Bounds on Amplitudes and EFTs Brando Bellazzini IPhT - CEA/Saclay - - PowerPoint PPT Presentation
Bounds on Amplitudes and EFTs Brando Bellazzini IPhT - CEA/Saclay based on 1605.06111 and work in progress Eltville, Burg Crass, EFTs for collider physics, Flavor phenomena and EWSB, Sept 15th 2016 Hierarchy of scales UV a blessing
Brando Bellazzini
IPhT - CEA/Saclay
Eltville, Burg Crass, ‘EFTs for collider physics, Flavor phenomena and EWSB’, Sept 15th 2016
based on 1605.06111 and work in progress
a blessing…
gρfπ
ΛUV ΛIR
a blessing…
LIR = L∆≤4 + X
O
O(x) Λ∆−4
UV
gρfπ
ΛUV ΛIR
E/ΛUV
a blessing…
LIR = L∆≤4 + X
O
O(x) Λ∆−4
UV
gρfπ
ΛUV ΛIR
E/ΛUV
a blessing…
LIR = L∆≤4 + X
O
O(x) Λ∆−4
UV
gρfπ
ΛUV ΛIR
E/ΛUV
large couplings from a strong sector may help
L = g2
∗
m2
∗
(∂H2)2
e.g. in CHM:
UV IR
EFT encodes UV-info via Finite set of C’s is needed at any order in Power counting = understanding = symmetries E/ΛUV
LIR = X
i
ci Oi(x) Λ∆−4
UV
ci
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
¯ ψi∂ψ − m∗ ¯ ψψ + . . .
(1)
Example
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
¯ ψi∂ψ − m∗ ¯ ψψ + . . .
(1)
Example
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
¯ i@ − ✏ · m∗ ¯ + . . .
χ–sym
¯ ψi∂ψ − m∗ ¯ ψψ + . . .
(1)
Example
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
¯ i@ − ✏ · m∗ ¯ + . . .
χ–sym
1-to-1 amplitude dominated by a less-relevant operator
M(1 → 1) ∼ 1 E
✏ · m∗ < E < m∗
(2)
¯ ψi∂ψ − gAµ ¯ ψγµψ + . . .
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
Example
(2)
¯ ψi∂ψ − gAµ ¯ ψγµψ + . . .
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
g ⌧ 1
¯ i@ − ✏ · g∗Aµ ¯ µ + g2
∗
m2
∗
( ¯ µ )2 + . . .
Example
(2)
¯ ψi∂ψ − gAµ ¯ ψγµψ + . . .
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
g ⌧ 1
¯ i@ − ✏ · g∗Aµ ¯ µ + g2
∗
m2
∗
( ¯ µ )2 + . . .
dominated by dim-6 at intermediate energy
✏ · m∗ < E < m∗
M(2 → 2) = g2
SM
E2 ✓ 1 + 1 ✏2 E2 m2
∗
◆
1/E2 gSM
gSM
1/m2
∗
g∗ g∗
Amplitude runs fast within the validity of EFT
Example
(∂π)2 − m2
∗π2 + g2 ∗π4 + . . .
(3)
Examples
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
(∂π)2 − m2
∗π2 + g2 ∗π4 + . . .
(3)
Examples
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
π → π + c
(@⇡)2 − ✏2(m2
∗⇡2 + ✏2g2 ∗⇡4) + g2 ∗
m4
∗
(@⇡)4 + . . .
(∂π)2 − m2
∗π2 + g2 ∗π4 + . . .
(3)
Examples
just suppress relevant, marginal and less-irrelevant operators by symmetries Higher dim-operators may dominate the amplitude within EFT
π → π + c
(@⇡)2 − ✏2(m2
∗⇡2 + ✏2g2 ∗⇡4) + g2 ∗
m4
∗
(@⇡)4 + . . .
Amplitude runs fast within the validity of EFT:
✏ · m∗ < E < m∗ M(2 → 2) = g2
∗✏4
✓ 1 + E4 m4
∗✏4
◆
M4
=
g∗ E
m∗
gSM
E2
E4
M4
=
g∗ E
m∗
gSM
E2
E4
LEP LHC
M4
=
g∗ E
m∗
gSM
E2
E4
LEP LHC
g2
∗
m4
∗
W 4
µν
∼ g2
∗
E4 m4
∗
VT VT
‘remedios’: strongly int. transv. vectors
1603.03064 Liu, Pomarol, Rattazzi, Riva
M4
=
g∗ E
m∗
gSM
E2
E4
LHC
g2
∗
m4
∗
W 4
µν
∼ g2
∗
E4 m4
∗
VT VT
‘remedios’: strongly int. transv. vectors
1603.03064 Liu, Pomarol, Rattazzi, Riva Freeze-out
M4
=
g∗ E
m∗
gSM
E2
E4
LHC
g2
∗
m4
∗
W 4
µν
∼ g2
∗
E4 m4
∗
VT VT
‘remedios’: strongly int. transv. vectors
1603.03064 Liu, Pomarol, Rattazzi, Riva
*
1607.02474 Brugisser, Riva, Urbano
DM as light pseudo-goldstino
Freeze-out
M4
=
g∗ E
m∗
gSM
E2
E4
g2
∗
m4
∗
W 4
µν
∼ g2
∗
E4 m4
∗
VT VT
‘remedios’: strongly int. transv. vectors
1603.03064 Liu, Pomarol, Rattazzi, Riva
*
1607.02474 Brugisser, Riva, Urbano
DM as light pseudo-goldstino
LHC FCC
(∂π)2π2
(∂σ)4
( ¯ ψγµψ)2
¯ ψ2⇤ψ2
…
F 4
µν
…
∼ E2
∼ E4
Goldstones 4-Fermions dilaton Goldstino remedios
can amplitudes be softer than E4
(within an EFT) ?
For spin-0 particles the answer is: No!
(well known) e.g. hep-th/0602178
Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi
For spin-0 particles the answer is: No!
Analyticity, Crossing, and Unitarity
M4 1 2 3 4 (well known) e.g. hep-th/0602178
Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi
For spin-0 particles the answer is: No!
Re s Im s
sIR=(m1+m2)2 uIR=(m1-m2)2
C
˜
C
s-plane
Analyticity, Crossing, and Unitarity
M4 1 2 3 4 (well known) e.g. hep-th/0602178
Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi
For spin-0 particles the answer is: No!
Re s Im s
sIR=(m1+m2)2 uIR=(m1-m2)2
C
˜
C
s-plane
small circle=big circle
Analyticity, Crossing, and Unitarity
M4 1 2 3 4 (well known) e.g. hep-th/0602178
Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi
For spin-0 particles the answer is: No!
Re s Im s
sIR=(m1+m2)2 uIR=(m1-m2)2
C
˜
C
s-plane
s ↔ u
Discs = Discu
small circle=big circle
Analyticity, Crossing, and Unitarity
M4 1 2 3 4 (well known) e.g. hep-th/0602178
Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi
For spin-0 particles the answer is: No!
Re s Im s
sIR=(m1+m2)2 uIR=(m1-m2)2
C
˜
C
s-plane
s ↔ u
Discs = Discu Disc ∼ sσT ot(s) > 0
t=0
small circle=big circle
Analyticity, Crossing, and Unitarity
M4 1 2 3 4 (well known) e.g. hep-th/0602178
Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi
For spin-0 particles the answer is: No!
Re s Im s
sIR=(m1+m2)2 uIR=(m1-m2)2
C
˜
C
s-plane
s ↔ u
Discs = Discu Disc ∼ sσT ot(s) > 0
t=0
small circle=big circle
Analyticity, Crossing, and Unitarity
M4 1 2 3 4
IR-side UV-side
M00(2 → 2)
Z 1 ds s3 σ12!anything(s) > 0
(well known) e.g. hep-th/0602178
Adams, Arkani-Hamed, Dubovsky, Nicolis, Rattazzi
E4
L = 1 2(∂µπ)2 + c Λ4 (∂µπ)4 + . . .
π → π + const
L = 1 2(∂µπ)2 + c Λ4 (∂µπ)4 + . . .
π → π + const
dispersion relation:
ππ → ππ
IR-side UV-side
M00
ππ!ππ(s = 0) = 4
π Z 1 ds s3 σππ!anything(s) > 0
L = 1 2(∂µπ)2 + c Λ4 (∂µπ)4 + . . .
π → π + const
calculable within the EFT
c > 0
dispersion relation:
ππ → ππ
IR-side UV-side
M00
ππ!ππ(s = 0) = 4
π Z 1 ds s3 σππ!anything(s) > 0
L = 1 2(∂µπ)2 + c Λ4 (∂µπ)4 + . . .
π → π + const
calculable within the EFT
c > 0
dispersion relation:
ππ → ππ
IR-side UV-side
M00
ππ!ππ(s = 0) = 4
π Z 1 ds s3 σππ!anything(s) > 0
This interacting theory can’t be softer than E4
(1) Amplitudes not Lorentz inv. but Little-group covariant
(1) Amplitudes not Lorentz inv. but Little-group covariant
s, t, u
εσ
µ
+polarizations
uσ
α vσ
α
…
1
(not amplitudes squared)
(1) Amplitudes not Lorentz inv. but Little-group covariant
s, t, u
εσ
µ
+polarizations
uσ
α vσ
α
…
1
(
)−1
αα1
uσ
α
x
hΦα1(p1)Φβ1(p2) . . .i α1
M4
1 2 3 4
σ1
σ2
σ3 σ4
=
(not amplitudes squared)
(1) Amplitudes not Lorentz inv. but Little-group covariant
s, t, u
εσ
µ
+polarizations
uσ
α vσ
α
…
1
(
)−1
αα1
uσ
α
x
hΦα1(p1)Φβ1(p2) . . .i α1
M4
1 2 3 4
σ1
σ2
σ3 σ4
=
(not amplitudes squared)
(2) polarizations carry non-analyticities
(1) Amplitudes not Lorentz inv. but Little-group covariant
s, t, u
εσ
µ
+polarizations
uσ
α vσ
α
…
1
(
)−1
αα1
uσ
α
x
hΦα1(p1)Φβ1(p2) . . .i α1
M4
1 2 3 4
σ1
σ2
σ3 σ4
=
(not amplitudes squared)
(2) polarizations carry non-analyticities
…
u(p) ∼ p pµσµ
✏L
µ(p) ∼ (pz, 0, 0,
p p2
z + m2)T
(3) crossing is not just s u t
(3) crossing is not just s u t ψ(1)X(2) → ψ(3)X(4)
(3) crossing is not just s u t
A
ψ(1)X(2) → ψ(3)X(4)
(3) crossing is not just s u t
p1 uσ1(p1)
A
ψ(1)X(2) → ψ(3)X(4)
(3) crossing is not just s u t
p1 uσ1(p1)
uσ3†(p3)
p3
A
ψ(1)X(2) → ψ(3)X(4)
(3) crossing is not just s u t
p1 uσ1(p1)
uσ3†(p3)
p3
A
ψ(1)X(2) → ψ(3)X(4)
ψ(1)X(2) → ψ(3)X(4)
(3) crossing is not just s u t
p1 uσ1(p1)
uσ3†(p3)
p3
A A
ψ(1)X(2) → ψ(3)X(4)
ψ(1)X(2) → ψ(3)X(4)
(−1)2S
fermions flip sign
(3) crossing is not just s u t
p1 uσ1(p1)
uσ3†(p3)
p3
A
−p3
v−σ3(p3)
A
ψ(1)X(2) → ψ(3)X(4)
ψ(1)X(2) → ψ(3)X(4)
(−1)2S
fermions flip sign
(3) crossing is not just s u t
p1 uσ1(p1)
uσ3†(p3)
p3
A
−p3
v−σ3(p3)
v−σ1†(p1)
−p1
A
ψ(1)X(2) → ψ(3)X(4)
ψ(1)X(2) → ψ(3)X(4)
(−1)2S
fermions flip sign
(3) crossing is not just s u t
p1 uσ1(p1)
uσ3†(p3)
p3
A
−p3
v−σ3(p3)
v−σ1†(p1)
−p1
A
ψ(1)X(2) → ψ(3)X(4)
ψ(1)X(2) → ψ(3)X(4)
(−1)2S
fermions flip sign
p1 ↔ −p3 uσ(p1) ↔ v−σ(p3)
Forward elastic scattering is special! All previous issues cancel against each other out
(1) Lorentz Invariance
(1) Lorentz Invariance
|p , σi ! eiσθ(W,p)|Λp , σi
forward elastic amp. is invariant
M = M(s)
(2) polarizations are all alike
(1) Lorentz Invariance
|p , σi ! eiσθ(W,p)|Λp , σi
forward elastic amp. is invariant
M = M(s)
(2) polarizations are all alike
(1) Lorentz Invariance
|p , σi ! eiσθ(W,p)|Λp , σi
forward elastic amp. is invariant
M = M(s)
UCPT|particle, σi ⇠ |antiparticle, σi
U †
CPTΨα(0)UCPT ∼ Ψ† α(0)
(2) polarizations are all alike
(1) Lorentz Invariance
|p , σi ! eiσθ(W,p)|Λp , σi
forward elastic amp. is invariant
M = M(s)
h0|Ψα(0)|p , σi = uσ
α(p)
hp , σ|Ψα(0)|0i = v−σ
α (p)
UCPT|particle, σi ⇠ |antiparticle, σi
U †
CPTΨα(0)UCPT ∼ Ψ† α(0)
(2) polarizations are all alike
(1) Lorentz Invariance
|p , σi ! eiσθ(W,p)|Λp , σi
forward elastic amp. is invariant
M = M(s)
h0|Ψα(0)|p , σi = uσ
α(p)
hp , σ|Ψα(0)|0i = v−σ
α (p)
UCPT|particle, σi ⇠ |antiparticle, σi
U †
CPTΨα(0)UCPT ∼ Ψ† α(0)
uσ(p) ∼ v−σ(p)
(3) Crossing symmetry
(3) Crossing symmetry
M(kσ1
1 . . . → kσ1 1 . . .) =
h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(k1, . . .)
kσ3
3
→ kσ1
1
(3) Crossing symmetry =density matrices ρσ(k1)
M(kσ1
1 . . . → kσ1 1 . . .) =
h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(k1, . . .)
kσ3
3
→ kσ1
1
(3) Crossing symmetry =density matrices ρσ(k1)
M(¯ k−σ1
1
. . . → ¯ k−σ1
1
. . .) = (−1)2Spin · h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(−k1, . . .)
M(kσ1
1 . . . → kσ1 1 . . .) =
h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(k1, . . .)
kσ3
3
→ kσ1
1
(3) Crossing symmetry =density matrices ρσ(k1)
M(¯ k−σ1
1
. . . → ¯ k−σ1
1
. . .) = (−1)2Spin · h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(−k1, . . .)
M(kσ1
1 . . . → kσ1 1 . . .) =
h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(k1, . . .)
kσ3
3
→ kσ1
1
ρ(−k1)σ1 ?
(3) Crossing symmetry =density matrices ρσ(k1)
M(¯ k−σ1
1
. . . → ¯ k−σ1
1
. . .) = (−1)2Spin · h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(−k1, . . .)
M(kσ1
1 . . . → kσ1 1 . . .) =
h uσ1
α (k1)uσ1 † β
(k1) . . . i Aαβ...(k1, . . .)
kσ3
3
→ kσ1
1
ρ(−k1)σ1 ?
Mparticles(s) = Mantiparticles(u = −s)
if so, crossing in elastic scattering t=0 would be become again s u=-s
(4) Locality analytically continue density matrices off-shell ρσ(k) → ρσ(k)
uσ(k)uσ †(k) ≡ ρ(k)
(4) Locality analytically continue density matrices off-shell ρσ(k) → ρσ(k) (−1)2Spin · ρσ(k) = ρσ(−k)
u(p) ∼ p pµσµ
e.g. spin-1/2
uσ(k)uσ †(k) ≡ ρ(k)
(4) Locality analytically continue density matrices off-shell ρσ(k) → ρσ(k) (−1)2Spin · ρσ(k) = ρσ(−k)
u(p) ∼ p pµσµ
e.g. spin-1/2
uσ(k)uσ †(k) ≡ ρ(k)
[ψ(x1), ψ†(x2)]± = ρ(−i∂) Z d3p (2π)32|p|
(4) Locality analytically continue density matrices off-shell ρσ(k) → ρσ(k) (−1)2Spin · ρσ(k) = ρσ(−k)
u(p) ∼ p pµσµ
e.g. spin-1/2
uσ(k)uσ †(k) ≡ ρ(k)
propagator’s numerator: its parity fixed by Spin-Statistics
hT (x1) †(x2)i = Z d4k (2⇡)4 e−ikx12 ⇢σ(k) k2 i✏
[ψ(x1), ψ†(x2)]± = ρ(−i∂) Z d3p (2π)32|p|
can amplitudes be softer than E4
(within an EFT) ?
M4
can amplitudes be softer than E4
(within an EFT) ?
No!
Universal statement irrespectively spins
M4
can amplitudes be softer than E4
(within an EFT) ?
For Spinning particles: Analyticity, Crossing, and Unitarity act like for spin-0 in the forward elastic scattering
No!
Universal statement irrespectively spins
M4
can amplitudes be softer than E4
(within an EFT) ?
For Spinning particles: Analyticity, Crossing, and Unitarity act like for spin-0 in the forward elastic scattering
No!
Universal statement irrespectively spins
Re s Im s
sIR=(m1+m2)2 uIR=(m1-m2)2C
˜
C
s-plane
IR-side UV-side
M00(2 → 2)
Z 1 ds s3 σ12!anything(s) > 0
E4
M4
(1) ‘Higher Derivatives partial compositeness’
ψ → ψ + ξ
Lmix = λ ∂µψOµ
(1) ‘Higher Derivatives partial compositeness’
ψ → ψ + ξ
Lmix = λ ∂µψOµ
(1) ‘Higher Derivatives partial compositeness’
Leff = g2
∗
m6
∗
(∂νψ†)2(∂µψ)2 + . . .
ψ → ψ + ξ
Lmix = λ ∂µψOµ
(1) ‘Higher Derivatives partial compositeness’
Leff = g2
∗
m6
∗
(∂νψ†)2(∂µψ)2 + . . .
M(2 → 2) = g2
∗(E/m∗)6
ψ → ψ + ξ
Lmix = λ ∂µψOµ
(1) ‘Higher Derivatives partial compositeness’
Leff = g2
∗
m6
∗
(∂νψ†)2(∂µψ)2 + . . .
M(2 → 2) = g2
∗(E/m∗)6
doesn’t admit a local unitary UV completion
(2) Goldstino ψ(x) → ψ0(x0) = ψ(x) + ξ
x → x0 = x − iξσθ + iθ†σξ
(2) Goldstino ψ(x) → ψ0(x0) = ψ(x) + ξ
x → x0 = x − iξσθ + iθ†σξ Leff = − 1 4F 2 G† 2⇤G2 + . . .
up to field red.
M(GG → GG)(s, t = 0) = s2 F 2
(2) Goldstino ψ(x) → ψ0(x0) = ψ(x) + ξ
x → x0 = x − iξσθ + iθ†σξ
+light fields
L = − aψ F 2 (G†ψ†)⇤(Gψ) + e aψ F 2 (∂νG†¯ σµ∂νG)(ψ†¯ σµψ)
+ iaπ 4F 2 ∂µπ∂νπ(G†¯ σµ∂νG) + h.c.
− iaA 2F 2 (G†¯ σµ∂νG)FµρF νρ + h.c.
Leff = − 1 4F 2 G† 2⇤G2 + . . .
up to field red.
M(GG → GG)(s, t = 0) = s2 F 2
(2) Goldstino ψ(x) → ψ0(x0) = ψ(x) + ξ
x → x0 = x − iξσθ + iθ†σξ
+light fields
L = − aψ F 2 (G†ψ†)⇤(Gψ) + e aψ F 2 (∂νG†¯ σµ∂νG)(ψ†¯ σµψ)
+ iaπ 4F 2 ∂µπ∂νπ(G†¯ σµ∂νG) + h.c.
− iaA 2F 2 (G†¯ σµ∂νG)FµρF νρ + h.c.
Leff = − 1 4F 2 G† 2⇤G2 + . . .
up to field red.
M(GG → GG)(s, t = 0) = s2 F 2 M(Gψ → Gψ)(s, t = 0) = aψ F 2 s2
M(GA → GA)(s, t = 0) = aA F 2 s2 M(Gπ → Gπ)(s, t = 0) = aπ 2F 2 s2
(2) Goldstino ψ(x) → ψ0(x0) = ψ(x) + ξ
x → x0 = x − iξσθ + iθ†σξ
+light fields
L = − aψ F 2 (G†ψ†)⇤(Gψ) + e aψ F 2 (∂νG†¯ σµ∂νG)(ψ†¯ σµψ)
+ iaπ 4F 2 ∂µπ∂νπ(G†¯ σµ∂νG) + h.c.
− iaA 2F 2 (G†¯ σµ∂νG)FµρF νρ + h.c.
Leff = − 1 4F 2 G† 2⇤G2 + . . .
up to field red.
M(GG → GG)(s, t = 0) = s2 F 2
aψ > 0
aA > 0
aπ > 0
M(Gψ → Gψ)(s, t = 0) = aψ F 2 s2
M(GA → GA)(s, t = 0) = aA F 2 s2 M(Gπ → Gπ)(s, t = 0) = aπ 2F 2 s2
(3) Goldstino and R-axion
(3) Goldstino and R-axion
X2
NL = 0
XNL(ANL − A†
NL) = 0
RNL = eiANL
constrained superfields Komargodski Seiberg 0907.2441
ANL → ANL + ✏
L = Z d4θ ⇣ X†
NLXNL + f 2 aR† NLRNL
⌘ + ✓Z d2θ FXNL + wR R2
NL + h.c.
◆
(3) Goldstino and R-axion
X2
NL = 0
XNL(ANL − A†
NL) = 0
RNL = eiANL
constrained superfields Komargodski Seiberg 0907.2441
ANL → ANL + ✏
L = Z d4θ ⇣ X†
NLXNL + f 2 aR† NLRNL
⌘ + ✓Z d2θ FXNL + wR R2
NL + h.c.
◆
L = LAV + f 2
a(∂µa)2 + 2i
✓ f 2
a
F 2 − 4w2
R
F 4 ◆ (∂µG†¯ σνG)∂µa∂νa + 2wR F 2 (G† 2 + G2)(∂µa)2 + . . .
(3) Goldstino and R-axion
X2
NL = 0
XNL(ANL − A†
NL) = 0
RNL = eiANL
constrained superfields Komargodski Seiberg 0907.2441
ANL → ANL + ✏
L = Z d4θ ⇣ X†
NLXNL + f 2 aR† NLRNL
⌘ + ✓Z d2θ FXNL + wR R2
NL + h.c.
◆
L = LAV + f 2
a(∂µa)2 + 2i
✓ f 2
a
F 2 − 4w2
R
F 4 ◆ (∂µG†¯ σνG)∂µa∂νa + 2wR F 2 (G† 2 + G2)(∂µa)2 + . . .
wR < 1 2faF
(3) Goldstino and R-axion
X2
NL = 0
XNL(ANL − A†
NL) = 0
RNL = eiANL
constrained superfields Komargodski Seiberg 0907.2441
ANL → ANL + ✏
L = Z d4θ ⇣ X†
NLXNL + f 2 aR† NLRNL
⌘ + ✓Z d2θ FXNL + wR R2
NL + h.c.
◆
L = LAV + f 2
a(∂µa)2 + 2i
✓ f 2
a
F 2 − 4w2
R
F 4 ◆ (∂µG†¯ σνG)∂µa∂νa + 2wR F 2 (G† 2 + G2)(∂µa)2 + . . .
slightly improved bound
Komargodski Festuccia Dine 0910.2527
wR < 1 2faF
(3) Goldstino and R-axion
X2
NL = 0
XNL(ANL − A†
NL) = 0
RNL = eiANL
constrained superfields Komargodski Seiberg 0907.2441
ANL → ANL + ✏
L = Z d4θ ⇣ X†
NLXNL + f 2 aR† NLRNL
⌘ + ✓Z d2θ FXNL + wR R2
NL + h.c.
◆
L = LAV + f 2
a(∂µa)2 + 2i
✓ f 2
a
F 2 − 4w2
R
F 4 ◆ (∂µG†¯ σνG)∂µa∂νa + 2wR F 2 (G† 2 + G2)(∂µa)2 + . . .
slightly improved bound
Komargodski Festuccia Dine 0910.2527
wR < 1 2faF
Γ(a → GG) < 1 32π ✓m5
a
F 2 ◆
(4) SM fermions as pseudo-Goldstini (revive old idea Bardeen and Visnjic NPB 1982)
w/ F. Riva, J. Serra and F. Sgarlata
(4) SM fermions as pseudo-Goldstini (revive old idea Bardeen and Visnjic NPB 1982)
shaded region violates positivity!
for illustration only! pedestrian bounds
w/ F. Riva, J. Serra and F. Sgarlata
(4) SM fermions as pseudo-Goldstini (revive old idea Bardeen and Visnjic NPB 1982)
shaded region violates positivity!
Λ & 9 ⇣ g∗ 4π ⌘1/2 TeV
for g∗ > 3
for illustration only! pedestrian bounds
w/ F. Riva, J. Serra and F. Sgarlata
(4) SM fermions as pseudo-Goldstini (revive old idea Bardeen and Visnjic NPB 1982)
shaded region violates positivity!
Λ & 9 ⇣ g∗ 4π ⌘1/2 TeV
for g∗ > 3
for illustration only! pedestrian bounds
N = 9
w/ F. Riva, J. Serra and F. Sgarlata
some live in the swampland and can’t be UV-completed
(low-energy constraints can’t be made arbitrarily irrelevant).
(R-axion, Goldstinos, fermionic shift sym…)
M4
= + + +…
O(M −2
P lanck)
bound on
elementary strong sector
m=0
O(g2
weak)
M4
= + + +…
O(M −2
P lanck)
bound on
elementary strong sector
m=0
O(g2
weak)
=
hTTTTistrong
if
hJJJJistrong
gravity YM
bound on higher powers (Froissart relaxed)