2PI solutions of the φ4 model across the complex
plane
Gergely Markó
Theoretical Physics Department
2017, 22nd of March, Budapest
• Motivation
• 2PI and model intro
• Minkowski solution
• Analytic continuation
• Summary
Collaborators: U. Reinosa (École Polytechnique, CPHT), Zs. Szép (MTA-ELTE Statistical and Biological Physics
Research Group)
Motivation 1/2
• The exact two-point function contains important information: pole masses,
spectral properties, possible complex poles of resonances.
• A truncation of 2PI approximates the exact two-point function.
• These truncations are usually solved (few counterexamples exist) in
Euclidean space.
• Two possible ways: solve the continued equations or numerically continue
the solution to the complex plane.
• Solving the equations in Minkowski: first step is at T = 0, then go to finite
temperature.
• We already have Euclidean data, we need a continuation method, and to test
it before applying.
Motivation 2/2
• We based our parametrization of the O(4) model on the curvature masses,
assuming that they are close to the pole masses. We need to assess the error
we made.
2.4
h/T3⋆
2
1.6 M̂L,0 [MeV]
273 304 334 364 394 424
1.2
0.8
0.4
1.6
1.4
M̂L,0 [MeV]
280 300 320 340 360 380 400 420
T⋆ [MeV]
1.2
170
1
0.4
2
m ⋆/
0.3
0.1
0
160
0.8
150
0.6
140
130
0.4
120
0.2
110
0.5
2
T⋆
0.2
10 5
30 25 20 15
0
50 45 40 35
λ⋆
h
T3⋆
0.4
Λp /T⋆ = 10
Λp /T⋆ =50(2
-lo op)
0.3
m⋆2/ 2 0.2
T⋆
0.1
0 50
40
30 λ⋆
20
10
0
• 2PI has an inherent ambiguity in the defintion of two-point functions (Ḡ vs.
δ 2Γ[φ, Ḡ]
(2)
Γ =
) in a given truncation.
2
δφ
• The most reasonable parametrization would be based on the pole mass
aquired from the external propagator Γ(2), but this needs a Bethe-Salpeter
equation to be solved.
• Solving the BS equation is considerably simpler in Euclidean space.
• Therefore one needs a reliable continuation method to aquire pole masses
from Euclidean propagators.
Introduction to 2PI
The two-particle irreducible formalism is a continuum functional method
which gives an infinite diagramatic expansion for the quantum effective
action in terms of the exact one- and two-point functions, while these
Green’s functions are determined by stationarity conditions prescribed on
the effective action.
A bilocal source is introduced in the generating functional
Z
W [J,K]
Z[J, K] = e
=
Dϕ exp − S0 − Sint + ϕ · J + ϕ · K · ϕ
The 2PI effective action defined through a double Legendre transform
Z
Γ[φ, G] = W [J, K] −
δW [J, K]
d x
J(x) −
δJ(x)
| {z }
4
Z
4
Z
d x
φ(x)
4
d y
δW [J, K]
K(x, y)
δK(x, y)
| {z }
[φ(x)φ(y)+G(x,y)]/2
The physical φ̄(x) and Ḡ(x, y) are determined from stationarity conditions at vanishing sources
(J, K → 0)
δΓ[φ, G] δΓ[φ, G] = 0,
=0
δφ(x)
δG(x, y)
φ̄(x)
Ḡ(x,y)
Introduction to 2PI
The diagramatic expansion has to be truncated in order for the theory to
be solved, leading to approximations of the exact theory. However, the
truncations can systematically include more and more diagrams. We expect
a convergence in the results, which has to be confirmed.
Γ[φ, G] can be written as shown in Cornwall et al., PRD 10, 2428 (1974)
1
1 −1
−1
Γ[φ, G] = S0[φ] + Tr log G + Tr G0 G − 1 + Γint[φ, G]
2
2
S0 is the free action,
G0 is the free propagator,
Γint[φ, G] contains all the 2PI graphs constructed with vertices from Sint(φ + ϕ).
The Tr is to be understood in all indices and as integration over coordinates.
The 1PI effective action is recovered: Γ1PI[φ] = Γ[φ, Ḡ].
Assuming a homogenous field φ and a propagator which only depends on the difference of
coordinates leads to the effective potential γ[φ, G]. In thermal field theory γ[φ̄, Ḡ] = −pV + c.
The model(s) we consider
We solve the φ4 scalar model in Minkowski space, because
• it is a simple field theory with spontaneous symmetry breaking;
• we already solved it in Euclidean;
• similar qualitative behaviour as the linear sigma model, which we considered
in the physical parametrization.
We then use the method of Padé approximants for analytic continuation, in the
• φ4 scalar model, to test quality and basic properties;
• perturbation theory, to look for complex poles;
• O(4) data from our previous studies.
The 2-loop truncation 1/2
G=
γ[φ, G]
=
1
Tr
2
T
Z
Q
λ0
+
3
δγ[φ, G] =0
δG(Q) G=Ḡ
δγ[φ, Ḡ] =0
δφ
φ=φ̄
δ 2γ[φ, Ḡ] δφ2 φ=φ̄
T [Ḡ] =
⇒
,φ =
h
i 1
λ4 φ4
−1
−1
2 2
log(G (Q)) + G0 (Q)G(Q) + m2φ +
2
24
λ2
+
3
λ2
−
12
2
2
m0
M̄ (Q) =
"
=
φ̄
=
M̂
,
2
m2
.
λ2 2 λ0
φ +
+
2
2
λ4 2 λ2
+
φ̄ +
6
2
λ2
+
6
λ2 2
φ
+
2
#
,
2
I[Ḡ](Q) =
,
S[Ḡ] =
.
,
The 2-loop truncation 2/2
Renormalization:
• Finite equations are obtained by expanding the 2PI propagator around a known G0
propagator. See Patkós & Szép, NPA811, 329 for details.
• Truncation artefact: ambiguity in the definition of n-point functions,
δ 2 Γint
δφδφ
6= 2
δΓint
δG ,
etc.
• 5 bare parameters (m2i = m2 + δm2i , i = 0, 2; λi = λ + δλi, i = 0, 2, 4) are determinde
in terms of 2 renormalized parameters (m2, λ) and a renormalization scale M02.
Finite equations:
iḠ
−1
2
2
(Q)=Q -M̄ (Q),
2
2 λ 2 λ
M̄l =m + φ + TF[Ḡ],
2
2
2
M̄ (Q)=M̄l +M̄nl(Q),
λ
2
M̄nl(Q)=
2
−1
2
2
iG0 (Q)=Q -M0
2
φ I[Ḡ](Q)-I[G0]
|
{z
}
2
2
2
IF [Ḡ](Q)
λ2
2 λ 2 λ
0=φ̄ m + φ̄ + TF[Ḡ]+ SF[Ḡ]
6
2
6
Z
2
2
2 λ
2
2
TF[Ḡ] =
Ḡr(Q),
Ḡr(Q)= Ḡ(Q)-G0(Q) +i Ml -M0 + φ IF[G0](Q) G0(Q)
{z
}
|
2
Q
δ Ḡ(Q)
Z
SF[Ḡ] = S[δ Ḡ] + 3S[δ Ḡ, δ Ḡ, G0] + 3
Ḡr(Q)IF[G0](Q)
Q
Building blocks
Spectral representation of the propagator
∞
Z
G(Q) =
0
Sum rule:
∞
Z
0
spectral
function:
i
dm2
ρ(m2) 2
2π
Q − m2 + iε
ρ(Q2) = −2=(−iG(Q))
ds
ρ(s) = 1
2π
Once subtracted dispersion relation (for subtraction scale µ we choose µ = 0)
Z ∞
=I[G](s0)
s−µ
0
2
P
ds 0
<I[G](s) = <I[G](µ ) +
π
(s − µ2)(s0 − s)
0
2
∞
ds ρ(s)
2π Q2E + s
0
to compute the local integrals TF[GE ], SF[GE ] using some technical cutoff
Continuation to Euclidean space-time: GE (QE ) =
Z
We study the case when a real pole exists −→ ρ(s) = 2πZδ(s − Mp2) + σ(s)
Dealing with the bubble
Imaginary part
Z∞
=IF [G](K) =
0
dm21
2π
Z∞
0
dm22
2
2
ρ(m1 )ρ(m2 )=I[G1 , G2 ](K),
2π
Gi (K) =
i
K 2 − m2i
, i = 1, 2
Θ(K 2 − (m1 + m2 )2 ) q 2
2 K 2 − (m − m )2
=I[G1 , G2 ](K) = −
K
−
(m
+
m
)
1
2
1
2
16πK 2
Real part (subtraction scale µ = 0)
Z∞
0
s
0 =IF [G](s )
<IF [G](s) = I[G] − I[G0 ] + P ds 0 0
,
|
{z
} π
s (s − s)
2
sth = 4Mp
sth
IF [G]
– Using the spectral reprezentation of G and working in dim. regularization:
Z∞
I[G] =
0
– Using
dm21
2π
0
dm22
2
2
ρ(m1 )ρ(m2 )I[G1 , G2 ],
2π
1
I[G0 ] =
16π 2
I[G]-I[G0 ] =
|
{z
}
IF [G]
Z∞
Z∞
0
m2
m2
2
2
1
m1 ln 2 − m2 ln 22 
1 
1
κ
κ 
- +γE -1+
I[G1 , G2 ]=


2
2
2
16π
m1 − m2


1
M02
− + γE + ln
4πκ2
"
Z∞
dm21
dm22
2π
0
2π
2
2
#
and the sum rule
ρ(m1 )ρ(m2 )IF [G1 , G2 ],
Z∞
0
dm2
2
ρ(m ) = 1:
2π
m2
m2
2
2
1
m1 ln
− m2 ln 22
2
eM0
eM0
1
IF [G1 , G2 ]=
16π 2
m21 − m22
Numerical Algorithm
λ2 2
λ 2
φ̄ + TF[Ḡ] + φ̄ IF[Ḡ] and obtain
First introduce M ≡ m +
2
2
2
λ
−1
2
2
2
gap equation:
iḠ (K) = K − M − φ̄ IF[Ḡ](K) − IF[Ḡ]
2
2
2
λ 2 λ 2
λ
2
field equation:
0 = φ̄ M − φ̄ + φ̄ IF[Ḡ] + SF[Ḡ]
3
2
6
2
2
−→ TF[G] is only needed to get the renormalized parameter m2
Gap and field equations are solved iteratively for a given M 2 > 0 and λ pair
M0 = 1 ⇒ dimensionfull quantities are in units of M0
(i)
(i)
(i) sum
(i)
F.E.
(i)
ρ(s) {Mp , σ(s) −−→ Zs } −−→ φ̄
rule
2
−→ =M̄ (s)
(i+1)
2
−→ <M̄ (s)
(i+1)
−→ ρ(s)
(i+1)
– [sth, ∞] → [tth, 1] with s = t/(1 − t)
J σ(t) is splined and the spline coefficients are stored
cubic t-grid to have high resolution in the IR, UV asymptotics of σ is exploited
– to compute <I[G](s) with disp. rel. =I 2[G](s) is splined
– TF[G] and SF[G] computed in Euclidean space after GE(QE )-G0,E(QE ) is splined
(0)
Initialization: σ(s)(0) = 0 −sum
−−→ Zs
rule
(0)
= 1; Mp = M =⇒ ρ(s)(0) = 2πδ(s − M 2)
Parameter space 1/2
• We are interested in SSB ⇒ our goal
is to map out the φ̄ 6= 0 regions of the
(m2, λ) plane.
Parameter space 1/2
0.14
λc
0.12
0.1
φ̄ 0.08
0.06
0.04
0.02
00
0.005
0.01
0.015
0.02
0.025
M2
0.03
0.035
0.04
0
5
10
15
20
25
30
35
λ
30
120
25
λ
80
40
20
λ
• We are interested in SSB ⇒ our goal
is to map out the φ̄ 6= 0 regions of the
(m2, λ) plane.
• Critical lines:
λ̄c : φ̄ = 0 & M̄ 2 ≡ M 2 = 0
λc : φ̄ = 0 & M̂ 2 = 0(M̄ 2 6= 0).
0
15
0
0.02
0.04
−m2
0.06
10
5
0
λc
λ̄c
0
0.01
0.02
0.03
−m2
0.04
0.05
0.06
Parameter space 1/2
0.14
λc
0.12
M2 =0.01
0.1
M2 =0.01
φ̄ 0.08
0.06
M2 =0.015
M2 =0.015
M2 =0.03
0.04
0.02
00
0.005
0.01
0.015
0.02
0.025
M2
0.03
0.035
0.04
0
5
10
15
20
25
30
35
λ
30
120
25
λ
80
40
20
λ
• We are interested in SSB ⇒ our goal
is to map out the φ̄ 6= 0 regions of the
(m2, λ) plane.
• Critical lines:
λ̄c : φ̄ = 0 & M̄ 2 ≡ M 2 = 0
λc : φ̄ = 0 & M̂ 2 = 0(M̄ 2 6= 0).
• The defintion of M 2 includes φ̄ ⇒
introduces intricacy in the mapping
(M 2, λ) ↔ (m2, λ).
0
15
0
0.02
0.04
−m2
0.06
10
5
0
λc
λ̄c
0
0.01
0.02
0.03
−m2
0.04
0.05
0.06
Parameter space 1/2
0.14
λc
0.12
M2 =0.01
0.1
M2 =0.01
φ̄ 0.08
0.06
M2 =0.015
M2 =0.015
M2 =0.03
0.04
0.02
00
0.005
0.01
0.015
0.02
0.025
M2
0.03
0.035
0.04
0
5
10
15
20
25
30
35
λ
30
120
25
λ
80
40
20
λ
• We are interested in SSB ⇒ our goal
is to map out the φ̄ 6= 0 regions of the
(m2, λ) plane.
• Critical lines:
λ̄c : φ̄ = 0 & M̄ 2 ≡ M 2 = 0
λc : φ̄ = 0 & M̂ 2 = 0(M̄ 2 6= 0).
• The defintion of M 2 includes φ̄ ⇒
introduces intricacy in the mapping
(M 2, λ) ↔ (m2, λ).
(
φ̄ = 0 : (m21, λ)
2
• (M , λ) →
φ̄ 6= 0 : (m22, λ)
and vice versa.
0
0
15
0.02
0.04
−m2
0.06
λc
λ̄c
2
M = 0.01
M2 = 0.01
M2 = 0.015
M2 = 0.015
M2 = 0.03
10
5
0
0
0.01
0.02
0.03
0.04
−m2
0.05
0.06
0.07
0.08
Parameter space 1/2
−0.08
φ̄ 6= 0
−0.09
−m2
λc
0.12
M2 =0.01
0.1
M2 =0.01
φ̄ 0.08
0.06
M2 =0.015
M2 =0.015
M2 =0.03
0.04
0.02
00
0.005
0.01
0.015
0.02
0.025
M2
0.03
0.035
0.04
0
5
10
25
20
0
0
15
0
−0.14
−0.15
0
0.005
0.01
0.015 0.02
M2
0.025
0.03
0.035
λ
40
0.02
0.04
−m2
0.06
λc
λ̄c
2
M = 0.01
M2 = 0.01
M2 = 0.015
M2 = 0.015
M2 = 0.03
5
−0.13
35
80
10
−0.12
20
30
120
−0.1
−0.11
15
25
30
λ
−0.07
0.14
λ
• We are interested in SSB ⇒ our goal
is to map out the φ̄ 6= 0 regions of the
(m2, λ) plane.
• Critical lines:
λ̄c : φ̄ = 0 & M̄ 2 ≡ M 2 = 0
λc : φ̄ = 0 & M̂ 2 = 0(M̄ 2 6= 0).
• The defintion of M 2 includes φ̄ ⇒
introduces intricacy in the mapping
(M 2, λ) ↔ (m2, λ).
(
φ̄ = 0 : (m21, λ)
2
• (M , λ) →
φ̄ 6= 0 : (m22, λ)
and vice versa.
• Insult to injury: non-physical solutions
0
0.01
0.02
0.03
0.04
−m2
0.05
0.06
0.07
0.08
Parameter space 2/2
We already faced similar problems in G.M., Reinosa & Szép, PRD 92, 125035 (2015). ⇒ Use
localized approxmation for qualitative understanding:
Ḡloc(Q) =
i
i
≡
.
Q2 − M̄ 2
Q2 − M 2
30
25
λ
20
λc
λ̄c
M2 = 0.01
M2 = 0.01
M2 = 0.015
M2 = 0.015
M2 = 0.03
15
10
5
0
0
0.01
0.02
0.03
0.04
−m2
0.05
0.06
0.07
0.08
Parameter space 2/2
We already faced similar problems in G.M., Reinosa & Szép, PRD 92, 125035 (2015). ⇒ Use
localized approxmation for qualitative understanding:
Ḡloc(Q) =
i
i
≡
.
Q2 − M̄ 2
Q2 − M 2
30
25
λ
20
15
10
5
0
0
0.01
0.02
0.03
0.04
−m2
0.05
0.06
0.07
0.08
Parameter space 2/2
Now we can explore a huge range:
Ḡloc(Q) =
i
i
≡
.
Q2 − M̄ 2
Q2 − M 2
120
100
λ
80
60
40
20
0
0
0.1
0.2
−m2
0.3
0.4
0.5
Parameter space 2/2
Now we can explore a huge range:
Ḡloc(Q) =
i
i
≡
.
Q2 − M̄ 2
Q2 − M 2
120
100
λ
80
60
40
20
0
0
0.1
0.2
−m2
0.3
0.4
0.5
Parameter space 2/2
And exclude a region where no solution exists:
Ḡloc(Q) =
i
i
≡
.
Q2 − M̄ 2
Q2 − M 2
120
100
λ
80
60
40
20
0
0
0.1
0.2
−m2
0.3
0.4
0.5
Parameter space 2/2
The final map:
i
i
Ḡloc(Q) = 2
≡ 2
.
Q − M̄ 2
Q − M2
120
Lo
ss
100
λ−
of
s
olu
tio
n
λ
80
λc
60
λ−
40
λ̄c
φ̄=0
λi
20
0
φ̄ 6= 0
0
0.1
0.2
−m2
0.3
0.4
0.5
Minkowski solution at φ̄ 6= 0
Spectral function
Real and imaginary part of M̄ 2(Q)
first iteration vs. converged result
0.02
0.06
M2 = 0.5, λ = 5
0.015
0.04
Re/Im M̄2
ρ
0.005
Re{M̄2 }−M2
Im{M̄2 }(1)
Im{M̄2 }
−
0.02
0.01
Re{M̄2 }(1) −M2
2
λ2 φ(1)
(
)
2 2
φ̄
− λ32π
32π
0
−0.02
λ = 5, M2 = 0.5
−0.04
0
0
0.5
1
1.5
2
√
s
2.5
3
– for small λ one has Mp2 ' M 2
3.5
4
−0.06
−0.08
0
1
2
√
3
4
s
– the algorithm converges quickly
5
Minkowski solution at φ̄ 6= 0
Spectral function
Real and imaginary part of M̄ 2(Q)
first iteration vs. converged result
0.02
0.15
M2 = 0.5, λ = 5
0.1
0.015
Re/Im M̄2
ρ
0.005
Re{M̄2 }−M2
Im{M̄2 }(1)
Im{M̄2 }
−
0.05
0.01
Re{M̄2 }(1) −M2
2
λ2 φ(1)
(
)
2 2
φ̄
− λ32π
32π
0
−0.05
−0.1
λ = 15, M2 = 0.5
−0.15
0
0
0.5
1
1.5
2
√
s
2.5
3
– for small λ one has Mp2 ' M 2
3.5
4
−0.2
−0.25
0
1
2
√
3
4
5
s
– the algorithm converges quickly
– the 2-loop 2PI result is close to a
perturbative one even at large λ
however, this doesn’t mean that a 1-loop
perturbative result would be enough, because
we have to know φ̄ and the local part of gap
mass
Quality of the results
Euclidean
m2=-0.5, λ=5, M02=2.1
Cutoff dependence
Minkowski
Residue of the pole: Zd vs. Zs
1.2 × 10−8
1.008
int prec = 10−7
int prec = 10−5
1 × 10−8
2
M̂ 2 /M̂Λ=90
1.006
∆γ(φ̄)/∆γ(φ̄)|Λ=200
1.004
8 × 10−9
Zs − Zd
φ̄/φ̄Λ=200
6 × 10−9
1.002
M2 = 0.1
1
4 × 10−9
0.998
2 × 10−9
0.996
a + b log c(x)/xd
0.994
a + b/xc
0
−2 × 10−9
0
2
4
6
8
10
λ
12
14
16
– discrepancy much smaller than
the integration precision
– the needed integration precision
increases with λ
18
20
fit: a + b log(x)/xc
0.992
0
50
100
150
200
Λ
– compared to Fejős & Szép, PRD84,
056001, the curvature mass and the
effective potential is now calculated
– the quantities “converge” with
increasing cutoff
Padé approximants
According to Wikipedia: the ”best” approximation of a function by a given order rational function.
Let the (n, m)-th
Pn orderi Padé approximant of the function f (x) be
aix
i=0
Pm
Pn,m(x) =
, then the first n+m Taylor-series coefficients of f (x)
j
1 + j=1 bj x
and Pn,m(x) are the same:
f (x) = x/(ex − 1)
f (0) = Pn,m(0)
0
0
f (0) = Pn,m(0)
00
(radius of convergence = 2π):
2
1
00
f (0) = Pn,m(0)
..
(n+m)
f (n+m)(0) = Pn,m
(0).
f
0
2
4
6
8
T4
P2,2
-1
-2
The n + m + 1 equations uniquely determine Pn,m(x), that is ai and bj . Various algorithms
exist for the computation of the (n, m)-th order Padé approximant of a function.
Padé approximants
According to Wikipedia: the ”best” approximation of a function by a given order rational function.
Let the (n, m)-th
Pn orderi Padé approximant of the function f (x) be
aix
i=0
Pm
, then the first n+m Taylor-series coefficients of f (x)
Pn,m(x) =
j
1 + j=1 bj x
and Pn,m(x) are the same:
f (x) = x/(ex − 1)
f (0) = Pn,m(0)
0
0
f (0) = Pn,m(0)
00
(radius of convergence = 2π):
2
1
00
f (0) = Pn,m(0)
..
(n+m)
f (n+m)(0) = Pn,m
(0).
f
0
2
4
6
8
T6
P3,3
-1
-2
The n + m + 1 equations uniquely determine Pn,m(x), that is ai and bj . Various algorithms
exist for the computation of the (n, m)-th order Padé approximant of a function.
Padé approximants
According to Wikipedia: the ”best” approximation of a function by a given order rational function.
Let the (n, m)-th
Pn orderi Padé approximant of the function f (x) be
aix
i=0
Pm
, then the first n+m Taylor-series coefficients of f (x)
Pn,m(x) =
j
1 + j=1 bj x
and Pn,m(x) are the same:
f (x) = x/(ex − 1)
f (0) = Pn,m(0)
0
0
f (0) = Pn,m(0)
00
(radius of convergence = 2π):
2
1
00
f (0) = Pn,m(0)
..
(n+m)
f (n+m)(0) = Pn,m
(0).
f
0
2
4
6
8
T8
P4,4
-1
-2
The n + m + 1 equations uniquely determine Pn,m(x), that is ai and bj . Various algorithms
exist for the computation of the (n, m)-th order Padé approximant of a function.
Multipoint Padé approximants
Rational approximation of a function known at N points (fi = f (xi), i = 1 . . . N ) instead of
N -th derivative at one point?
CN (x) =
aN (x − xN −1)
a1 a2(x − x1)
···
,
1+
1+
1
a finite continued fraction with the notation
1
1
x≡
.
1+
1+x
• Find ai from the conditions CN (xi) = f (xi).
• Can be done recursively, by constructing
g1(zi)
=
ui ,
i = 1, . . . , N ,
gp(z)
=
gp−1(zp−1) − gp−1(z)
, p > 2.
(z − zp−1)gp−1
• ai = gi(xi), the diagonal of an upper triangular matrix, where each row can be determined
from the previous one.
• CN can be converted into rational form → P N −1 , N −1 (N odd) or PN/2−1,N/2 (N even).
2
2
Analytic continuation using multipoint Padé approximants
Testing:
• Take F known on both the Euclidean and Minkowski frequency axis.
• Evaluate F for N imaginary frequencies QE,i (Euclidean result).
• Construct CN (QE).
• Continue to the Minkowski axis, writing F (QM ) := lim CN (QE = −iQM + ε).
ε→0
• Quantify the error evaluating
Z
Q=
QM,max
2
dq (F (q) − F (q)) ,
QM,min
• Two examples: 2PI selfenergy M̄ 2 and similar perturbative selfenergy containing the
perturbative bubble diagram, both with M 2 = 0.5, λ = 5, φ̄ ≈ 0.5566 .
• Vary N and Λ ≡ QE,N to check dependence.
0.08
N
2PI Minkowski
N = 10
N = 50
0.04
2
5
10
15
20
25
30
35
1.5
105 × Q
−0.04
−0.08
45
50
Q0 (Λ)
Q(Λ)
Q0 (N)
Q(N)
√
ℜM̄ 2 ( s) − M 2
0
40
1
0.5
√
ℑM̄ 2 ( s)
0
2
4
√
s
6
8
10
0
2
3
4
5
Λ
6
7
8
Complex poles using multipoint Padé approximants
• Real poles are very well reproduced, even with relatively small N .
• O(4) linear sigma model, perturbation theory at T = 0, following Chiku et al., PRD 52,
076001 (1998) and Hidaka et al., PRD 67, 056004 (2003).
√
• Exact calculation leads to a complex pole for the σ propagator s = 569 − i · 119 MeV.
• Following the multipoint Padé continuation recipe introduced on the test cases, we find the
√
complex pole at s = 566 − i · 109 MeV.
• Word of caution: the Padé approximation leads to spurious poles in the selfenergy, which in
this case do not disturb significantly the placement of the complex pole. In general one must
be careful though.
0
ℜGσ−1 = 0
Exact
Padé
−50
10−5
−100
−1
√
ℑ s [MeV]
ℑG σ
=0
ρ
−150
10−6
−200
−250
10−7
0
200
400
600
800 1000
√
s [MeV]
1200
1400
1600
−300
300
Exact
Padé
400
500
600
700
√
ℜ s [MeV]
800
900
1000
Applications 1/2
Questions:
2
1. What is the relation of M̄σ2 and M̂σ2 to the pole mass Mσ,p
in the O(4) 2-loop G.M., Reinosa
& Szép, PRD 87, 105001 (2013) truncation?
2. And in the O(4) O(λ2) G.M., Reinosa & Szép, PRD 92, 125035 (2015) truncation?
3. Is there a possibility for a complex σ pole in these truncations?
Method:
• Take the existing data for the selfenergies at small T .
• Extrapolate to Nτ → ∞.
• Use multipoint Padé approximants to continue to the complex plane (Minkowski axis).
Illustration (σ and π spectral functions in O(λ2), m2 = 0.4, λ = 36):
0.025
0.0012
0.020
0.0010
0.015
0.0008
ΡΣ
0.010
ΡΠ
0.0006
0.0004
0.005
0.0002
s
s
1
2
3
4
5
6
1
2
3
4
5
6
Applications 2/2
Answers:
2
2
(≈ 1%), while Mp,σ
≈
/ M̂σ2 (10% − 100%)
1. 2-loop: M̄σ2 ≈ Mp,σ
2
2
2. O(λ2): M̄σ2 ≈ Mp,σ
(≈ 1%), while Mp,σ
≈
/ M̂σ2 (10% − 30%)
3. If M̄σ > 2M̄π the pole is complex, however =Mp,σ /<Mp,σ 1
But! A correct comparison would be between M̂ 2 and the pole of the external
propagator (Γ(2) ≡ Ĝ) (in the O(λ2) case)
Ĝ
−1
δ 2Γ
(Q) ≡ 2 (Q) = Q2 − M̄ 2(Q) + O(φ̄2 log Q) .
δφ
2
Therefore the difference between M̂p,σ
and M̂σ2 can be expected in the order of
2
the difference between Mp,σ
and M̄σ2.
⇒ Reinforces the original parametrization procedure based on M̂σ2.
Summary
• We solved the 2-loop 2PI truncation of the scalar φ4 model in Minkowski
space at T = 0.
• Using this solution we tested Padé approximants as a method of analytic
continuation to find spectral functions and real poles from the Euclidean
two-point function.
• We found that even complex poles of resonances may be reproduced using
Padé approximants.
• Analytic continuation of pre-existing Euclidean data was used to find the
(practically real) pole of the internal propagator Ḡ, and the estimation of the
pole of the external propagator Ĝ.