Neutrinoless Double Beta Decay: Neutrino Physicsprac.us.edu.pl/~us2011/talks/Rodejohann.pdfT1/2 = 5...

Neutrinoless Double Beta Decay: Neutrino Physics

Werner Rodejohann

(MPIK, Heidelberg)

Ustron, 15/09/11

1

Outline

(A, Z) → (A, Z + 2) + 2 e− (0νββ) ⇒ Lepton Number Violation

• Standard Interpretation: light Majorana neutrinos

– general properties

– testing the inverted ordering

– distinguishing neutrino models

– light sterile neutrinos

• Non-Standard Interpretations: something else

– heavy Majorana neutrinos

– SUSY

– RH currents

– . . .

review on 0νββ and particle physics: W.R., 1106.1334

2

Why should we probe Lepton Number Violation (LNV)?

• L and B accidentally conserved in SM

• effective theory: L = LSM + 1Λ LLNV + 1

Λ2 LLFV, BNV, LNV + . . .

• baryogenesis: B is violated

• B, L often connected in GUTs

• GUTs have seesaw and Majorana neutrinos

• chiral anomalies: ∂µJµB,L = c Gµν Gµν 6= 0 with JB

µ =∑

qi γµ qi and

JLµ =

∑ℓi γµ ℓi

⇒ Lepton Number Violation as important as Baryon Number Violation

(0νββ is much more than a neutrino mass experiment)

3

Experimental Aspects: existing limits

Isotope T 0ν1/2 [yrs] Experiment

48Ca 5.8 × 1022 CANDLES

76Ge 1.9 × 1025 HDM

1.6 × 1025 IGEX

82Se 3.2 × 1023 NEMO-3

96Zr 9.2 × 1021 NEMO-3

100Mo 1.0 × 1024 NEMO-3

116Cd 1.7 × 1023 SOLOTVINO

130Te 2.8 × 1024 CUORICINO

136Xe 5.0 × 1023 DAMA

150Nd 1.8 × 1022 NEMO-3

4

Experimental Aspects: upcoming experiments

Name Isotope source = detector; calorimetric with source 6= detector

high energy res. low energy res. event topology event topology

CANDLES 48Ca – X – –

COBRA 116Cd (and 130Te) – – X –

CUORE 130Te X – – –

DCBA 82Se or 150Nd – – – X

EXO 136Xe – – X –

GERDA 76Ge X – – –

KamLAND-Zen 136Xe – X – –

LUCIFER 82Se or 100Mo or 116Cd X – – –

MAJORANA 76Ge X – – –

MOON 82Se or 100Mo or 150Nd – – – X

NEXT 136Xe – – X –

SNO+ 150Nd – X – –

SuperNEMO 82Se or 150Nd – – – X

XMASS 136Xe – X – –

next generation: mass from 10 → 100 kg y and background from 0.1 → 0.01

counts/keV/kg/y and lifetime from 1025 → 1026 y

5

Experiment Isotope Mass of Sensitivity Status Start of

Isotope [kg] T0ν1/2

[yrs] data-taking

GERDA 76Ge 18 3 × 1025 running 2011

40 2 × 1026 in progress ∼ 2012

1000 6 × 1027 R&D ∼ 2015

CUORE 130Te 200 6.5 × 1026∗ in progress ∼ 2013

2.1 × 1026∗∗

MAJORANA 76Ge 30-60 (1 − 2) × 1026 in progress ∼ 2013

1000 6 × 1027 R&D ∼ 2015

EXO 136Xe 200 6.4 × 1025 in progress ∼ 2011

1000 8 × 1026 R&D ∼ 2015

SuperNEMO 82Se 100-200 (1 − 2) × 1026 R&D ∼ 2013-2015

KamLAND-Zen 136Xe 400 4 × 1026 in progress ∼ 2011

1000 1027 R&D ∼ 2013-2015

SNO+ 150Nd 56 4.5 × 1024 in progress ∼ 2012

500 3 × 1025 R&D ∼ 2015

6

Experimental Aspects

(T 0ν1/2)

−1 ∝

a M ε t without background

a ε

√

M t

B ∆Ewith background

with

• B is background index in counts/(keV kg yr)

• ∆E is energy resolution

• ǫ is efficiency

• (T 0ν1/2)

−1 ∝ (particle physics)2

Note: factor 2 in particle physics is combined factor of 16 in M × t × B × ∆E

7

Interpretation of Experiments

Master formula:

Γ0ν = Gx(Q, Z) |Mx(A, Z) ηx|2

• Gx(Q, Z): phase space factor

• Mx(A, Z): nuclear physics

• ηx: particle physics

8

Interpretation of Experiments

Master formula:

Γ0ν = Gx(Q, Z) |Mx(A, Z) ηx|2

• Gx(Q, Z): phase space factor; calculable

• Mx(A, Z): nuclear physics; problematic

• ηx: particle physics; interesting

9

Interpretation of Neutrino-less Double Beta Decay

• Standard Interpretation:

Neutrinoless Double Beta Decay is mediated by light and massive Majorana

neutrinos (the ones which oscillate) and all other mechanisms potentially

leading to 0νββ give negligible or no contribution

• Non-Standard Interpretations:

There is at least one other mechanism leading to Neutrinoless Double Beta

Decay and its contribution is at least of the same order as the light neutrino

exchange mechanism

10

Why do we need neutrino mass?

• in all seesaws (type I, II, III): neutrino mass inverse proportional to its origin

• GUTs: normal hierarchy. . .

• IH and QD neutrinos: special flavor symmetries required. . .

• QD: HDM, strong RG effects, leptogenesis,. . .

• mass hierarchy is moderate:

NH :m2

m3≥

√

∆m2⊙

∆m2A

>∼1

5≃

√mµ

mτ≃

√ms

mb≃

√√

mc

mt

⇒ Neutrino masses are strange and can tell us a lot

11

∆L 6= 0: Neutrinoless Double Beta Decay

W

νi

νi

W

dL

dL

uL

e−L

e−L

uL

Uei

q

UeiIm

Rem

mm

ee

ee

ee

(1)

(3)

(2)

| |

| || | e

e.

.

eem

2iβ2iα

Amplitude proportional to coherent sum (“effective mass”):

|mee| ≡∣∣∑

U2ei mi

∣∣ =

∣∣|Ue1|2 m1 + |Ue2|2 m2 e2iα + |Ue3|2 m3 e2iβ

∣∣

7 out of 9 parameters of mν !

12

• Standard Interpretation:

W

νi

νi

W

dL

dL

uL

e−L

e−L

uL

Uei

q

Uei

• Non-Standard Interpretations: e.g.

W

WR

NR

NR

νL

W

dL

dL

uL

e−R

e−L

uL

dR

χ/g

dRχ/g

dc

dc

uL

e−L

e−L

uL

W

W

∆−−

dL

dL

uL

e−L

e−L

uL

√2g2vL hee

uL

uL

χ/g

χ/g

dc

dc

e−L

uL

uL

e−L

W

W

dL

dL

uL

e−L

χ0

χ0

e−L

uL

13

Schechter-Valle theorem: no matter what process, neutrinos are Majorana:

Blackbox diagram is 4 loop:

mν ∼ 1

(16π2)4MeV5

m4W

<∼ 10−23 eV

14

The usual plot

15

Crucial points

• NH: can be zero!

• IH: cannot be zero! |mee|inhmin =

√

∆m2A c2

13 cos 2θ12

• gap between |mee|inhmin and |mee|nor

max

• QD: cannot be zero! |mee|QDmin = m0 c2

13 cos 2θ12

• QD: cannot distinguish between normal and inverted ordering

16

Plot against other observables

0.01 0.1mβ (eV)

10-3

10-2

10-1

100

<m

ee>

(e

V)

Normal

CPV(+,+)(+,-)(-,+)(-,-)

0.01 0.1

Inverted

CPV(+)(-)

10-3

10-2

10-1

100

<m

ee>

(e

V)

0.1 1

Σ mi (eV)

Normal

CPV(+,+)(+,-)(-,+)(-,-)

0.1 1

Inverted

CPV(+)(-)

17

Neutrino Mass Matrix

18

Which mass ordering with which life-time?

Σ mβ |mee|NH

√

∆m2A

√

∆m2⊙ + |Ue3|2∆m2

A

∣∣∣

√

∆m2⊙ + |Ue3|2

√

∆m2Ae2i(α−β)

∣∣∣

≃ 0.05 eV ≃ 0.01 eV ∼ 0.003 eV ⇒ T 0ν1/2

>∼ 1028−29 yrs

IH 2√

∆m2A

√

∆m2A

√

∆m2A

√

1 − sin2 2θ12 sin2 α

≃ 0.1 eV ≃ 0.05 eV ∼ 0.03 eV ⇒ T 0ν1/2

>∼ 1026−27 yrs

QD 3m0 m0 m0

√

1 − sin2 2θ12 sin2 α

>∼ 0.1 eV ⇒ T 0ν1/2

>∼ 1025−26 yrs

19

From life-time to particle physics: Nuclear Matrix Elements

20

SM vertex

Nuclear Process Nucl

Σi

νiUei

e

W

νi

e

W

Uei

Nucl

• 2 point-like Fermi vertices; “long-range” neutrino exchange; momentum

exchange q ≃ 1/r ≃ 0.1 GeV

• NME ↔ overlap of decaying nucleons. . .

• different approaches (QRPA, NSM, IBM, GCM, pHFB) imply uncertainty

• plus uncertainty due to model details

• plus convention issues

21

48Ca

76Ge

82Se

94Zr

96Zr

98Mo

100Mo

104Ru

110Pd

116Cd

124Sn

128Te

130Te

136Xe

150Nd

154Sm

0

2

4

6

8

M0ν

(R)QRPA (Tü)SMIBM-2PHFBGCM+PNAMP

0

2

4

6

8

10

48Ca 76Ge 82Se 96Zr 100Mo 110Pd 116Cd 124Sn 130Te 136Xe 150Nd

Isotope

NSMQRPA (Tue)

QRPA (Jy)IBMIBM

GCMPHFB

Pseudo-SU(3)

M′0

νFaessler, 1104.3700 Dueck, W.R., Zuber, PRD 83

22

0νββ and Neutrino physics

Isotope T 0ν1/2 [yrs] Experiment |mee|limmin [eV] |mee|limmax [eV]

48Ca 5.8 × 1022 CANDLES 3.55 9.91

76Ge 1.9 × 1025 HDM 0.21 0.53

1.6 × 1025 IGEX 0.25 0.63

82Se 3.2 × 1023 NEMO-3 0.85 2.08

96Zr 9.2 × 1021 NEMO-3 3.97 14.39

100Mo 1.0 × 1024 NEMO-3 0.31 0.79

116Cd 1.7 × 1023 SOLOTVINO 1.22 2.30

130Te 2.8 × 1024 CUORICINO 0.27 0.57

136Xe 5.0 × 1023 DAMA 0.83 2.04

150Nd 1.8 × 1022 NEMO-3 2.35 5.08

23

Best limit on |mee| from best life-time?

0.01

0.1


Isotope

T1/2 = 5 x 1025 y

<m>IHmax

<m>IHmin, sin2 θ12 = 0.27


IH rangeNSMTue

JyIBMIBM

GCMPHFB

Pseudo-SU(3)

〈mν〉[

eV]

0.01

0.1


Isotope

T1/2 = 1026 y

<m>IHmax



IH rangeNSMTue

JyIBMIBM

GCMPHFB

Pseudo-SU(3)〈m

ν〉[

eV]

24

Inverted Ordering

Nature provides 2 scales:

〈mν〉IHmax ≃ c213

√

∆m2A and 〈mν〉IHmin ≃ c2

13

√

∆m2A cos 2θ12

if we know that ordering is inverted: ruling out Majorana nature

if limit below 〈mν〉IHmin: inverted ordering ruled out (if Majorana is assumed)

25

Ruling out Inverted Hierarchy

|mee|IHmin =(1 − |Ue3|2

)√

|∆m2A|

(1 − 2 sin2 θ12

)=

(0.015 . . . 0.020) eV 1σ

(0.010 . . . 0.024) eV 3σ

• small |Ue3|

• large |∆m2A|

• small sin2 θ12

Current 3σ range of sin2 θ12 gives factor of 2 uncertainty for |mee|IHmin

⇒ combined factor of 16 in M × t × B × ∆E

⇒ need precision determination of θ12

Dueck, W.R., Zuber, PRD 83

26

Ruling out Inverted Hierarchym3 = 0.001 eV

IH, 3σIH, BF

0.01

0.1

0.28 0.3 0.32 0.34 0.36 0.38

0.125

0.25

0.5

1

2

4

8

1

2

4

8

16

32

0.25

0.5

1

2

4

8

16

〈mν〉[

eV]

sin2 θ12

T0ν

1/2

[102

7y]


27


0.1

1

10


T1

/2 [

10

27 y

]

Isotope

0.27 < sin2(θ12) < 0.38

spread due to NMEs and due to θ12!!

IH will begin to be tested within this decade

28


0.1

1

10


T1/2

[10

27 y

]

Isotope

sin2(θ12) = 0.27NSMTue

JyIBM

GCMPHFB

Pseudo-SU(3)

0.1

1

10


T1/2

[10

27 y

]

Isotope

sin2(θ12) = 0.38

NSMTue

JyIBM

GCMPHFB

Pseudo-SU(3)

sin2 θ12 = 0.27 sin2 θ12 = 0.38

spread due to NMEs and due to θ12!!

IH will begin to be tested within this decade

29

Testing Inverted Hierarchy

lifetime to enter the IH regime

0.01

0.1

1


T1

/2 [

10

27 y

]

Isotope

mihmaxNSMTue

JyIBM

GCMPHFB

Pseudo-SU(3)

30

Testing neutrino models

UTBM =

√23

√13 0

−√

16

√13 −

√12

−√

16

√13

√12

with mass matrix

(mν)TBM = U∗TBM mdiag

ν U †TBM =

A B B

· 12 (A + B + D) 1

2 (A + B − D)

· · 12 (A + B + D)

A =1

3

(2 m1 + m2 e−2iα

), B =

1

3

(m2 e−2iα − m1

), D = m3 e−2iβ

⇒ Flavor symmetries. . .

Recent reviews: Altarelli, Feruglio 1002.0211; Ishimori; Kobayashi; Ohki;

Okada; Shimizu; Tanimoto, 1003.3552

31

The Zoo

Barry, W.R., PRD 81, updated regularly on

http://www.mpi-hd.mpg.de/personalhomes/jamesb/Table_A4.pdf

32

How to distinguish?

• LFV

• low scale scalars: Higgs, LFV

• compatible with GUTs?

• leptogenesis possible?

• neutrino mass observables

33

correlation with observables

12

PRD78:093007 (2008)

PRL 99 (2007) 151802

PRD72, 091301 (2005)

A4

0-nu DBD & FLAVOR0-nu DBD & FLAVOR

1/

Slide by J. Valle

34

Sum-rules in Models and 0νββ

constrains masses and Majorana phases

Barry, W.R., NPB 842

35

10-3

10-2

10-1

<m

ee>

(eV

)

0.01 0.1

mβ (eV)

10-3

10-2

10-1

0.01 0.1

3σ 30% error3σ exactTBM exact

2m2 +

m3 =

m1

m1 +

m2 =

m3

InvertedNormal

10-3

10-2

10-1

<m

ee>

(eV

)

0.01 0.1

mβ (eV)

10-3

10-2

10-1

0.01 0.1

3σ 30% error3σ exactTBM exact

Normal Inverted2/m

2 + 1/m

3 = 1/m

11/m

1 + 1/m

2 = 1/m

3

m1 + m2 − m3 = ǫ mmax

stable: new solutions not before ǫ ≃ 0.2

36

Sterile Neutrinos??

• LSND/MiniBooNE

• cosmology

• BBN

• reactor anomaly (Mention et al., PRD 83)

∆m241[eV

2] |Ue4| |Uµ4| ∆m251[eV

2] |Ue5| |Uµ5|3+2/2+3 0.47 0.128 0.165 0.87 0.138 0.148

1+3+1 0.47 0.129 0.154 0.87 0.142 0.163

or ∆m241 = 1.78 eV2 and |Ue4|2 = 0.151

Kopp, Maltoni, Schwetz, 1103.4570

37

Mass Orderings

ν4

m2

ν1,2,3

∆m241

m2

ν4

ν1,2,3

∆m241

ν4

ν5

∆m241

∆m251

m2

ν1,2,3

ν4

m2

ν5

ν1,2,3

∆m241

∆m251

νk

∆m2

k1

m2

νj

ν1,2,3

∆m2j1

3 active neutrinos can be normally or inversely ordered

1+3+1 scenarios first noticed in Goswami, W.R., JHEP 0710

38

Which one is sterile?

39

Sterile Neutrinos and 0νββ

• recall |mee|actNH can vanish and ||mee|actIH | ∼ 0.02 eV cannot vanish

• |mee| = | |Ue1|2m1 + |Ue2|2m2 e2iα + |U2e3|m3 e2iβ

︸︷︷︸

|mee|act+ |Ue4|2m4 e2iΦ1

︸︷︷︸

|mee|st|

• ∆m2st ≃ 1 eV2 and |Ue4| ≃ 0.15

• sterile contribution to 0νββ:

||mee|st| ≃√

∆m2st |Ue4|2 ≃ 0.02 eV

≫ |mee|NH

≃ |mee|IH

• ⇒ |mee|NH cannot vanish and |mee|IH can vanish!

Barry, W.R., Zhang, JHEP 1107

40

0.001 0.01 0.1

mlight

(eV)

10-3

10-2

10-1

100

<m

ee>

(eV

)

1+3, Normal, SN 1+3, Inverted, SI

3 ν (best-fit)3 ν (2σ)1+3 ν (best-fit)1+3 ν (2σ)

0.001 0.01 0.1


0.001 0.01 0.1

mlight

(eV)

10-3

10-2

10-1

100

<m

ee>

(eV

)

2+3, Normal, SSN 2+3, Inverted, SSI


0.001 0.01 0.1


41

An exotic modifications of the neutrino picture

one neutrino could be (Pseudo-)Dirac, the other Majorana!

“Schizophrenic neutrinos”

For instance: inverted hierarchy, m2 Dirac:

|mee| = c212 c2

13

√

∆m2A

factor 2 larger than lower limit

Mohapatra et al., PLB 695

42

10-3

10-2

10-1

100

10-3

10-2

10-1

100

<m

ee>

(eV

)

10-3

10-2

10-1

100

0.1 1

Σ mν (eV)0.1 1

Normal Inverted

ν1

ν2

ν3

10-3

10-2

10-1

100

10-3

10-2

10-1

100

<m

ee>

(eV

)10

-3

10-2

10-1

100

0.1 1

Σ mν (eV)0.1 1

Normal Inverted

ν2 & ν

3

ν1 & ν

3

ν1 & ν

2

Barry, Mohapatra, W.R., PRD 83

43

Summary

• Standard Interpretation

– probes 1/Λseesaw

– neutrino parameters: m0, phase, mass ordering

– inverted hierarchy and θ12 !

– testing models!

– sterile neutrinos mix things up!

Note: 0νββ is a broad particle physics (not only neutrino!) experiment

44

W.R., 1106.1334

45

Summary

46

Neutrinoless Double Beta Decay Goldhaber-Grodzins-Sunyar Celebration

Sensitivity and backgrounds

T"0" = ln(2)N$t/UL(%)1-tonne 76Ge Example

slide by J.F. Wilkerson

47

M0ν =( gA

1.25

)2(

M0νGT − g2

V

g2A

M0νF

)

with

M0νGT = 〈f |∑

lk

σl σk τ−l τ−

k HGT(rlk, Ea)|i〉

M0νF = 〈f |∑

lk

τ−l τ−

k HF(rlk, Ea)|i〉

• rlk ≃ 1/p ≃ 1/(0.1 GeV) distance between the two decaying neutrons

• Ea average energy

• sum over all multipolarities!

• ’neutrino potential’ HGT,F(rlk, Ea) integrates over the virtual neutrino

momenta

• (if heavy particles exchanged: nucleon structure: gA = gA(0)/(1 − q2/M2A)2)

48

The 2νββ matrix elements can be written as

M2νGT =

∑

n

〈f |P

aσa τ−

a |n〉〈n|P

b

σb τ−b |i〉

En−(Mi−Mf )/2

M2νF =

∑

n

〈f |P

aτ−

a |n〉〈n|P

b

τ−b |i〉

En−(Mi−Mf )/2

• no direct connection to 0νββ. . .

• sum over 1+ states (low momentum transfer)

• adjust some parameters to reproduce 2νββ-rates

• 2νββ observed in 48Ca, 76Ge, 82Se, 96Zr, 100Mo (plus exc. state), 116Cd,128Te, 130Te, 150Nd (plus exc. state) with half-lives from 1018 to 1024 yrs

• can partly be tested with forward angle (= low momentum transfer) (p, n)

charge exchange reactions

49

A typical spectrum will look like. . .

⇒ first reason for multi-isotope determination

50

Testing NMEs

1

T 0ν1/2(A, Z)

= G(Q, Z) |M(A, Z) η|2

if you measured two isotopes:

T 0ν1/2(A1, Z1)

T 0ν1/2(A2, Z2)

=G(Q2, Z2) |M(A2, Z2)|2G(Q1, Z1) |M(A1, Z1)|2

systematic errors drop out, ratio sensitive to NME model

⇒ second reason for multi-isotope determination

51

plus uncertainty due to model details:

• Short range correlations:

– Jastrow function

– Unitary Correlation Operator method

– Coupled Cluster method

• model parameters

– correlated: e.g., gA, gpp, SRC

– uncorrelated: e.g., model space of single particle states

– some include errors, some don’t. . .

52

Convention issues

Through convention:

G0ν ∝ g4A

R2A

with RA = r0A1/3

• careful when comparing NMEs with different gA

• r0 = 1.1 fm or r0 = 1.2 fm

• different G0ν values are out there

Smolnikov, Grabmayr, PRC 81; Cowell, PRC 73;


53

Faessler, Fogli et al., PRD 79

ellipse major axis: SRC (blue, red) and gA

ellipse minor axis: gpp

54

Faessler, Fogli et al., PRD 79

ellipse major axis: SRC (blue, red) and gA

ellipse minor axis: gpp

56

Experimental Aspects

Number of events (measuring time t ≪ T 0ν1/2 life-time):

N = ln 2 a M t NA (T 0ν1/2)

−1

suppose there is no background:

• if you want 1026 yrs you need 1026 atoms

• 1026 atoms are 103 mols

• 103 mols are 100 kg

From now on you can only loose: efficiency, background, natural abundance,. . .

57

Need to forbid single β decay:

•• ⇒ even/even → even/even

• either direct (0νββ) or two simultaneous decays with virtual (energetically

forbidden) intermediate state (2νββ)

58

• 35 candidate isotopes

• 9 are interesting: 48Ca, 76Ge, 82Se, 96Zr, 100Mo, 116Cd, 130Te, 136Xe, 150Nd

• Q-value vs. natural abundance vs. reasonably priced enrichment

vs. association with a well controlled experimental technique vs.. . .

⇒ no superisotope

0

5

10

15

20

25

30

35


Natu

ral abundance [%

]

Isotope

Natural abundance of different 0νββ candidate Isotopes

0

2

4

6

8

10

12

14

16

18

20


G0

ν [10

-14 y

rs-1

]

Isotope

G0ν for 0νββ-decay of different Isotopes

T 0ν1/2 ∝ 1/a T 0ν

1/2 ∝ Q−5

59

typical model for NME: set of single particle states with a number of possible

wave function configurations; solve H in a mean background field

Approaches:

• Quasi-particle Random Phase Approximation (QRPA) (many single particle states, few

configurations)

• Nuclear Shell Model (NSM) (many configurations, few single particle states)

• Interacting Boson Model (IBM)

• Generating Coordinate Method (GCM)

• projected Hartree-Fock-Bogoliubov model (pHFB)

gives uncertainty due to different approach. . .

. . . plus uncertainty due to model details . . .

. . .plus convention issues

60

Statistical Analysis

consider scenario with “true values”:

Scenario m3 [eV] |mee| [eV] mβ [eV] Σ [eV]

QD 0.3 0.11 − 0.30 0.30 0.91

Combine measurements of 2, or all 3, mass approaches

Maneschg, Merle, W.R., EPL 85

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QD with |mee|exp = 0.20 eV

• if ζ = 0: σ(m3) ≃ 15% at 3σ

• if ζ = 0.25: σ(m3) ≃ 25%

• if Σ wrong: reconstruction of m3 wrong by one order of magnitude

• leave Σ out: σ(m3) ≃ 50%

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Neutrinoless Double Beta Decay: Neutrino Physicsprac.us.edu.pl/~us2011/talks/Rodejohann.pdfT1/2 = 5...

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Transcript of Neutrinoless Double Beta Decay: Neutrino Physicsprac.us.edu.pl/~us2011/talks/Rodejohann.pdfT1/2 = 5...