasiparticle

text: random

text: phase

text: ap-

text: performed

text: in

text: this

text: work,

text: as

text: well

text: as

text: the

text: 0νββ

text: decay

text: transi-

text: proximation

text: (QRPA)

text: and

text: the

text: interacting

text: boson

text: tion

text: operator

text: employed.

text: A

text: more

text: extensive

text: description

text: of

text: Preprint

text: submitted

text: to

text: Elsevier

text: May

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text: ##LTLine##

text: ##LTLine##