Non-perturbative methodsand
Chiral Perturbation Theoryapplied to
Meson–Meson and Nucleon–Nucleoninteractions
Seminar for JLab & Indiana U. for Amplitude Analysis
Miguel Albaladejo
IFIC, U. Valencia
January 18, 2013
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Short biographical presentation
Name Miguel Albaladejo SerranoBorn August 6, 1981, Murcia (Spain)
Undergrad. U. of MurciaGrad. U. of Valencia
Master Th. Meson-Meson scattering up to√
s = 2 GeV and its spectroscopyDate June 15, 2007Advisors E. Oset (Valencia), J. A. Oller (Murcia)
Ph.D. U. of MurciaPh.D. Th. Non-perturbative methods and Chiral Perturbation Theory
applied to Meson-Meson and Nucleon-Nucleon interactionsDate December 4, 2012Advisor J. A. Oller (Murcia)
Postdoc IFIC, Valencia U.Date January 1, 2013 - July 31, 2013
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Main publications
I authors: M. Albaladejo and J.A. Ollertitle: Identification of a Scalar Glueballjournal: Phys. Rev. Lett. 101, 252002 (2008)
I authors: M. Albaladejo, J.A. Oller and L. Rocatitle: Dynamical generation of pseudoscalar resonancesjournal: Phys. Rev. D 82 094019 (2010)
I authors: M. Albaladejo and J.A. Ollertitle: Nucleon-Nucleon Interactions from Dispersion Relations: Elastic Partial Wavesjournal: Phys. Rev. C 84, 054009 (2011)
I authors: M. Albaladejo and J.A. Ollertitle: Size of the σ meson and its naturejournal: Phys. Rev. D 86, 034003 (2012)
I authors: M. Albaladejo, J.A. Oller, E. Oset, G. Rios and L. Rocatitle: Finite volume treatment of ππ scattering and limits to phase shifts extractionfrom lattice QCDjournal: JHEP 1208, 071 (2012)
I authors: M. Albaladejo and J.A. Ollertitle: Nucleon-Nucleon Interactions from Dispersion Relations: Coupled Partial Wavesjournal: Phys. Rev. C 86, 034005 (2012)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Outline of the seminar
Part I: The Scalar Sector and the Scalar Glueball
Part II: Scalar-Pseudoscalar scattering andheavy pseudoscalar resonances
Part III: On the size and the nature of the σ meson
Part IThe Scalar sector and the scalar glueball
[M. Albaladejo and J. A. Oller, Phys. Rev. Lett. 101, 252002 (2008)]
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Introduction
Objective and tools:Problem: Scalar mesons identification. How many? Where?σ, f0(980), f0(1370), f0(1500), f0(1710)...
Very broad resonances, strongly coupled channels open up in the nearby ofresonances, which have very different natures: dynamically generated, qq,glueballs. . .Our objective: study of strongly interacting channels with quantum numbersI = 0, I = 1/2 JPC = 0++ for
√s 6 2 GeV.
We use Chiral Lagrangians, implementing Unitarity through N/D (UChPT).Coupled channel dynamics. Channels to be considered:
I = 0: ππ, KK , ηη, σσ, ηη′, ρρ, ωω, η′η′, ωφ, φφ, K∗K∗, a1(1260)π, π?πI = 1/2, I = 3/2: Kπ, Kη and Kη′[Jamin,Oller,Pich,NPB,587,331(’00)][NPB,622,279(’02)]
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Lagrangians and AmplitudesU(3) O(p2) Lagrangian
Gasser and Leutwyler
L = L2 + LS8 + LS1
N
Coupled channel partial wave: N/D[Oller,Oset,PR,D60,074023(’99)]
T = (I + N(s)g(s))−1 N(s)
T N N T= +
Large Nc limit implies η′ becomes the ninth Goldstone boson: SU(3)→ U(3).
L2 =f 2
4〈DµU†DµU〉+
f 2
4〈χ†U + χU†〉 −
12
M21η
21
DµU = ∂µU − irµU + iUlµ = ∂µU − ig [vµ,U]
In our case, rµ = lµ = gvµ: vectorial resonances nonet. Yang-Mills fields,minimal coupling, local gauge symmetry.Mixing: η1, η8 → η, η′. Mixing angle is θ ≈ −20.
N(s): Nij amplitudes matrix, obtained from Lagrangian, i , j = 1, . . . 13g(s): Unitarization loops (diagonal) matrix (including all intermediate states).
gi (s) = gi (s0)− s − s08π2
∫ ∞sth,i
ds ′ pi (s ′)/√
s ′(s ′ − s0)(s ′ − s + iε)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Lagrangians (II)
Vector field vµ is given by:
vµ =
ρ0√2
+ ω8√6
ρ+ K∗+
ρ− − ρ0√2
+ ω8√6
K∗0
K∗− K∗0 − 2ω8√6
+1√3ω1
We assume ideal mixing between ω8, ω1 to generate physical φ, ω.Derivative piece of L2 has interactions of: ΦΦ, V ΦΦ y VV ΦΦ:
LΦΦ2 = f 2
4 〈∂µU∂µU†〉 LVV ΦΦ2 = g2
⟨Φ2vµvµ − vµΦvµΦ
⟩LV ΦΦ2 = − igf 2
4 〈∂µU[vµ,U†] + [vµ,U]∂µU†〉g is determined through decay width ρ→ ππ, from LV ΦΦ
2 , being g = 4.23
We introduce explicit resonances (JPC = 0++) from RChPT[Ecker et al.,NPB,321,311(’88)][PLB,223,425(’89)].
LS8 = cd 〈S8uµuµ〉+ cm 〈S8χ+〉 , LS1 = cd S1 〈uµuµ〉+ cmS1 〈χ+〉
χ+ = u†χu† + uχ†u, U(x) = u(x)2,uµ = iu†DµUu† = u†µ
S(i)1 singlet and S(i)
8 octet scalar resonance, with M, cd and cm fitted to data.
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
σσ channel amplitudes. Pion rescatteringWe want to obtain σσ amplitudes starting from our lagrangian. σ is S-wave ππinteraction, |σ〉 = |ππ〉0, [Oller,Oset,NPA,620,438(’97)]
Ti
=⇒Ti
D−1
D−1
=⇒
1s−sσ
1s−sσ
Ti
gσ
gσPion rescattering, given by factor D−1(s) = (1 + t2G(s))−1, with:
t2 =s −m2
π/2f 2π
(4π)2G(s) = α+ logm2π
µ2− σ(s) log
σ(s)− 1σ(s) + 1
To isolate transition amplitude Ni→σσ :
limsi→sσ
Ti→(ππ)0(ππ)0
DII (s1)DII (s2)=
Ni→σσg2σππ
(s1 − sσ)(s2 − sσ)=⇒ Ni =
g2σππ
t2(sσ)2Ti→(ππ)0(ππ)0
Now, close to the σ pole:
t2(s1)
DII (s1)=
t2(s1)
1 + t2(s1)GII (s1)' −
g2σππ
s1 − sσ+ regular terms
Further, gσππ can be calculated through:
1g2σππ
= −d
ds1
(t2(s1)−1 + GII (s1)
)=
1t22(s1)f 2
−dGII (s1)
ds1
∣∣∣sσ
=⇒g2σ
t2(sσ)2' f 2
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Results: free parametersM (MeV) cd (MeV) cm (MeV)
S(1)8 1290± 5 25.8± 0.5 25.8± 1.1
S(2)8 1905± 13 20.3± 1.4 −13.9± 2.0
S(1)1 894± 13 14.4± 0.3 46.6± 1.1
First octet (M(1)8 , c(1)
d , c(1)m ) fixed to:
[Jamin,Oller,Pich,NPB,587,331(’00)][NPB,622,279(’02)].
Second octet has only fixed its mass, M(2)8 , but not its couplings.
Singlet mass M(2)1 is more difficult to fix, but for lower values, we can obtain
the same physics with higher couplings.
Since SU(3) breaking is milder in the vector sector, we take just onesubstraction constant for the whole set of vector channels,
aρρ = aωω = aK?K? = aωφ = aφφ
Together, we have aππ, aKK , aηη, aηη′ , aη′η′ and aσσ.
Total: just 12 free parameters, for about 370 data points.
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Results: Comparison with experimental data
0
90
180
270
360
450
500 750 1000 1250 1500 1750
δ0 0
(deg
) 0102030
275 350 425
00.5
11.5
500 750 1000 1250 1500 1750
η0 0
√s (MeV)
180
270
3601000 1250 1500 1750
δ1,
2(d
eg)
Etkin et al.Cohen et al.
0
0.5
1
1000 1250 1500 1750
| S1,
2|
√s (MeV)
0
0.1
0.2
1000 1250 1500 1750 2000
| S1,
3|2
√s (MeV)
0
0.04
0.08
0.12
1500 1600 1700 1800 1900
∣ ∣ S 1,5∣ ∣2
√s (MeV)
0306090
120150180210240
750 1000 1250 1500 1750
φ(d
eg)
√s (MeV)
50
100
150
200
250
750 1000 1250 1500 1750
A
√s (MeV)
ππ → ππ: Although reaching energies of 2GeV, description of low energy data is stillrather goodππ → KK : Near threshold, Cohen data arefavored.ππ → ηη,ηη′: In good agreement for a lowweight on χ2. In addition, the data areunnormalized.I = 1/2 K−π+ → K−π+ amplitude andphase from LASS.
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Spectroscopy. Pole content: Summary
I = 0√
s (MeV) Resonance M (PDG) Γ (PDG)456± 6− i 241± 7 σ ≡ f0(500)983± 4− i 25± 4 f0(980) 980± 10 40− 1001466± 15− i 158± 12 ← f L
0 f0(1370) 1200− 1500 200− 5001602± 15− i 44± 15 ← f R
0 f0(1500) 1505± 6 109± 71690± 20− i 110± 20 f0(1710) 1724± 7 137± 81810± 15− i 190± 20 f0(1790) 1790+40
−30 270+30−60
I = 1/2√
s (MeV) Resonance M (PDG) Γ (PDG)708± 6− i 313± 10 κ ≡ K∗0 (800) − −1435± 6− i 142± 8 K∗0 (1430) 1414± 6 290± 211750± 20− i 150± 20 K∗0 (1950) − −
About the σ meson, we will discuss laterRemarkable agreement in f0(980) width with Madrid Group[García-Martín et al.,PRL,107,072001(’11)].We discuss now in detail each resonance/pole in detail
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Spectroscopy. Pole content: f0(1370)
f0(1370)
f L0 :√
s = 1466± 15− i 158± 12 MeV
Shift in mass peak:Mf0(1370) ' 1370 MeV.Strong coupling to σσ, ππΓ(f0(1370)→4π)Γ(f0(1370)→ππ)
= 0.30± 0.12, in goodagreement with the interval0.10− 0.25 of [Bugg,EPJC,52,55(’07)]
M (MeV) Γ (MeV)PDG 1200− 1500 200− 500Our 1370± 15 316± 24
1000 1100 1200 1300 1400 15000
50
100
150
200
250
Re√s (MeV)
Im√s (MeV)
0
500
1000
1500
2000
2500
3000
1000 1100 1200 1300 1400 1500
|Tσσ→σσ|2
Re√s (MeV)
√s ' 1.37 GeV
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Spectroscopy. Pole content: f0(1500)
f0(1500)
f R0 :√
s = 1602± 15− i 44± 15 MeVThe mass peak is at 1.5 GeV, due toηη′ threshold. Effect similar toa0(980) with the KK threshold[Oller,Oset,PR,D60,074023(’99)].The width is Γ ≈ 1.2×88 ' 105 MeV,because a BW at (1.6− i0.04) GeV iscut by ηη′ thresold.Complicated region. Importantcoupled channel dynamics. Threeinterfering effects give raise tof0(1500):
1 f R0 : Pole at (1.6− i0.04) GeV
2 f0(1370): Pole at (1.47− i0.16) GeV3 Nearby thresholds: ηη′, ωω
M (MeV) Γ (MeV)PDG 1505± 6 109± 7Our 1502 105± 36
1300 1400 1500 1600 17000
50
100
150
200
250
fR0
fL0
Re√s (MeV)
Im√s (MeV)
0
1000
2000
3000
4000
5000
6000
1300 1400 1500 1600 1700
|Tηη′ →
ηη′ |2
Re√s (MeV )
ηη′ threshold
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Spectroscopy. Pole content: f0(1710)
f0(1710)
√s = 1690± 20− i 110± 20 MeV
Shifted in mass, bump at√
s ≈ 1.7GeV.The width “measured” on the real axisis Γeff ' 160 MeV.The effective width can depend on theconcrete process.
M (MeV) Γ (MeV)PDG 1724± 7 137± 8Our 1700± 20 160± 40BR Our value PDG
Γ(KK)Γ
0.36± 0.12 0.38+0.09−0.19
Γ(ηη)Γ
0.22± 0.12 0.18+0.03−0.13
Γ(ππ)
Γ(KK)0.32± 0.14 0.41+0.11
−0.17Γ(ηη)
Γ(KK)0.64± 0.38 0.48± 0.15
1500 1550 1600 1650 1700 1750 18000
50
100
150
200
f0(1710)
Re√s (MeV)
Im√s (MeV)
0
6000
12000
18000
24000
1500 1600 1700 1800
|Tη′ η
′ →η′ η
′ |2
Re√s (MeV)
Γ ' 160 MeV
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Spectroscopy. Pole content: f0(1790)
f0(1790)
√s = 1810± 15− i 190± 20 MeV
Weak signal on the real axisIt couples weakly to KK , a majordifference with respect to f0(1710), asalso observed by BESII.It is the partner of the pole at1.75− i0.15 GeV in I = 1/2.These poles originate from the higherbare octet, S(2)
8 , with M(2)8 = 1905,
c(2)d = 20.3, c(2)
m = −13.9 MeV.
M (MeV) Γ (MeV)BESII 1790+40
−30 270+30−60
Our 1810± 15 380± 40
1700 1750 1800 1850 19000
100
200
300
400
Re√s (MeV)
Im√s (MeV)
0
400
800
1200
1600
2000
1700 1750 1800 1850 1900
|Tηη′ →
ηη′ |2
Re√s (MeV)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
WA102 and CBC data
0
2
4
6
8
500 1000 1500
Eve
nts/
0.02
GeV
√s (MeV)
×104
WA102 pp→ ppπ+π−
0
1
2
3
1000 1250 1500 1750
Eve
nts/
0.04
GeV
√s (MeV)
×103
WA102 pp→ ppK+K−
0
1
2
3
4
1250 1500 1750
Eve
nts/
0.08
GeV
√s (MeV)
×102
WA102 pp→ ppηη
0
3
6
9
12
1250 1500 1750
Eve
nts/
9.2
MeV
√s (MeV)
×103
CBC pp→ π0ηη
2
4
6
8
10
12
1500 1550 1600 1650 1700 1750E
vent
s/12
MeV
√s (MeV)
×101
CBC pp→ π0ηη ′
Similarly as done by WA102 Collaboration, we put a coherent sum of Breit–Wigners:Ai (√
s) = NRi (√
s) +∑
jaj eiθj gj;i
M2j −s−iMj Γj
, j =
σ, f0(980), f0(1370), f R
0√
s 6 mηη′σ, f0(980), f0(1710), f0(1790)
√s > mηη′
Note that Mj , Γj and gj;i are fixed from the previous study on scattering (the poles onprevious slide), without changing them.A constant is included to ensure continuity.
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Couplings of f R0 and f0(1710): the lightest scalar glueball
[Chanowitz,PRL,95,172001(’05)]: Chiral suppression mechanism of the coupling of theglueball to qq. Coupling ∝ mq. This implies |gss | |gnn|The coupling pattern of f0(1710) suggests an enhancement in ss production.
Coupling (GeV) f R0 f0(1710)
gπ+π− 1.31± 0.22 1.24± 0.16gK0K0 2.06± 0.17 2.00± 0.30gηη 3.78± 0.26 3.30± 0.80gηη′ 4.99± 0.24 5.10± 0.80gη′η′ 8.30± 0.60 11.70± 1.60
Coupling (GeV) f R0 f0(1710)
gss 11.5± 0.5 13.0± 1.0gns −0.2 2.1gnn −1.4 1.2
gss/6 1.9± 0.1 2.1± 0.2
We can obtain gss , gns and gnn from gηη, gηη′ and gη′η′ (mixing anglesinβ = −1/3):
3gη′η′ = 2gss + gnn + 2√2gns
3gηη′ = −√2gss +
√2gnn + gns
3gηη = gss + 2gnn − 2√2gns
√3η = −ηs +
√2ηn√
3η′ =√2ηs + ηn
ηs = ss ηn = (uu + dd)/√2
Together with the OZI rule requires |gss | |gns |.This is precisely what we obtain from the previous couplings and equations.A similar reasoning on KK leads to expect gK0K0 = gss/6.
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Lattice support to the Chiral Supression Mechanism
Couplings of the lightest pseudoscalar glueball from quenched latticecalculations [Sexton et al.,PRL,75,4563(’95)]:
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8
λ mρ
(mps
mρ
)2
0++ glueballChiral suppression
This result is compatible with the chiral suppression mechanism[Chanowitz,PRL,95,172001(’05)]
Mass [Chen et al.,PRD,73,014516(’06)]:
Mgb0++ = 1.66± 0.05 GeV
Mf0(1710) = 1.69± 0.02 GeV
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Summary
We have performed the first coupled channel study of the meson-mesonS-waves for I = 0 and I = 1/2 up to 2 GeV with 13 coupled channels.Interaction kernels are determined from Chiral Lagrangians and implemented inN/D–type equations, avoiding ad-hoc parameterizationsWe have fewer free parameters, compared with previous works in the literature,for a vast quantity of data.All scalar resonances below 2 GeV are generated:
I = 0: σ, f0(980), f0(1370), f0(1500), f0(1710) and f0(1790)I = 1/2: κ, K∗0 (1430) and K∗0 (1950)
The structure of the couplings to pairs of pseudoscalars implies that:f0(1370), K∗0 (1430) (and a0(1450)) remain as pure octet.f R0 and f0(1710), which are the same pole reflected on different sheets, have adecay pattern showing an enhanced ss production, in agreement with the chiralsuppression mechanism of glueball decay, soIt should be considered as the lightest unmixed 0++ glueball.
Part IIDynamical generation of pseudoscalar resonances
[M. Albaladejo, J. A. Oller and L. Roca, Phys. Rev. D 82, 094019 (2010)]
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Introduction – MotivationExcited Pseudoscalars above 1 GeV
I = 1 π(1300), π(1800)
I = 1/2 K(1460), K(1630)
I = 0
η(1295), η(1405), η(1475)︸ ︷︷ ︸η(1440)?
[Masoni et al.,JPG,32(’06)][Klempt,Zaitsev,PR,454(’07)][Wu et al.,PRL,108(’11)]
I = 0 X(1835)?Exotic quantum numbers? Glueballs?[Chanowitz,PRL,46,981(’81)][Ishikawa,PRL,46,978(’81)][Close,PRD,55,5749(’97)]
π+
K+K 0
π−
K− K 0
π0
η1η8
General Idea
K
K
K
f0(980)
K
K(1460)
Scalars dynamically generatedin PS–PS interactions.
Is it possible to generatedynamically heavier PS withS–PS interactions?
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
FormalismOur approach
S1(p1) S2(p2)
P1(k1) P2(k2)
g1 g2
T (sin)
q
p1 − q p2 − q
S–PS interaction kernels are given by thetriangle diagram. Internal lines are K[Alarcon et al.,PR,D80;PR,D82(’10)]
Basic point: this diagram is enhancedbecause MSi ' 2mK (f0(980), a0(980))
Kernel
LK = 2ig1 g2
∫d4q
(2π)4T ((P − q)2)
(q2 − m2K + iε)((p1 − q)2 − m2
K + iε)((p2 − q)2 − m2K + iε)
Dependence of the vertices is known:sin = (P − q)2 ⇔ Inserting a dispersion relationg1, g2 ⇔ couplings of S1,2 to KK . Residues. No (new) free parameter.
Dispersion Relation
T (sin) = T (sA) +∑
i
sin − sA
sin − si
Resi
si − sA+
sin − sA
π
∫ ∞sth
ds′ImT (s′)
(s′ − sin)(s′ − sA)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Loop functions from the DR and S-wave projection
The insertion of the Dispersion Relation induces the appearence of three and fourpoint one loop functions, that we project onto S partial wave
p1 p2
p4 p3
` + p1
` ` + p1 + p2
p1 p2
p4 p3
` + p1
`− p3 − p4
`− p4
`
C3 =i2
∫ +1
−1d cos θ
∫d4`
(2π)41
((`+ p1)2 −m21)(`2 −m2
2)((`+ p1 + p2)2 −m23)
C4 =i2
∫ +1
−1d cos θ
∫d4`
(2π)41
((`+ p1)2 −m21)(`2 −m2
2)((`− p3 − p4)2 −m23)((`− p4)2 −m2
4)
These loop functions are worked out analytically and numerically, and checked,when possible, against OneLOop [van Hameren, Papadopoulos, Pittau,JHEP0909,106(’09)]
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Coupling the channels
IG Channels0+ (η) a0π, f0η, f0η′1/2 (K) f0K , a0K1− (π) f0π, a0η, a0η′1+ a0π3/2 a0K
T =TL
1 + TLG
As an example, the I = 1/2 amplitudes areconstructed as:
TL(f0K → f0K) =g2
f02
LK [3T I=1KK→KK + T I=0
KK→KK ]
TL(f0K → a0K) =
√3gf0ga02
LK [T I=1KK→KK − T I=0
KK→KK ]
TL(a0K → a0K) =g2
a02
LK [3T I=0KK→KK + T I=1
KK→KK ] .
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ResultsI = 1/2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
1.2 1.4 1.6 1.8
|T(f
0K→f 0K
)|2/1
05
√s (GeV)
K(1460)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
1.2 1.4 1.6 1.8
|T(a
0K→a0K
)|2/1
05
√s (GeV)
K(1460)
f0(980)K and a0(980)K channelsClear resonant peak M ' 1460 MeVΓ ' 100 MeV (Exp. Γ ' 250 MeV)Generated in three-body interactionsalso [Martinez-Torres et al.,PR,D84(’11)]
asub = −0.5 (Natural sized)|gf0K/ga0K | ' ( 20.2
5.7 )1/4 ' 1.4K?(892)π not included (but couplesto K(1460))
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ResultsI = 1
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
1 1.2 1.4 1.6 1.8 2
|T(a
0η′→a0η′ )|2/105
√s (GeV)
π(1800)
0.0
0.5
1.0
1.5
2.0
2.5
3.0
1 1.2 1.4 1.6 1.8
|T(a
0η→a0η)|2/1
05
√s (GeV)
π(1300)
f0(980)π, a0(980)η and a0(980)η′
a0η′ almost elastic (higher threshold)aa0η′ = −1.3⇒ π(1800)
Γ ' 200 MeV (Exp. Γ = 208± 12 MeV)
af0π,a0η = −2 ⇒ π(1300)
M ' 1.2 GeV, Γ ' 200 MeV.Generated in three-body interactionsalso [Martinez-Torres et al.,PR,C83(’11)]
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ResultsI = 0
0
2
4
6
8
10
12
14
16
1.2 1.4 1.6 1.8 2
|T(f
0η′→f 0η′ )|2/1
05
√s (GeV)
X(1835)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 1.2 1.4 1.6 1.8
|T(f
0η→f 0η)|2/105
√s (GeV)
η(1475)
a0(980)π, f0(980)η and f0(980)η′
a0η′ almost elastic (higher threshold)aa0η′ = −1.2⇒ X(1835)
Γ ' 70 MeV (Exp.Γ = 67.7± 20.3± 7.7 MeV)
aa0π,f0η ' −1 ⇒ η(1475) ?M ' 1.45 GeVΓ ' 150 MeV (Exp. 85± 9 MeV)K?(892)K not included (but couplesto η(1475))
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ResultsI = 0
0
2
4
6
8
10
12
14
16
1.2 1.4 1.6 1.8 2
|T(f
0η′→f 0η′ )|2/1
05
√s (GeV)
X(1835)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 1.2 1.4 1.6 1.8
|T(f
0η→f 0η)|2/105
√s (GeV)
η(1475)
Regarding η(1475):Resonant peak at M ' 1.45 GeVCouples mostly to a0π, as η(1475) ⇒ identify it as η(1475)
Low mass and absence of η(1405) ⇒ possible interpretation as single η(1440)[Wu et al.,PRL,108(’11)]
We do not find any signal for η(1295)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ResultsI = 3/2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
1 1.2 1.4 1.6 1.8 2
∣ ∣ T(a± 0K
±→
a± 0K
±)∣ ∣2 /1
05
√s (GeV)
Single channel: a0(980)±K± (pure I = 3/2).Exotica1 = −0.5 (as in I = 1/2)Resonant behavior, in accordance with aprediction of Longacre [Longacre,PRD,42,874(’90)]
This channel was not isolated in I = 1/2(K(1460)) experiments[Daum et al.,NPB,187,1(’81)][Brandenburg et al.,PRL,36,1239(’76)]
The other exotic channel IG = 1+ shows noresonant behavior
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Summary
We have studied the S-PS interactions in the region 1− 2 GeVScalars: f0(980), a0(980) dynamically generated in PS-PS interactionsWe have studied the π,K ,η-alike channels, as well as the exotic channelsThese interactions dynamically generate some resonances experimentally seenWhence, a large contribution to these physical signals has a dynamical origin
Resonance IG Width (MeV) PropertiesK(1460) I = 1
2 Γ & 100 |gf0K/ga0K | ' 1.4π(1800) IG = 1− Γ ' 200 a0η′ elasticπ(1300) IG = 1− Γ & 200 a0π, f0η coupled channelsX(1835) IG = 0+ Γ ' 70 f0η′ elasticη(1475) IG = 0+ Γ ' 150 f0η elasticExotic I = 3
2 Γ ' 200 a0K threshold
Part IIIOn the size and the nature of the σ meson
[M. Albaladejo and J. A. Oller, Phys. Rev. D 86, 034003 (2012)]
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Introduction
We now make a specific study of the σ meson nature and sizeElastic, low energy ππ interactions are considered (SU(2) Chiral Lagrangians)O(p4) ππ amplitude (T4) is calculated. It involves LECs (li) as well as loopfunctions:
T (s) =V (s)
1 + V (s)G(s)
' V2(s) + V4(s)− V 22 (s)G(s) + · · ·
≡ T2(s) + T4(s) + · · ·V2(s) = T2(s)
V4(s) = T4(s) + T 22 (s)G(s)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Comparison with other determinations
Γσ/2
(MeV
)
Mσ (MeV)
ABCDEFGH
220
240
260
280
300
420 440 460 480 500
Ref. Mσ (MeV) Γ/2 (MeV) a00 b00M2
π
A 441+16−8 272+9
−13 - -B 484± 17 255± 10 0.233± 0.013 0.285± 0.012C 457+14
−13 279+11−7 - -
D 463± 6+31−17 254± 6+33
−34 0.218± 0.014 0.276± 0.013E 452± 12 260± 15 - -F 470± 50 285± 25 - -G 456± 12 241± 14 - -This work 440± 10 238± 10 0.219± 0.005 0.281± 0.006Average 453± 5 258± 5 0.220± 0.003 0.279± 0.003Mean 458± 14 261± 17 0.223± 0.007 0.280± 0.004New PDG 400− 550 200− 350 − −
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Dependence of sσ with mπ
400
500
600
700
800
900
1000
1100
1200
0 100 200 300 400 500 600
Mσ
(MeV
)
mπ (MeV)
FuIAM
LONLO
Dyn. n = 1Dyn. n = 2Que. n = 1Que. n = 2
0
50
100
150
200
250
300
350
0 50 100 150 200 250 300 350
Γσ/2
(MeV
)
mπ (MeV)
IAMLO
NLO
0
2
4
6
8
10
12
0 100 200 300 400 500 600 700
s σ/m
2 π
mπ (MeV)
Re(sσ ) LO
Re(sσ ) NLO
−Im(sσ ) LO
−Im(sσ ) NLO
3.7
3.8
3.9
4
300 400 500 600 700
Re(
s σ/m
2 π)
mπ (MeV)
fπ, li depend on mπ (ChPT).Agreement with IAM and Lattice.Evolution of the σ pole:
1 Physical pion mass: pole at II RS(resonance).
2 Mσ < 2mπ , Γσ → 0 (II RS still)mπ ' 310–330 MeV.
3 Mσ . 2mπ (threshold) and jumpsto I RS (bound state).
References:[Prelovsek et al.,PRD,82,094507(’10)](Lattice)[Pelaez,Rios,PRD,82,114002(’10)] (IAM)[Fu,CPL,28,081202(’11)] (Lattice)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Scalar form factor and quadratic scalar radius of the σ
Now, we study the scalar form factor of the σ meson, FσFrom this form factor we can extract the quadratic scalar radiusWe need the amplitude for ππ scattering in the presence of a scalar source,H(q)
=⇒H(q)
=⇒H(q)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ππH → ππ amplitudeKinematics
π(p1)
π(p2)
π(p3)
π(p4)
H(q)
Mandelstam-like variables:
s = (p3 + p4)2 s ′ = (p1 + p2)2
t = (p1 − p3)2 t ′ = (p2 − p4)2
u = (p1 − p4)2 u′ = (p2 − p3)2
s + t + u + s ′ + t ′ + u′ = q2 + 8m2π
Perturbative Amplitude
Calculated from O(p4) chiral lagrangians:
li , i = 1, . . . , 4.Loops: A0, B0(p2), C0(s, s ′, q2)
Amplitude An(s, s ′, t, t ′, u, u′, q2)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ππH → ππ amplitudeResummed Amplitude
TS (s, s′, q2) =W (s, s′, q2)(
1 + V (s)G(s))(
1 + V (s′)G(s′))
= W2 + W4
−W2V (s)G(s)−W2V (s′)G(s′) + · · ·= Φ2 + Φ4 + · · ·
• Resummed amplitude• Kernel• Perturbative amplitude
Matching both series, we have:W2 = Φ2
W4 = Φ4 + W2V (s)G(s) + W2V (s′)G(s′)Φn is the angular projection of the amplitude An:
Φn =
∫d2ΩAn(s, s′, t, t′, u, u′, q2)
Perturbative Amplitude
Calculated from O(p4) chiral lagrangians:
li , i = 1, . . . , 4.Loops: A0, B0(p2), C0(s, s ′, q2)
Amplitude An(s, s ′, t, t ′, u, u′, q2)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Scalar form factor of the σ meson
H(q)
11+V (s′)G(s′)
11+V (s)G(s)
W (s, s′, q2)
=⇒ gσ gσ
Fσ(q2)
1s′−sσ
1s−sσ
H(q)
Fσ(q2) g2σ
(s − sσ)(s ′ − sσ)= lim
s,s′→sσ
W (s, s ′, q2)
(1 + V (s)G(s)) (1 + V (s ′)G(s ′))
Close to the σ pole, sσ, we have:
V (s)
1 + V (s)G(s)
s'sσ= − g2σ
s − sσ+ · · ·
Scalar Form Factor of the σ meson
Fσ(q2) =g2σ
V (sσ)2W (sσ, sσ, q2)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Radius of the σ meson
From Fσ(q2) we can get 〈r 2〉σS :
Fσ(q2) = Fσ(0)
(1 +
q2
6 〈r2〉σS + · · ·
)⇐⇒ 〈r 2〉σS =
6Fσ(0)
∂Fσ(q2)
∂q2
∣∣∣∣q2=0
0
0.5
1
1.5
2
2.5
3
50 100 150 200 250 300 350 400 450 500 550 600
√ 〈r2〉σ s
(fm
)
mπ (MeV)
Re√〈r2〉σs
−Im√〈r2〉σs
[1] [Jamin et al.,PRD,74(’06)]
[2] [Oller,Roca,PL,B651(’07)]
[3] [Amendolia et al.,NPB,277(’86)]
[4] [Dally et al.,PRL,48(’82)]
[5] [de Forcrand,Liu,PRL,69(’92)][Hou et al..PRD,64(’01)]
Diverges:in the chiral limit as logmπ (as〈r2〉πS,V ): infinite pion cloudwhere mσ = 2mπ : zero binding energybound state
Imaginary part is small, despite large σwidthAround 0.5 fm in a wide range of mπ
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Radius of the σ meson
From Fσ(q2) we can get 〈r 2〉σS :
Fσ(q2) = Fσ(0)
(1 +
q2
6 〈r2〉σS + · · ·
)⇐⇒ 〈r 2〉σS =
6Fσ(0)
∂Fσ(q2)
∂q2
∣∣∣∣q2=0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
50 100 150 200 250 300 350
〈r2 〉
σ s(f
m2 )
mπ (MeV)
Re 〈r2〉σs−Im 〈r2〉σs
[1] [Jamin et al.,PRD,74(’06)]
[2] [Oller,Roca,PL,B651(’07)]
[3] [Amendolia et al.,NPB,277(’86)]
[4] [Dally et al.,PRL,48(’82)]
[5] [de Forcrand,Liu,PRL,69(’92)][Hou et al..PRD,64(’01)]
At physical mπ :
〈r2〉σS = (0.19± 0.02)− i (0.06± 0.02) fm2√〈r2〉σS = (0.44± 0.03)− i (0.07± 0.03) fm
For comparison:
〈r2〉Kπs = 0.1806± 0.0049 fm2[1]
〈r2〉πs = 0.65± 0.05 fm2[2]
〈r2〉π±
V = 0.439± 0.008 fm2[3]
〈r2〉K±
V = 0.28± 0.07 fm2[4]
Scalar glueballs (expected): 0.1− 0.2 fm[5]
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Naive view of the 4q vs. ππ molecule question
Molecule
π π
q
q q
q
4q
σq
q q
q
Intermediate
π π
q
q q
q
〈r 2〉σS is an estimation of σ’s size.Pion would overlap so much thatthey are tightly packed. σ is acompact state.Hence, a 4q nature seems moreadequate than a molecule picture.For large mπ values, mσ . 2mπ, andσ becomes a 2π molecule.
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Naive view of the 4q vs. ππ molecule question
Molecule
π π
q
q q
q
4q
σq
q q
q
0
0.5
1
1.5
2
2.5
3
50 100 150 200 250 300 350 400 450 500 550 600
√ 〈r2〉σ s
(fm
)
mπ (MeV)
Re√〈r2〉σs
−Im√〈r2〉σs
〈r 2〉σS is an estimation of σ’s size.Pion would overlap so much thatthey are tightly packed. σ is acompact state.Hence, a 4q nature seems moreadequate than a molecule picture.For large mπ values, mσ . 2mπ, andσ becomes a 2π molecule.
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Summary
We have performed a detailed calculation of ππ scattering at low energyThe σ meson pole (440± 10− i238± 10 MeV) is well reproduced andcompared with other valuesWe have studied the dependence of the pole position with the pion mass,finding remarkable agreement with lattice and IAM resultsThe σ meson quadratic scalar radius is calculated, finding an small value for it(4q vs. molecule question)The Feynman–Hellmann theorem has been checked for the σ meson
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Other works I have done. . .
NN interactionsfrom Dispersion Relations
[Albaladejo, Oller, PR,C84(’11)][Albaladejo, Oller, PR,C86(’12)]
0
10
20
30
40
50
60
70
80
0 50 100 150 200 250 300
deg
rees
|p| (MeV)
1S0-25
-20
-15
-10
-5
0
0 50 100 150 200 250 300
deg
rees
|p| (MeV)
1P1
0
10
20
30
40
50
60
70
0 50 100 150 200 250 300
|p| (MeV)
3P0
-25
-20
-15
-10
-5
0
0 50 100 150 200 250 300
|p| (MeV)
3P1
Finite volume effects in ππ interactions[M. Albaladejo et al., JHEP 1208,071(’12)]
200
300
400
500
600
700
800
900
1.5 2 2.5 3 3.5 4 4.5 5
E(M
eV)
Lmπ
I = 0
IAMN/D
BS
1
2
3
4
0
10
20
30
40
50
60
70
80
90
100
300 400 500 600 700 800
δ(d
eg)
√s (MeV)
I = 0
L = ∞ IAMn = 2 IAML = ∞ N/Dn = 2 N/D
Now working, among other things, at heavy-meson molecules (DD,. . . ) in a FiniteVolume and possibilities of finding them in Lattice QCD
Non-perturbative methodsand
Chiral Perturbation Theoryapplied to
Meson–Meson and Nucleon–Nucleoninteractions
Seminar for JLab & Indiana U. for Amplitude Analysis
Miguel Albaladejo
IFIC, U. Valencia
January 18, 2013
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References III
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References IV
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References V
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Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
σ width effects on g(s)With sσ complex, Ni→σσ would be complex, thus violating unitarity. So we take sireal, but varying according to a mass distribution.
Lehman propagator representation (dispersion relation):
P(s) = − 1π
∫ ∞√
sth
ds ′ =P(s ′)s − s ′ + iε ,
In a first approach:
=P(s ′) ∝ =(
1s ′ −m2
σ + i mσΓσ(s ′)
)Γσ(s ′) = Γσ
√1− 4m2
π/s ′1− 4m2
π/m2σ∫ ∞
√sth
ds ′=P(s ′) = 1
g(s) (unitarity loop) for σ channel can be written as:∫ ∞√
sth
ds1∫ ∞√
sth
ds2=P(s1)=P(s2) g4(s; s1, s2) ,
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Spectroscopy, poles and Riemann sheets
When parameters are fitted to data, we extrapolate our amplitudes to thes-complex plane, finding poles, and we identify them with resonances through√s0 ≈ M − iΓ/2.Analytical extrapolations are needed in the Tij to the different Riemann sheets,which appear because of the cuts in G(s) at opening thresholds:
p(s) =
√s − (ma + mb)2
√s − (ma −mb)2
2√
s
Using continuity, for√
s real, > ma + mb, we have:
G II (s + iε) = G(s − iε) = G(s + iε)− 2i=G(s + iε) = G(s + iε) +i4π
p(s)√s
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Spectroscopy. Couplings of f0(1370)
f0(1370) (I = 0)Coupling bare (MeV) final (MeV)gπ+π− 3.9 3.59± 0.18gK0K0 2.3 2.23± 0.18gηη 1.4 1.70± 0.30gηη′ 3.7 4.00± 0.30gη′η′ 3.8 3.70± 0.40
K∗0 (1430) (I = 1/2)
Coupling bare (MeV) final (MeV)gKπ 5.0 4.8gKη 0.7 0.9gKη′ 3.4 3.8
Bare coupling are very similar to the physical ones.Bare: those of S(1)
8 , with M(1)8 = 1.29 GeV, c(1)
d = c(1)m = 25.8 MeV.
The first scalar octet is a pure one, not mixed with the nearby f0(1500) norf0(1710)
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ππ scattering at NLO in SU(2)
δ0 0
(deg
)
√s (MeV)
LONLO
AverageFroggatt
S118E865
NA48/20
20
40
60
80
100
120
250 300 350 400 450 500 550 600 650 700 750 800
-505
10152025
275 300 325 350 375
δ2 0
(deg
)
√s (MeV)
LONLO
Hoogland et al.Losty et al.
-30
-25
-20
-15
-10
-5
0
300 400 500 600 700 800 900 1000
We consider I = 0, 2 S-wave. Freeparameters: li (i = 1, . . . , 4) and a(subtraction constant)
Fit √sσ (MeV)LO 465± 2− i 231± 7
NLO 440± 10− i 238± 10Fit a00 b0
0M2π
LO 0.209± 0.002 0.278± 0.005NLO 0.219± 0.005 0.281± 0.006
Fit a l1 l2 l3 l4 χ2
LO −1.36± 0.12 - - - - 1.6NLO −1.2± 0.4 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5 0.7
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
ππ scattering at NLO in SU(2)
f π(M
eV)
mπ (MeV)
NLOBeane et al.
JLQCD
85
90
95
100
105
110
115
120
0 100 200 300 400 500 600
a2 0
mπ (MeV)
NLOBeane et al.Feng et al.
Fu
-0.5
-0.4
-0.3
-0.2
-0.1
0
0 100 200 300 400 500 600
We consider I = 0, 2 S-wave. Freeparameters: li (i = 1, . . . , 4) and a(subtraction constant)
Fit √sσ (MeV)LO 465± 2− i 231± 7
NLO 440± 10− i 238± 10Fit a00 b0
0M2π
LO 0.209± 0.002 0.278± 0.005NLO 0.219± 0.005 0.281± 0.006
Fit a l1 l2 l3 l4 χ2
LO −1.36± 0.12 - - - - 1.6NLO −1.2± 0.4 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5 0.7
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Comparing with other determinations of the LECsRef. l1 l2 l3 l4GL −2.3± 3.7 6.0± 1.3 2.9± 2.4 4.6± 0.9CGL −0.4± 0.6 4.31± 0.11 − 4.4± 0.2ABT 0.4± 2.4 4.9± 1.0 2.5+1.9
−2.4 4.20± 0.18PP −0.3± 1.1 4.5± 0.5 − −GKMS 0.37± 1.96 4.17± 0.47 − −BCT − − − 4.4± 0.3OR − − − 4.5± 0.3DFGS − − −15± 16 4.2± 1.0NA48/2 − − 2.6± 3.2 −RBC-UKQCD − − 2.57± 0.18 3.83± 0.9PACS-CS − − 3.14± 0.23 4.04± 0.19ETM − − 3.70± 0.07± 0.26 4.67± 0.03± 0.1JLQCD/TWQCD − − 3.38± 0.40± 0.24+0.31
−0 4.09± 0.50± 0.52MILC − − 2.85± 0.81+0.37
−0.92 3.98± 0.32+0.51−0.28
This work 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5
l1
GL
CGL
ABT
PP
GKMS
This work
-6 -5 -4 -3 -2 -1 0 1 2 3
l2
GL
CGL
ABT
PP
GKMS
This work
3.5 4 4.5 5 5.5 6 6.5 7 7.5
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Comparing with other determinations of the LECsRef. l1 l2 l3 l4GL −2.3± 3.7 6.0± 1.3 2.9± 2.4 4.6± 0.9CGL −0.4± 0.6 4.31± 0.11 − 4.4± 0.2ABT 0.4± 2.4 4.9± 1.0 2.5+1.9
−2.4 4.20± 0.18PP −0.3± 1.1 4.5± 0.5 − −GKMS 0.37± 1.96 4.17± 0.47 − −BCT − − − 4.4± 0.3OR − − − 4.5± 0.3DFGS − − −15± 16 4.2± 1.0NA48/2 − − 2.6± 3.2 −RBC-UKQCD − − 2.57± 0.18 3.83± 0.9PACS-CS − − 3.14± 0.23 4.04± 0.19ETM − − 3.70± 0.07± 0.26 4.67± 0.03± 0.1JLQCD/TWQCD − − 3.38± 0.40± 0.24+0.31
−0 4.09± 0.50± 0.52MILC − − 2.85± 0.81+0.37
−0.92 3.98± 0.32+0.51−0.28
This work 0.8± 0.9 4.6± 0.4 2± 4 3.9± 0.5
l3
GL
ABT
RBC/UKQCD
MILC
PACS-CS
ETM
JLQCD/TWQCD
NA48/2
This work
-2 -1 0 1 2 3 4 5 6
l4
GLCGLABTBCTDFGSORRBC/UKQCDMILCPACS-CSETMJLQCD/TWQCDThis work
3 3.5 4 4.5 5 5.5
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
An issue with the amplitude. . .
→ 1D1
= 1(p1+q)2−m2
π= 1
q22 −2|~q||~p| cos θ
. Diverges on integration
Upon resummation, one should have: → +
Because on-shell factorization, one really has: → 2×
W(1 + VG)2
' W − 2WVG
Indeed, the perturbative calculation of this diagram gives half this result.This means that this kind of terms should be resummed with a factor(1 + VG)−1 and not (1 + VG)−2. In that case, they do not contribute to thedouble σ-pole necessary for the form factor.Diagrammatically:
=⇒
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
1/Γ vs. r
0
0.5
1
1.5
2
0 100 200 300 400 500 600 700
r(f
m)
mπ (MeV)
√r2
1/Γ1/Γ
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Feynman–Hellmann theorem for the σ meson
1
2
3
4
5
6
7
8
100 200 300 400 500 600
Re
s σ
mπ (MeV)
Re(sσ ) LO
Re(sσ ) NLO
−Re(
Fσ (0)
2B
)NLO
0
1
2
3
4
50 100 150 200 250 300
Ims σ
mπ (MeV)
Im(sσ ) LO
Im(sσ ) NLO
−Im(
Fσ (0)
2B
)NLO
sσ =dsσdm2 =
dsσdm2
π
dm2π
dm2︸︷︷︸ChPT
Feynman–Hellmann theorem
dsσdm2 = −Fσ(0)
2B
It is worth noting that:
These two quantities are obtained fromthe expansion of two differentinteracting kernelssσ : the derivative does not act onexternal legs in ππ diagrams Spares
Scalar sector and glueball Pseudoscalar resonances Size and nature of σ Spares
Feynman–Hellmann theorem for the σ meson
1.4
1.6
1.8
2
2.2
2.4
2.6
50 100 150 200 250 300
Re
s σ
mπ (MeV)
Re(sσ ) LO
Re(sσ ) NLO
−Re(
Fσ (0)
2B
)NLO
0
1
2
3
4
50 100 150 200 250 300
Ims σ
mπ (MeV)
Im(sσ ) LO
Im(sσ ) NLO
−Im(
Fσ (0)
2B
)NLO
sσ =dsσdm2 =
dsσdm2
π
dm2π
dm2︸︷︷︸ChPT
Feynman–Hellmann theorem
dsσdm2 = −Fσ(0)
2BIt is worth noting that:
These two quantities are obtained fromthe expansion of two differentinteracting kernelssσ : the derivative does not act onexternal legs in ππ diagrams Spares
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