Scale Setting for Self-consistent Quantum
Backgrounds
Benjamin Kocha
in collaboration with
Carlos Contrerasc , and Paola Riosecoa
[email protected]
c
a PUC, Chile
UFSM Valparaiso, Chile
Sussex
January 2015
Benjamin Koch (PUC, Chile)
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Outline
I Introduction Γk
II Improving solutions
III Self-consistent scale setting
IV Example: “inoffensive”
V Examples: “offensive”
VI Summary and outlook
Based on arXiv:1409.4443 (accepted by PRD) and arXiv:1501.00904
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I
Introduction I
I
Introduction
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I Introduction Γk
Ideal case
Γk (φ)
(1)
In “Pleasantville”
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I Introduction Γk
Effective action
Generating functional
Z
Z [J] =
Connected diagrams
Z
Dφ exp −i dx(L(φ) − φJ)
(2)
W [J] = ln Z [J]
(3)
Z
Γ[φ] = supJ ( Jφ − W [J])
(4)
Effective action (1PI)
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I Introduction Γk
Wilsons flow
What happens if one integrates out only certain Momentum shell ?
k → k + δk?
⇒ RG-Flow
∂ k Γk = . . .
(5)
Solve:
Running couplings λk and
effective action
Γk = Γk (λk , φ)
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I Introduction Γk
Wilsons flow dangers
gauge redundance
φ → e iα φ
Fadeev Popov
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I Introduction Γk
Wilsons flow dangers
Infrared divergencies
k → 0, Γ0 =?
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I Introduction Γk
Wilsons flow dangers
Infrared divergencies
k → 0, Γ0 =?
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I Introduction Γk
Wilsons flow dangers
Ultra violet divergencies
k → ∞, Γ∞ =?
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I Introduction Γk
Wilsons flow dangers
Ultra violet divergencies
k → ∞, Γ∞ =?
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I Introduction Γk
Wilsons flow dangers
Renormalization:
Live with the problems
Absorb Infinities ∼ Λ in
definition of couplings λ at
scale M0
λi,0 = λi (M0 )
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I Introduction Γk
Wilsons flow after Renormalization
AFTER renormalization can
pretend to be back in
“Pleasantville”
Coupling flow λi (k) and
effective action Γk
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I Introduction Γk
Wilsons flow
At the end effective quantum action:
Z
√
Γk (φi (x), λn (k)) = d 4 x −g Lk (φi (x), λn (k))
(8)
Quantum background?
δΓk
=0
δφi
.
(9)
“GAP EQUATION”
solutions?
quantum solitons?
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I Introduction Γk
Wilsons flow
At the end effective quantum action:
Z
√
Γk (φi (x), λn (k)) = d 4 x −g Lk (φi (x), λn (k))
(8)
Quantum background?
δΓk
=0
δφi
.
(9)
“GAP EQUATION”
solutions?
quantum solitons?
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II
Improving solutions II
II
Improving solutions
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Improving solutions II
Improving solutions II
δΓk
δφi
= 0 is too hard
⇒ first solve classical eom
δS
=0
δφi
.
(10)
and take λ0 → λk as small correction of those solutionsi,ii
i ) Uehling potential: in QED textbooks
ii ) Improved black holes: B.K. and Frank Saueressig; Int.J.Mod.Phys. A29 (2014) 8, 1430011 ...
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Improving solutions II
Improving solutions II
δΓk
δφi
= 0 is too hard
⇒ first solve classical eom
δS
=0
δφi
.
(10)
and take λ0 → λk as small correction of those solutionsi,ii
i ) Uehling potential: in QED textbooks
ii ) Improved black holes: B.K. and Frank Saueressig; Int.J.Mod.Phys. A29 (2014) 8, 1430011 ...
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Improving solutions II
More general consideration
“Improved solution” - fully consistent?
No
Problem: Scale setting
k →? k(r )
(typically k ∼ 1/r )
(11)
Implies
Improved classical solution does not solve “gap equations”
Typically not even T µν conserved
⇒ Propose different scale setting
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III
Self-consistent scale setting III
III
Self-consistent scale setting
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Self-consistent scale setting III
Proposal
Effective action and running couplings
Z
√
Γk (φi (x), λn (k)) = d 4 x −g Lk (φi (x), λn (k)) ,
(12)
“Gap equations” for quantum background
δΓk
=0
δφi
.
(13)
Scale setting for this background?
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Self-consistent scale setting III
Proposal
Scale setting for this background k = k(r )?
Idea: Minimal k sensitivity
(just like Callan-Symanzik equations
d
dk hT φ1 (x1 )φ2 (x2 ) . . . ik k=kopt
≡ 0)
Realization: Promote k to field in Γk
Γk (φi (x), λn (k)) → Γ(φi (x), k(x), λn (k)) .
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Self-consistent scale setting III
Proposal
Realization:
“Scale field” k(x)
coupled equations of motion
δΓ(φi (x), k(x))
=0
δφi
,
d
L(φi (x), k(x), λn (k))
=0
dk
k=kopt
.
(15)
The simultaneous solution of (15) assures optimal scale setting
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Self-consistent scale setting III
Proposal
First questions on proposal
Consistent set of equations?
Solves ∇µ T µν ↔ k(r ) problem?
Examples?
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IV
Examples IV
IV
Examples: “inoffensive”
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IV
Examples IV
Inoffensive:
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Examples IV
φ4
Effective action
Z
m̃k2 2 g̃k 4
αk
2
4
(∂φ) −
φ − φ
Γ= d x
2
2
4!
Eom δφ:
g̃k 3
φ =0
6
(17)
1 0 4
g̃ φ = 0 ,
12 k
(18)
∂µ (αk ∂µ φ) + m̃k2 φ +
eom k:
αk0 (∂φ)2 − (m̃k2 )0 φ2 −
implies conservation
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∇µ T µν = 0
Scale Setting
(19)
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Examples IV
φ4 loops
Example: Self-consistent scale setting for φ4 at 1 loop:
γZ
=
βg
=
βm 2
d ln Zk
=0 ,
d ln k 2
dgk
3
g2
=
d ln k
16π 2 k
dm2
k
=
d ln k
= (−2 +
,
gk
)mk2
16π 2
.
Run proposed machine for spherical symmetry ...
!
r
14 k0
c1
k(r ) = ki + exp −2
m0 r ·
3 ki
r
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Scale Setting
....
.
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Examples IV
φ4 loops
Example: Self-consistent scale setting for φ4 at 1 loop:
Self-consistent
Like classical for small r
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IV
Examples IV
V
Examples: “offensive”
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IV
Examples IV
Offensive:
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Examples V
Gravity and gauge
Effective action
Z
Γk [gµν , Aα ] =
M
√
d 4 x −g
1
R − 2Λk
− 2 Fµν F µν
16πGk
4ek
,
(21)
Einstein-Hilbert-Maxwell
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Examples V
Gravity and gauge
Equations of motion
eom δgµν :
Gµν = −gµν Λk − ∆tµν + 8πGk Tµν
with
∆tµν = Gk gµν − ∇µ ∇ν
and
Dµ
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1
Gk
1
Tµν = Fνα Fµα − gµν Fµν F µν
4
eom δAµ :
1 µν
F
ek2
,
(22)
.
(23)
.
(24)
=0
Scale Setting
,
(25)
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Examples V
Gravity and gauge
Conservation and symmetry
Symmetry coordintates:
Symmetry U(1):
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∇µ Gµν = 0
(26)
∇[µ Fαβ] = 0 .
(27)
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Examples V
Gravity and gauge
Consistency:
Using everybody . . . one can actually show that
Scale setting eom δk:
Λ(k)
4π
1
αβ
Fαβ F
· (∂µ k) = 0 . (28)
R∇µ
− 2∇µ
− ∇µ
Gk
Gk
ek2
is actually self-consistent consequence of eoms and conservation laws
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Examples V
Gravity and gauge
Background solutions?
Actually, yes:
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Examples V
Gravity and gauge
Background solutions?
Actually, yes:
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Examples V
de Sitter black hole
De Sitter case:
eom gµν :
Gµν = −gµν Λk − ∆tµν
eom k:
R∇µ
1
Gk
− 2∇µ
Λk
Gk
,
(29)
=0
.
(30)
unknown functions: g00 (r ), g11 (r ), and k(r )
known (with caveat): Λk , and Gk
Don’t like caveat ...
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Examples V
de Sitter black hole
Trade (trick):
g00 (r ), g11 (r ), k(r ), Λk , and Gk
⇒
g00 (r ), g11 (r ), Λ(r ), and G (r )
Too many unknowns: Schwarzschild ansatz: g00 = 1/g11 ≡ f
⇒
f (r ), Λ(r ), and G (r )
Can be solved
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Examples V
de Sitter black hole
generalized de Sitter solution:
G (r )
=
f (r )
=
Λ(r )
=
G0
εr + 1
(31)
2G0 M0
Λ0 r 2
c4 (εr + 1)
1 + 3G0 M0 ε −
− (1 + 6εG0 M0 )εr −
+ r 2 ε 2 (6εG0 M0 + 1) ln
r
3
r
1 G M ε + 1 ln c4 (εr +1) + 4r 3 Λ ε 2 + 12ε 3 + 6Λ ε + 72ε 4 G M r 2
−72ε 2 r (εr + 1) εr + 2
0
0 0
0
0 0
6
r
2r (εr + 1)2
72ε 3 G
0 M0
+
+ 11ε 2
+ 2Λ0 r
2r (εr + 1)2
+ 6ε 2 G
0 M0
(32)
(33)
.
Constants of integration: G0 , M0 , Λ0 , ε, c4
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Examples V
de Sitter black hole
Classical limit:
G (r )|ε→0 = G0
Λ(r )|ε→0 = Λ0
f (r )|ε→0 = −
(34)
.
Λ0 r 2 2G0 M0
−
+1
3
r
(35)
,
(36)
ε parametrizes scale dependence of the couplings
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Examples V
de Sitter black hole
Asymptotics and horizons:
+r →0
Rµναβ R µναβ =
48G02 M02 48G02 M02 ε
−
+ O(r −4 ) .
r6
r5
(37)
⇒ Singularity persists
+r →∞
f (r ) = −r 2
1
Λ0 − 3ε 2 (6εG0 M0 + 1) ln (c4 ε) + O(r ) ,
3
(38)
⇒ shift of effective cosmological constant
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Examples V
de Sitter black hole
f (r ) for var. ε
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Horizons for var. ε
Scale Setting
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Examples V
Reissner Nordstrom black hole
Reissner Nordstrom case:
eom gµν :
Gµν = −∆tµν + 8π
eom Aµ
Dµ
eom k:
R∇µ
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1
Gk
1 µν
F
ek2
− ∇µ
4π
ek2
Gk
Tµν
ek 2
,
(39)
=0
Fαβ F
Scale Setting
.
αβ
(40)
· (∂µ k) = 0
.
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Examples V
Reissner Nordstrom black hole
Unknown functions:
g00 (r ) ≡ f (r ), g11 (r ), k(r ), F01 = q(r ),
Known
Gk , ek
Same trade as before g11 (r ) ≡ 1/f (r ) and k(r ) versus G (r ), e(r ),
⇒
f (r ), q(r ), G (r ), e(r )
Can be solved ...
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Examples V
Reissner Nordstrom black hole
Solution:
G (r )
=
f (r )
=
e 2 (r )
=
q(r )
=
G0
εr + 1
(42)
r 4 ε 2 e0 2 + 4εr 3 e0 2 + 4(1 − G0 M0 ε)e0 2 r 2 − 8rG0 M0 e0 2 + 16πG0 Q0 2
4r 2 (εr + 1)2 e0 2
r 6 ε 4 e0 2 + 3r 5 ε 3 e0 2 + 3r 4 e0 2 − 4r 3 e0 2 G0 M0 + 48r 2 πG0 Q0 2 ε 2 + 48εr πG0 Q0 2 + 16πG0 Q0 2 e0 2 π
Q0 2 G0 (εr + 1)3
Q0
4πe02
e 2 (r )
r2
.
Integration constants: G0 , Q0 , e0 , and ε
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Examples V
Reissner Nordstrom black hole
Classical limit:
f (r )|ε→0 = 1 −
2G0 M0 4πG0 Q02
+ 2 2
r
r e0
,
(43)
e 2 (r )|ε→0 = (4π)2 e02
(44)
4πQ0
.
r2
ε parametrizes scale dependence of the couplings (again)
q(r )|ε→0 =
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Examples V
Reissner Nordstrom black hole
Asymptotics:
+r →0
Rαβγδ R αβγδ = 27 7
G02 π 2 Q04
+ O 1/r 7
4
8
e0 r
.
(46)
⇒ Singularity persists
+r →∞
line element
1
2
ds∞
= − dt 2 + 4dr 2 + r 2 d θ 2 + r 2 sin2 θd φ2
4
.
⇒ like global monopole
Z
Q=
(still charge is unchanged)
q
Q0
F µν
2
d z γ S2 nµ σν 2 = 2 ,
e
e0
∂Σ
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(48)
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Examples V
Reissner Nordstrom black hole
f (r ) for var. ε
Horizons r± (M0 ) for var. ε
Observe that cosmic censureship unchanged
√ Q0
M0 = 2 π
e0 G0
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Scale Setting
,
(49)
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Examples V
Reissner Nordstrom black hole
Temperature slightly increased for ε 6= 0 ∼ ε 2
s

G0 G0 M0 2 + M0
T = T0 + ε 2
s
8π 
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G0 e0 2 G0 M0 2 −4πQ0 2
e0 2
G0 e0 2 G0 M0 2 −4πQ0 2
e0 2
Scale Setting
− 4π

Q0 2
2
e0


+ O(ε 3 ) .
(50)
+ G0 M0 
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V
Summary
V
Summary
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Summary
Context: Self-consistent quantum backgrounds δΓk /δφi = 0
Problem: Scale setting and consistency
Proposal: Self-consistent scale setting k → k(x)
δΓk /δφi = 0 and δΓk /δk = 0
Examples: Studied φ4 , and de Sitter-, RN black holes
Outlook: Hopefully to be applied in many more contexts ;-)
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Summary
Thank you
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