Strongly Interacting Dark Matter at Fixed-Target Experiments€¦ · CMB ⇠ p 3 x f T CMB m DM ⇠...

Strongly Interacting Dark Matter at Fixed-Target Experiments

ASHER BERLIN

Collaboration with Nikita Blinov, Stefania Gori, Philip Schuster, & Natalia Toro

TeVPA, Ohio State University August 11, 2017

Hidden Sector

Hidden Sector

DM + …

Th

Visible Sector

SM + … + SUSY + …ε ≪ 1

T

100% Hidden

Carlson, Machacek, and Hall (1992)

χ

χ

χ

χ

χ

100% Hidden


χ

χ

χ

χ

χ KE (Need to expel heat)

100% Hidden


χ

χ

χ

χ

χ KE (Need to expel heat)

⇒ mχ ≪ keV

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)

m� ⇠ ↵�

�T 2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (2)

m⇡ ⇠ ↵�

�T 2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

pions “⇡” ⌘ ⇡0 , ⌘0 , K0 , K0 , ⇡± , K± (4)

vector mesons “V ” ⌘ ⇢0 , ! , � , K⇤0 , K⇤0 , ⇢± , K⇤± (5)

�(WD ! e+e�) ⇠ ↵em ✏2 ✓2 mWD (6)

(7)

vCMB ⇠p

3xfTCMB

mDM⇠ 10�8 ⇥ 1 GeV

mDM(8)

(9)

mA0 > mDM (10)

(11)

�v / v2 (12)

(13)

Th < m� =) sh ' ⇢�Th

' m�n�

Th/ e�m�/Th (14)

(15)

Sh / e�m�/Th a3 ⇠ constant =) Th ⇠ m�

log a� m�

a(16)

(17)

⇢h ' sh Th ⇠ 1

a3 log a� 1

a3/2 ea(18)

(19)

2Nc

15⇡2 f5⇡

✏µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)

(21)

SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N2f � 1 pions, ⇡a T a (22)

(23)

�(3 ! 2) = n2⇡ h�v2i , h�v2i ⇠

⇣m⇡

f⇡

⌘10 1

m5⇡

(24)

(25)✏

2 cos ✓WA0

µ⌫ Bµ⌫ (26)

(27)

iM ⇠ ✏2 Tr⇥Q2 T a

⇤(28)

(29)

SU(N1)⇥ SU(N2)⇥ U(1)D ⇢ SU(Nf )L+R (30)

(31)m⇡

f⇡⇠ 2⇡ xf

N1/2c

⇥xf/2 + log (

pmpl Teq /m⇡)

⇤ (32)

(33)

eD f⇡mV

A0µ⌫ Tr(QV µ⌫) (34)

(35)

�⇡ > Hf =) sink for entire DM abundance (36)

(37)

�⇡ < Hf =) potential issues with BBN + CMB (38)

(39)

iM ⇠ ✏2 Tr(QT a T b)⇥ Tr(QT b) (40)

Location Timeline Ebeam (GeV) POT Baseline (m)

SeaQuest Fermilab April, 2017 120 1.44⇥ 1018 ! 1020 ? 5� 25SHiP CERN 2026 ? 400 2⇥ 1020 60� 110

Q =(41)

1

90% Hidden


χ

SM

χ

SM

Th = T

(heat dumped into SM)

χ

χ

χ

χ

χ

{ L � g� h �� }8

<

:

L � g� H ��+ g� HX

f

yf f̄f

9

=

;

8

<

:

L � g� A � i�5 �+ g� AX

f

yf f̄ i�5f

9

=

;

�

L � �c� �2 |H|2

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

M�̃ =

0

B

B

B

B

B

B

B

B

@

0 µ �g2 vu /p2 g1 vu /

p2

µ 0 g2 vd /p2 �g1 vd /

p2

�g2 vu /p2 g2 vd /

p2 M2 0

g1 vu /p2 �g1 vd /

p2 0 M1

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

n

H̃u , H̃d , W̃ , B̃o

(1)

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (2)

m� ⇠ ↵�

�

T 2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

(4)

1

The SIMP Miracle

Hochberg, Kuflik, Volansky, Wacker arXiv:1402.5143

π

π

π

π

π

{ L � g� h �� }8

<

:

L � g� H ��+ g� HX

f

yf f̄f

9

=

;

8

<

:

L � g� A � i�5 �+ g� AX

f

yf f̄ i�5f

9

=

;

�

L � �c� �2 |H|2

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

M�̃ =

0

B

B

B

B

B

B

B

B

@

0 µ �g2 vu /p2 g1 vu /

p2

µ 0 g2 vd /p2 �g1 vd /

p2

�g2 vu /p2 g2 vd /

p2 M2 0

g1 vu /p2 �g1 vd /

p2 0 M1

1

C

C

C

C

C

C

C

C

A

9

>

>

>

>

>

>

>

>

=

>

>

>

>

>

>

>

>

;

n

H̃u , H̃d , W̃ , B̃o

(1)

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (2)

m� ⇠ ↵�

�

T 2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

m⇡ ⇠ ↵�

�

T 2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (4)

(5)

1

The SIMP Miracle

Hochberg, Kuflik, Volansky, Wacker arXiv:1402.5143

A Theory of Pions

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)

m� ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (2)

m⇡ ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

pions “⇡” ⌘ ⇡

0, ⌘

0, K

0, K

0, ⇡

±, K

± (4)

vector mesons “V ” ⌘ ⇢

0, ! , � , K

⇤0, K

⇤0, ⇢

±, K

⇤± (5)

�(WD ! e

+e

�) ⇠ ↵em ✏

2✓

2mWD (6)

(7)

vCMB ⇠p

3xfTCMB

mDM⇠ 10�8 ⇥ 1 GeV

mDM(8)

(9)

mA0> mDM (10)

(11)

�v / v

2 (12)

(13)

Th < m� =) sh ' ⇢�

Th' m�n�

Th/ e

�m�/Th (14)

(15)

Sh / e

�m�/Tha

3 ⇠ constant =) Th ⇠ m�

log a(16)

(17)

⇢h ' sh Th ⇠ 1

a

3 log a� 1

a

3/2e

a(18)

(19)

2Nc

15⇡2f

5⇡

✏

µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)

(21)

SU(Nc) confines at ⇤ =) SU(Nf )L ⇥ SU(Nf )R ! SU(Nf )L+R =) N

2f � 1 pions, ⇡a

T

a (22)



1

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)

m� ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (2)

m⇡ ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

pions “⇡” ⌘ ⇡

0, ⌘

0, K

0, K

0, ⇡

±, K

± (4)


0, ! , � , K

⇤0, K

⇤0, ⇢

±, K

⇤± (5)

�(WD ! e

+e

�) ⇠ ↵em ✏

2✓

2mWD (6)

(7)

vCMB ⇠p

3xfTCMB

mDM⇠ 10�8 ⇥ 1 GeV

mDM(8)

(9)

mA0> mDM (10)

(11)

�v / v

2 (12)

(13)

Th < m� =) sh ' ⇢�

Th' m�n�

Th/ e

�m�/Th (14)

(15)

Sh / e

�m�/Tha


log a(16)

(17)

⇢h ' sh Th ⇠ 1

a

3 log a� 1

a

3/2e

a(18)

(19)

2Nc

15⇡2f

5⇡

✏

µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)

(21)



T

a (22)



1

(Wess-Zumino-Witten)

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)

m� ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (2)

m⇡ ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

pions “⇡” ⌘ ⇡

0, ⌘

0, K

0, K

0, ⇡

±, K

± (4)


0, ! , � , K

⇤0, K

⇤0, ⇢

±, K

⇤± (5)

�(WD ! e

+e

�) ⇠ ↵em ✏

2✓

2mWD (6)

(7)

vCMB ⇠p

3xfTCMB

mDM⇠ 10�8 ⇥ 1 GeV

mDM(8)

(9)

mA0> mDM (10)

(11)

�v / v

2 (12)

(13)

Th < m� =) sh ' ⇢�

Th' m�n�

Th/ e

�m�/Th (14)

(15)

Sh / e

�m�/Tha


log a(16)

(17)

⇢h ' sh Th ⇠ 1

a

3 log a� 1

a

3/2e

a(18)

(19)

2Nc

15⇡2f

5⇡

✏

µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)

(21)



T

a (22)

(23)

�(3 ! 2) = n

2⇡ h�v2i , h�v2i ⇠

⇣m⇡

f⇡

⌘10 1

m

5⇡

(24)

(25)✏

2 cos ✓WA

0µ⌫ B

µ⌫ (26)

(27)

iM ⇠ ✏

4 Tr⇥Q

2T

a⇤

(28)

(29)

SU(N1)⇥ SU(N2)⇥ U(1)D ⇢ SU(Nf )L+R (30)



Q =(31)

+1

. . .

+1

�1

. . .

�1

0

BBBBBBBBBBB@

1

CCCCCCCCCCCA

N

1

N

2

1

π

SM

π

SM

A’

ε

Th = T

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)

m� ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (2)

m⇡ ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

pions “⇡” ⌘ ⇡

0, ⌘

0, K

0, K

0, ⇡

±, K

± (4)


0, ! , � , K

⇤0, K

⇤0, ⇢

±, K

⇤± (5)

�(WD ! e

+e

�) ⇠ ↵em ✏

2✓

2mWD (6)

(7)

vCMB ⇠p

3xfTCMB

mDM⇠ 10�8 ⇥ 1 GeV

mDM(8)

(9)

mA0> mDM (10)

(11)

�v / v

2 (12)

(13)

Th < m� =) sh ' ⇢�

Th' m�n�

Th/ e

�m�/Th (14)

(15)

Sh / e

�m�/Tha


log a(16)

(17)

⇢h ' sh Th ⇠ 1

a

3 log a� 1

a

3/2e

a(18)

(19)

2Nc

15⇡2f

5⇡

✏

µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)

(21)



T

a (22)

(23)

�(3 ! 2) = n

2⇡ h�v2i , h�v2i ⇠

⇣m⇡

f⇡

⌘10 1

m

5⇡

(24)

(25)✏

2 cos ✓WA

0µ⌫ B

µ⌫ (26)



1

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)

m� ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (2)

m⇡ ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

pions “⇡” ⌘ ⇡

0, ⌘

0, K

0, K

0, ⇡

±, K

± (4)


0, ! , � , K

⇤0, K

⇤0, ⇢

±, K

⇤± (5)

�(WD ! e

+e

�) ⇠ ↵em ✏

2✓

2mWD (6)

(7)

vCMB ⇠p

3xfTCMB

mDM⇠ 10�8 ⇥ 1 GeV

mDM(8)

(9)

mA0> mDM (10)

(11)

�v / v

2 (12)

(13)

Th < m� =) sh ' ⇢�

Th' m�n�

Th/ e

�m�/Th (14)

(15)

Sh / e

�m�/Tha


log a(16)

(17)

⇢h ' sh Th ⇠ 1

a

3 log a� 1

a

3/2e

a(18)

(19)

2Nc

15⇡2f

5⇡

✏

µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)

(21)



T

a (22)

(23)

�(3 ! 2) = n

2⇡ h�v2i , h�v2i ⇠

⇣m⇡

f⇡

⌘10T

2

m

7⇡

(24)

(25)✏

2 cos ✓WA

0µ⌫ B

µ⌫ (26)



1

(Kinetic mixing)

The SIMP Miracle

Hochberg, Kuflik, Murayama, Volansky, Wacker arXiv: 1411.3727

SU(4), Nf = 3

SU(5), Nf = 3

SU(10), Nf = 3

10-2 10-1 1 100

2

4

6

8

10

10-2

10-1

1

10

102

mπ [GeV]

mπ/f π

SU(Nf )×SU(Nf ) / SU(Nf )

σ scatter/m

π[cm2 /g

]

4

8

12

16

20

The SIMP Miracle

Hochberg, Kuflik, Murayama, Volansky, Wacker arXiv: 1411.3727

mπ/fπ ≫ 1 ⇒ vector mesons nearby in mass mV ~ 4π fπ / Nc

1/2

SU(4), Nf = 3

SU(5), Nf = 3

SU(10), Nf = 3

10-2 10-1 1 100

2

4

6

8

10

10-2

10-1

1

10

102

mπ [GeV]

mπ/f π

SU(Nf )×SU(Nf ) / SU(Nf )

σ scatter/m

π[cm2 /g

]

4

8

12

16

20

A’

Mass Spectrum

Vector Mesons, V

Pions, π

{Prevent π π → A’ A’

{mπ / fπ ≫ 1

~ GeV

~ 100 MeV

Bullet Cluster

(CMB)

A’ DecaysHidden Sector

0 2 4 6 8 10 1210-3

10-2

10-1

1

m pê fp

BRHA¢ Æ

XL

A¢ Decays, m p ê mA¢ = 1ê3mV =1.8 m pmV =1.4 m p

w p0 + V ± p°

r0 h + f h

p + p -

V +V -

π+π- invisible

everything else semi-visible

V Decays

(mV > 2 mπ )

(mV < 2 mπ )

(Model requires mV ~ mπ)

V 0

A′∗

e−

e+V ±

A′∗

e−

e+

π±

V 0, V ±

π0,π±

π0V ±

π±

π0

V 0

A′∗

e−

e+V ±

A′∗

e−

e+

π±

V 0, V ±

π0,π±

π0V ±

π±

π0

V 0

A′∗

e−

e+V ±

A′∗

e−

e+

π±

V 0, V ±

π0,π±

π0V ±

π±

π0

(longer lived)

V Decays

10�2 10�1 100

mV [GeV]

10�5

10�4

10�3

10�2

10�1

100

101

102

103

104

c⌧V

[cm

]

" = 10�3, ↵D = 10�2

V 0

V ±

A′

Vπ

mass

e−

e−

A′

Z

V 0

π0

A′∗

e−

e+

e−

e−

A′

Z

π−

π+

e−

e−

A′

Z

V ±

π∓

A′∗

e−

e+

π±

Signal Examples

missing energy

missing energy + displaced resonant leptons

missing energy + displaced leptons

Parameter Space

Berlin, Blinov, Gori, Schuster, Toro arXiv:171X.XXXX

10-2 10-1 110-810-710-610-510-410-310-210-1

m p @GeVD

e

SIMP, m p : mV : mA¢ = 1 : 1.8 : 3, m p ê fp = p, aD = aem

Wp>WDM

Belle-II

BaBar Hinv.L

BaBar Hvis.L LEP

Hg-2Le,m

LSNDêE137êMiniBooNE

LDMX Hinv.LE137 Hdec.L

ELDER

BulletCluster

Kin. Eq.

Summary

• SIMP cosmology favors mπ / fπ ≫ 1, i.e., mπ ~ mV parametrically true.

• Semi-Visible decays of A’ and V can be tested extensively in low-energy accelerators.

Back Up Slides

Invisible or Visible Decays

10-40 cm

2

10-40 cm

2

10-41 cm

2

10-41 cm

2

10-42 cm

2

10-42 cm

2

10-1 1 101 102 10310-810-710-610-510-410-310-210-11

10-1 1 101 102 10310-810-710-610-510-410-310-210-11

Hochberg, Kuflik, Murayama arXiv: 1512.07917

(upper bound on mA’ from thermalization)

(lower bound on mA’ from CMB)

mA’ mA’

LDMX Reach

10�2 10�1 100 101

mA0 [GeV]

10�7

10�6

10�5

10�4

10�3

10�2

"

HPS 2018

LDMX P2

LDMX inv.

Belle II 20 fb�1

⌦⇡

=⌦

cdm

m⇡ : mV : mA0 = 1 : 1.8 : 3

↵D = 10�2, m⇡/f⇡ = 3

CCD 100g yr

SuperC

DMS

V Decays

(m < 2 mπ )V0

(m > 2 mπ )V±

radiative corrections under

U(1)D

V 0

A′∗

e−

e+V ±

A′∗

e−

e+

π±

V 0, V ±

π0,π±

π0V ±

π±

π0

V 0

A′∗

e−

e+V ±

A′∗

e−

e+

π±

V 0, V ±

π0,π±

π0V ±

π±

π0

LDMX Reach

10�2 10�1 100

mA0 [GeV]

10�7

10�6

10�5

10�4

10�3

10�2

"

E137

OrsayBaBar inv.

LSND

LDMX inv.

10cm

2/g

1cm

2/g

⌦⇡

=⌦

cdm

HPS 2016

HPS 2018

LDMX P1

LDMX P1

↵D = 10�2, m⇡/f⇡ = 3

π

π

π

V

Forbidden Semi-Annihilation

m� ⇠ ↵� (Teqmpl)1/2 ⇠ ↵� ⇥ 10 TeV (1)

m� ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (2)

m⇡ ⇠ ↵�

�T

2eqmpl

�1/3 ⇠ ↵� ⇥ 1 GeV (3)

pions “⇡” ⌘ ⇡

0, ⌘

0, K

0, K

0, ⇡

±, K

± (4)


0, ! , � , K

⇤0, K

⇤0, ⇢

±, K

⇤± (5)

�(WD ! e

+e

�) ⇠ ↵em ✏

2✓

2mWD (6)

(7)

vCMB ⇠p

3xfTCMB

mDM⇠ 10�8 ⇥ 1 GeV

mDM(8)

(9)

mA0> mDM (10)

(11)

�v / v

2 (12)

(13)

Th < m� =) sh ' ⇢�

Th' m�n�

Th/ e

�m�/Th (14)

(15)

Sh / e

�m�/Tha


log a(16)

(17)

⇢h ' sh Th ⇠ 1

a

3 log a� 1

a

3/2e

a(18)

(19)

2Nc

15⇡2f

5⇡

✏

µ⌫⇢� Tr [⇡@µ⇡@⌫⇡@⇢⇡@�⇡] (20)

(21)



T

a (22)

(23)

�(3 ! 2) = n

2⇡ h�v2i , h�v2i ⇠

⇣m⇡

f⇡

⌘10 1

m

5⇡

(24)

(25)✏

2 cos ✓WA

0µ⌫ B

µ⌫ (26)

(27)

iM ⇠ ✏

4 Tr⇥Q

2T

a⇤

(28)

(29)

SU(N1)⇥ SU(N2)⇥ U(1)D ⇢ SU(Nf )L+R (30)

(31)m⇡

f⇡⇠ 2⇡ xf

N

1/2c

⇥xf/2 + log (

pmpl Teq /m⇡)

⇤ (32)



Q =(33)

+1

. . .

+1

�1

. . .

�1

0

BBBBBBBBBBB@

1

CCCCCCCCCCCA

N

1

N

2

1

+1

. . .

+1

�1

. . .

�1

0

BBBBBBBBBBB@

1

CCCCCCCCCCCA

N1

N2

⇡aT a ⇠

0

@N1 ⇥N1 N2 ⇥N1

N1 ⇥N2 N2 ⇥N2

1

A ⇠

0

@(1, 1, 0) + (A, 1, 0)�N1, N2,+2

�

�N1, N2,�2

�(1, 1, 0) + (1, A, 0)

1

A

h�vi ⇠ e�(mV �m⇡)/T

m2⇡

⇠ e�(f⇡�m⇡)/T

m2⇡

� ⇠ n⇡e�(f⇡�m⇡)/m⇡

m2⇡

⇠ H ⇠ m2⇡

mpl

n⇡ ⇠ m4⇡

mple(m⇡/f⇡)

�1�1

⇢eq ⇠ m4⇡

mple(m⇡/f⇡)

�1�1

✓Teq

m⇡

◆3

⇠ T 4eq

m⇡

f⇡⇠

✓1 + log

(Teq mpl)1/2

m⇡

◆�1

(42)

2

Forbidden Semi-Annihilation

10-2 10-1 11

5

10

4p

m p @GeVD

mpêf p

Semi-Annihilation

mVêm p >>1

mVêm p =

2.2

mVêm p =

2

mVêmp = 1

.8

mV êmp = 4 p

Hmp ê fpL Nc1ê2

relaxes self-interaction constraints

3→2

DecaysHidden Sector

Here, for reference, we list the relevant decays in the specific model under consideration, using the KSFRrelation when applicable:

�(A0 ! `+`�) =↵em ✏2

3

�1� 4 r2

`

�1/2 �1 + 2 r2

`

�m

A

0

�(A0 ! hadrons) = R(ps = m

A

0) �(A0 ! µ+µ�)

�(A0 ! ⇡⇡) =2↵

D

3

(1� 4r2⇡

)3/2

(1� r�2V

)2m

A

0

�(A0 ! ⌘0 ⇢) =↵D

r2V

256⇡4

✓m

⇡

/f⇡

r⇡

◆4 h1� 2(r2

⇡

+ r2V

) + (r2⇡

� r2V

)2i3/2

mA

0

�(A0 ! ⌘0 �) =↵D

r2V

128⇡4

✓m

⇡

/f⇡

r⇡

◆4 h1� 2(r2

⇡

+ r2V

) + (r2⇡

� r2V

)2i3/2

mA

0

�(A0 ! ⇡0 !) =3↵

D

r2V

256⇡4

✓m

⇡

/f⇡

r⇡

◆4 h1� 2(r2

⇡

+ r2V

) + (r2⇡

� r2V

)2i3/2

mA

0

�(A0 ! K0 K⇤0 , K0 K⇤0) =3↵

D

r2V

128⇡4

✓m

⇡

/f⇡

r⇡

◆4 h1� 2(r2

⇡

+ r2V

) + (r2⇡

� r2V

)2i3/2

mA

0

�(A0 ! ⇡± ⇢⌥) =3↵

D

r2V

128⇡4

✓m

⇡

/f⇡

r⇡

◆4 h1� 2(r2

⇡

+ r2V

) + (r2⇡

� r2V

)2i3/2

mA

0

�(A0 ! K± K⇤⌥) =3↵

D

r2V

128⇡4

✓m

⇡

/f⇡

r⇡

◆4 h1� 2(r2

⇡

+ r2V

) + (r2⇡

� r2V

)2i3/2

mA

0

�(A0 ! V V ) =↵D

6

(1� 4r2V

)1/2(1 + 16r2V

� 68r4V

� 48r6V

)

(1� r2V

)2m

A

0

�(⇢ ! `+`�) =32⇡ ↵em ↵

D

✏2

3

✓r⇡

m⇡

/f⇡

◆2

(r2V

� 4r2`

)1/2 (r2V

+ 2r2`

) (1� r2V

)�2 mA

0

�(� ! `+`�) =16⇡ ↵em ↵

D

✏2

3

✓r⇡

m⇡

/f⇡

◆2

(r2V

� 4r2`

)1/2 (r2V

+ 2r2`

) (1� r2V

)�2 mA

0

�(! ! `+`�) = 0

�(V ! hadrons) = R(ps = m

V

) �(V ! µ+µ�)

�(! ! `+`�⇡0) ⇠ �(⇢± ! `+`�⇡±) ⇠ �(K⇤± ! `+`�K±) ⇠ ✏2↵D

m5V

/m4A

0 . (131)

In the last line of the box above, we have given a parametric form for the 3-body decay of vector mesons,which is especially relevant for the vectors that are unable to mix with the photon. In the limit that m

`

' 0,

22

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