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QCD with axial hemial p otential and p ola rization of dilepton p ro dution in heavy-ion ollisions Alexander A. Andrianov Saint-P etersburg State Universit y HSQCD 2014 Axial hemial p otential 1 Outline
SLIDE 1QCD with axial hemi al p
tential
and p
la
rization
f
dilepton p ro du tion in heavy-ion
llisions
Alexander A. Andrianov Saint-P etersburg State Universit y HSQCD 2014 Axial hemi al p
tential
1
SLIDE 2Outline
◮
Phenomenology
f
lo al pa rit y b reaking (LPB)
◮
Axial ba ry
n
ha rge T 5 ⇔ axial hemi al p
tential µ
5
◮
Ee tive meson theo ry in a medium with LPB V e to r Meson Dominan e (VMD) app roa h to LPB: Massive Maxw ell-Chern-Simons ele tro dynami s (CFJ mo del) A.Andrianov, V.Andrianov, D. Esp riu & X. Planells, Phys. Lett. B 710:230 (2012) Ee tive s ala r/pseudos ala r meson theo ry with µ 5 A. A. Andrianov, D. Esp riu & X. Planells, Eur. Phys. J. C , 73:2294 (2013) Phase stru ture
f
the NJL qua rk mo del with µ 5 A. A. Andrianov, D. Esp riu & X. Planells, Eur. Phys. J. C , 74:2776 (2014)
◮
Manifestation
f
LPB in heavy ion
llisions
(HIC): dilepton p
la
rization asymmetry A. A. Andrianov, V. A. Andrianov, D. Esp riu & X. Planells, a rXiv:1402.2147 HSQCD 2014 Axial hemi al p
tential
2
SLIDE 3Phenomenology
f
lo al pa rit y b reaking P a rit y: w ell established global symmetry
f
strong intera tions. Reasons to b elieve it ma y b e b rok en in a nite volume?! quantum u tuations
f θ
pa rameter (P
dd
bubbles [T. D. Lee and G. C. Wi k . . . ℄: their manifestation in Chiral Magneti Ee t (CME))[D. E. Kha rzeev, A.Zhitnitsky , L. D. M Lerran, K.F ukushima, H. J. W a rringa (an ea rlier p rop
sal:
A.Vilenkin, 1980)℄ New QCD phase ha ra terized b y a sp
ntaneous
pa rit y b reaking due to fo rmation
f
neutral pion-lik e ba kground [A.A.Anselm . . . . . . A. A., V. A. Andrianov & D. Esp riu℄ High energy p ro du tion
f
pseudos ala r gluelumps ⇔ pa rit y-o dd bun hes
f
gluon jets ⇒ then a PB ba kground remains inside a hot dense nu lea r reball in HIC !? HSQCD 2014 Axial hemi al p
tential
3
SLIDE 4Phenomenology
f
lo al pa rit y b reaking P a rit y: w ell established global symmetry
f
strong intera tions. Reasons to b elieve it ma y b e b rok en in a nite volume?! quantum u tuations
f θ
pa rameter (P
dd
bubbles [T. D. Lee and G. C. Wi k . . . ℄: their manifestation in Chiral Magneti Ee t (CME))[D. E. Kha rzeev, A.Zhitnitsky , L. D. M Lerran, K.F ukushima, H. J. W a rringa (an ea rlier p rop
sal:
A.Vilenkin, 1980)℄ New QCD phase ha ra terized b y a sp
ntaneous
pa rit y b reaking due to fo rmation
f
neutral pion-lik e ba kground [A.A.Anselm . . . . . . A. A., V. A. Andrianov & D. Esp riu℄ High energy p ro du tion
f
pseudos ala r gluelumps ⇔ pa rit y-o dd bun hes
f
gluon jets ⇒ then a PB ba kground remains inside a hot dense nu lea r reball in HIC !? HSQCD 2014 Axial hemi al p
tential
3
SLIDE 5Phenomenology
f
lo al pa rit y b reaking P a rit y: w ell established global symmetry
f
strong intera tions. Reasons to b elieve it ma y b e b rok en in a nite volume?! quantum u tuations
f θ
pa rameter (P
dd
bubbles [T. D. Lee and G. C. Wi k . . . ℄: their manifestation in Chiral Magneti Ee t (CME))[D. E. Kha rzeev, A.Zhitnitsky , L. D. M Lerran, K.F ukushima, H. J. W a rringa (an ea rlier p rop
sal:
A.Vilenkin, 1980)℄ New QCD phase ha ra terized b y a sp
ntaneous
pa rit y b reaking due to fo rmation
f
neutral pion-lik e ba kground [A.A.Anselm . . . . . . A. A., V. A. Andrianov & D. Esp riu℄ High energy p ro du tion
f
pseudos ala r gluelumps ⇔ pa rit y-o dd bun hes
f
gluon jets ⇒ then a PB ba kground remains inside a hot dense nu lea r reball in HIC !? HSQCD 2014 Axial hemi al p
tential
3
SLIDE 6Phenomenology
f
lo al pa rit y b reaking P a rit y: w ell established global symmetry
f
strong intera tions. Reasons to b elieve it ma y b e b rok en in a nite volume?! quantum u tuations
f θ
pa rameter (P
dd
bubbles [T. D. Lee and G. C. Wi k . . . ℄: their manifestation in Chiral Magneti Ee t (CME))[D. E. Kha rzeev, A.Zhitnitsky , L. D. M Lerran, K.F ukushima, H. J. W a rringa (an ea rlier p rop
sal:
A.Vilenkin, 1980)℄ New QCD phase ha ra terized b y a sp
ntaneous
pa rit y b reaking due to fo rmation
f
neutral pion-lik e ba kground [A.A.Anselm . . . . . . A. A., V. A. Andrianov & D. Esp riu℄ High energy p ro du tion
f
pseudos ala r gluelumps ⇔ pa rit y-o dd bun hes
f
gluon jets ⇒ then a PB ba kground remains inside a hot dense nu lea r reball in HIC !? HSQCD 2014 Axial hemi al p
tential
3
SLIDE 7Phenomenology
f
lo al pa rit y b reaking Lo al la rge u tuations in the top
logi al
ha rge p resumably exist in a hot environment. F
r
p eripheral heavy ion
llisions
they lead to the Chiral Magneti Ee t (CME): La rge B ⇒ la rge E ⇒ ha rge sepa ration. D. Kha rzeev, R. D. Pisa rski & M. H. G. T ytgat, Phys. Rev. Lett. 81, 512 (1998) K. Bu kley , T. F ugleb erg, & A. Zhitnitsky , Phys. Rev. Lett. 84 (2000) 4814 D. E. Kha rzeev, L. D. M Lerran and H. J. W a rringa, Nu l. Phys. A 803, 227 (2008) F
r
entral
llisions
(and light qua rks) they
rresp
nd
to an ephemeral phase with axial hemi al p
tential µ
5 = lo ated in "u tons"
f
few-F ermi size. A. A. Andrianov, V. A. Andrianov, D. Esp riu & X. Planells, Phys. Lett. B 710 (2012) 230. HSQCD 2014 Axial hemi al p
tential
4
SLIDE 8Axial ba ry
n
ha rge and axial hemi al p
tential
QCD has a non-trivial va uum stru ture with dierent top
logi al
se to rs. T
p
logi al
ha rge T 5 ma y a rise in a nite volume due to quantum u tuations in a hot medium due to sphaleron transitions [Manton, M Lerran, Rubak
v,
Shap
shnik
v℄
T 5 = 1 8π 2
vol.
d 3 xε jkl T r
G
j∂ k G l − i 2 3 G j G k G l
and
survive fo r a sizeable lifetime in a heavy-ion reball
∆
T 5 = fo r
∆
t ≃ τ reball ≃ 5 ÷ 10 fm. HSQCD 2014 Axial hemi al p
tential
5
SLIDE 9Axial ba ry
n
ha rge and axial hemi al p
tential
Latti e simulation
f
lo al u tuations
f
the top
logi al
ha rge in the QCD va uum [Leinw eb er℄. HSQCD 2014 Axial hemi al p
tential
6
SLIDE 10Axial ba ry
n
ha rge and axial hemi al p
tential
F
r
the reball lifetime
ne
an
ntrol
the value
f ∆
T 5 intro du ing into the QCD Lagrangian a top
logi al
hemi al p
tential µθ
in a gauge inva riant w a y via ∆L top = µθ∆ T 5 , where
∆
T 5 = T 5( t f ) − T 5( 0) = 1 8π 2
t
f dt
vol.
d 3 x T r
G µν
Gµν
.
The pa rtial
nservation
f
axial urrent (b rok en b y gluon anomaly)
∂µ
Jµ 5 − 2im q J 5 = N f 8π 2 T r
G µν
Gµν
p
redi ts the indu ed axial ha rge (in the hiral limit m q ≃ 0) d dt ( Q q 5 − 2N f T 5) ≃ 0, Q q 5 =
vol.
d 3 x¯ qγ 0γ 5 q = N L − N R to b e
nserved
during τ reball . HSQCD 2014 Axial hemi al p
tential
7
SLIDE 11Axial ba ry
n
ha rge and axial hemi al p
tential
The ha ra teristi left-right
s illation
time is governed b y inverse qua rk masses. F
r
u, d qua rks 1/ m q ∼ 1/ 5 Me V− 1 ∼ 40 fm ≫ τ reball and the left-right qua rk mixing an b e negle ted. F
r
s qua rk 1/ m s ∼ 1/ 150 Me V− 1 ∼ 1 fm ≪ τ reball and Q s 5 ≃ due to left-right
s illations.
F
r
u, d qua rks QCD with a top
logi al
ha rge ∆ T 5 = an b e equally des rib ed at the Lagrangian level b y top
logi al
hemi al p
tential µθ
r
b y axial hemi al p
tential µ
5
∆
T 5 ≃ 1 2N f
Q
q 5 ⇐
⇒ µ
5 ≃ 1 2N f
µθ, ∆L
top = µθ∆ T 5 ⇐
⇒ ∆L
q = µ 5 Q q 5 HSQCD 2014 Axial hemi al p
tential
8
SLIDE 12Axial ba ry
n
ha rge and axial hemi al p
tential
LPB is investigated in e.m. intera tions
f
leptons and photons with hot/dense nu lea r matter via heavy ion
llisions.
e.m. intera tion implies Q q 5 → ˜ Q 5 = Q q 5 − T em 5 , T em 5
=
N 8π 2
vol.
d 3 x ε jkl T r
ˆ
A j∂ k ˆ A l
. µ
5 is
njugated
to (nea rly)
nserved ˜
Q 5 Bosonization
f ˜
Q 5 with hiral Lagrangian, VMD...
∆ T
5
⇐ ⇒ µ
5 The ⇐
=
sense means that if
ne
is able to t µ 5 from phenomenology ,
∆
T 5 an b e found using kno wn te hniques
n
the latti e. Ho w do es ˜ Q 5 ae t the hadroni phenomenology? HSQCD 2014 Axial hemi al p
tential
9
SLIDE 13Ee tive meson theo ry in a medium with LPB V e to r mesons Lo w energy QCD an b e des rib ed with the help
f
V e to r Meson Dominan e
L
int = ¯ qγµ ˆ V µ q;
ˆ
Vµ ≡ − eAµ Q + 1 2 gωωµI + 1 2 gρρ
µτ
3,
( Vµ,a) ≡
Aµ, ωµ, ρ
µ
where
Q = τ 3 2 + 1 6, gω ≃ gρ ≡ g ≃ 6. In this framew
rk,
the follo wing term is generated in the ee tive lagrangian fo r ve to r mesons
∆L ≃ εµνρσ
T r
ˆ
ζµ
Vν Vρσ
with ˆ
ζµ = ˆ ζδµ
fo r a spatially homogeneous and isotropi ba kground (ˆ ≡ isospin
ntent)
and ζ ∝ µ 5 . T w
dierent
ases
f
isospin stru ture fo r µ 5 :
◮
Isosinglet pseudos ala r ba kground (T ≫ µ ) [RHIC, LHC℄
◮
Pion-lik e (isotriplet) ba kground (µ ≫ T ) [F AIR, NICA℄ HSQCD 2014 Axial hemi al p
tential
10
SLIDE 14Ee tive meson theo ry in a medium with LPB Massive MCS ele tro dynami s fo r ve to r mesons
L
MCS = − 1 4 F αβ( x) Fαβ( x) + 1 2 m 2 Aν( x) Aν( x) + 1 2 ζµ Aν( x)˜ F µν( x) + g. f. In momentum spa e w ave Eqs. g λν k 2 − m 2
−
k λ k ν + i ε λναβ ζα kβ
aλ( k) =
k λ aλ( k) = Energy sp e trum: T ransversal p
la
rizations K µ
ν ε ν ±( k)
=
k
2 − m 2 ±
√
D
ε µ
±( k);
ω
k , ±
=
k
2 + m 2 ± ζ 0| k |;
ζµ = (ζ
0, 0, 0, 0) Longitudinal p
la
rization
ω
k , L
=
k
2 + m 2 HSQCD 2014 Axial hemi al p
tential
11
SLIDE 15V e to r Meson sp e trum in PB medium After diagonalization
f
mass matrix m 2 V ,ǫ = m 2 V − ǫζ| k|
= ⇒ |ζ|,
where ǫ = 0, ± 1 is the meson p
la
rization. The photon itself happ ens to b e unae ted b y a singlet ˆ
ζ
. The p
sition
f
the p
les
fo r ± p
la
rized mesons is hanging with w ave ve to r | k|. Massive ve to r mesons split into three p
la
rizations with masses m 2 V ,+ < m 2 V , L < m 2 V ,− . This splitting unambiguously signies LPB. Can it b e measured?
→
dilepton p ro du tion in HIC from the de a ys ρ, ω → e+ e− HSQCD 2014 Axial hemi al p
tential
12
SLIDE 16Manifestation
f
LPB in heavy ion
llisions
L, ±
ntribution
fo r ve to r mesons b efo re a eptan e
rre tions:
dNǫ ee dM ≃ V
α
2Γ V m 2 V 3π 2 g 2 M 2
M
2 − n 2 V m 2
π
m 2 V − n 2 V m 2
π
3/
2
×
ǫ
∞
M dk
k
2 0 − M 2 e k 0/ T − 1 m 4 V ,ǫ
M
2 − m 2 V ,ǫ
2
+
m 4 V ,ǫ
Γ
2 V m 2 V
,
where n V = 2, 0; | k| =
k
2 0 − M 2 and M 2 > n 2 V m 2
π
. V abso rbs
mbinato
rial fa to rs dierent fo r ρ and ω , µ , nite volume supp ression. Empiri ally fo r µ 5 = the ratio ρ/ ω ∼ 10 holds. Simulations implemented with PHENIX a eptan e: | y ee| < 0. 35,
|
p e t | > 200 Me V, gaussian M ee smea ring (width=10 Me V) HSQCD 2014 Axial hemi al p
tential
13
SLIDE 17Manifestation
f
LPB in heavy ion
llisions
ρ
sp e tral fun tion P
la
rization splitting in ρ sp e tral fun tion fo r LPB ζ = 400 Me V (µ 5 = 290 Me V)
mpa
red with ζ = (shaded region). POLARIZA TION ASYMMETRY!! HSQCD 2014 Axial hemi al p
tential
14
SLIDE 18Manifestation
f
LPB in heavy ion
llisions
ρ
sp e tral fun tion P
la
rization splitting in ρ sp e tral fun tion fo r LPB ζ = 400 Me V (µ 5 = 290 Me V)
mpa
red with ζ = (shaded region). POLARIZA TION ASYMMETRY!! HSQCD 2014 Axial hemi al p
tential
14
SLIDE 19Manifestation
f
LPB in heavy ion
llisions
Disto rted sp e tral fun tions In-medium ρ and ω hannels (solid and dashed line) and their va uum
ntributions
(light and da rk shaded regions) fo r ζ = 400 Me V. In-medium ρ is enhan ed b y a fa to r 1.8 due to ππ regeneration into ρ. ENHANCEMENT OF DILEPTON YIELD!! HSQCD 2014 Axial hemi al p
tential
15
SLIDE 20Manifestation
f
LPB in heavy ion
llisions
Disto rted sp e tral fun tions In-medium ρ and ω hannels (solid and dashed line) and their va uum
ntributions
(light and da rk shaded regions) fo r ζ = 400 Me V. In-medium ρ is enhan ed b y a fa to r 1.8 due to ππ regeneration into ρ. ENHANCEMENT OF DILEPTON YIELD!! HSQCD 2014 Axial hemi al p
tential
15
SLIDE 21Manifestation
f
LPB in heavy ion
llisions
PHENIX/ST AR anomaly
Abno
rmal e+ e− ex ess in entral HIC HSQCD 2014 Axial hemi al p
tential
16
SLIDE 22Manifestation
f
LPB in heavy ion
llisions
PHENIX/ST AR anomaly
Abno
rmal e+ e− ex ess in entral HIC HSQCD 2014 Axial hemi al p
tential
17
SLIDE 23P
la
rization asymmetry: dilepton angula r distributions A. A. Andrianov, V. A. Andrianov, D. Esp riu & X. Planells, a rXiv:1402.2147 Case A: the angle θ A b et w een the t w
utgoing
leptons in the lab
rato
ry frame. The ρ sp e tral fun tion is p resented fo r xed µ 5 = 300 Me V.
s θ
A ∈ [− 0. 2, 0], [ 0, 0. 2], [ 0. 2, 0. 4], [ 0. 4, 0. 6] and [ 0. 6, 0. 8] in the left panel, and
s θ
A ≥ − 0. 2, 0, 0. 2, 0. 4 in the right
ne.
Results a re
rre ted
fo r PHENIX exp erimental a eptan e. HSQCD 2014 Axial hemi al p
tential
18
SLIDE 24P
la
rization asymmetry: dilepton angula r distributions Case B: the angle θ B b et w een
ne
f
the t w
utgoing
leptons in the lab
rato
ry frame and the same lepton in the dilepton rest frame The ρ sp e tral fun tion is p resented fo r xed µ 5 = 300 Me V.
s θ
B ∈ [ 0. 3, 0. 4], [ 0. 4, 0. 5], [ 0. 5, 0. 6] and [ 0. 6, 0. 7] in the left panel, and
s θ
B ≤ 0. 4, 0. 5, 0. 6, 0. 7 in the right
ne.
Results a re
rre ted
fo r PHENIX exp erimental a eptan e. HSQCD 2014 Axial hemi al p
tential
19
SLIDE 25Con lusions LPB not fo rbidden b y any physi al p rin iple in QCD at nite temp erature/densit y . T
p
logi al
u tuations transmit their inuen e to hadroni physi s via an axial hemi al p
tential.
LPB leads to unexp e ted mo di ations
f
the in-medium meson p rop erties. LPB ma y help explaining the enhan ed dilepton yield in the LMR
f
PHENIX and ST AR. Measurements
f
the lepton p
la
rization asymmetry ma y reveal in an unambiguous w a y the existen e
f
LPB. HSQCD 2014 Axial hemi al p
tential
20
SLIDE 26Ee tive s ala r/pseudosala r meson theo ry with µ 5 A. A. Andrianov, D. Esp riu & X. Planells, Eur. Phys. J. C , 73:2294 (2013) The s ala r se to r an b e estimated b y using the spurion te hnique in the hiral Lagrangian with an isosinglet µ 5 Dν =
⇒
Dν − i{ I qµ 5δ 0ν, ·} = Dν − 2i I qµ 5δ 0ν. T w
new
p ro esses a re lik ely to app ea r inside the reball: the de a ys η, η′ → ππ that a re stri tly fo rbidden in QCD
n
pa rit y grounds. In a medium where pa rit y is b rok en: a re these p ro esses relevant within the reball? Can these pa rti les rea h thermal equilib rium?! HSQCD 2014 Axial hemi al p
tential
21
SLIDE 27Ee tive s ala r/pseudosala r meson theo ry with µ 5 No O( p 2) terms in the hiral Lagrangian involving verti es ηππ .
O(
p 4) terms lead to
uplings
su h as
Lηππ ∼
16µ 5 Fη f 2
π
L ∂η ∂π∂π, where L ∼ 10−3 . A rough estimate
f
the pa rtial width sho ws a strong dep enden e
n µ
5 as Γη→ππ ∝ µ 2 5 and gives values simila r
r
higher than Γρ→ππ = 150 Me V. An ee tive Lagrangian is needed to in lude the lightest isos ala r degrees
f
freedom su h as σ( 600) and a 0( 980) , whi h will b e mixed with their pseudos ala r pa rtners η, η′ and π , resp e tively . HSQCD 2014 Axial hemi al p
tential
22
SLIDE 28Ee tive s ala r/pseudosala r meson theo ry with µ 5 Generalized Σ mo del Ee tive Lagrangian:
L =
1 4 T r
Dµ
HDµ H†
+
b 2 T r
M(
H + H†)
+
M 2 2 T r
HH†
−λ
1 2 T r
(
HH†) 2
− λ
2 4
T
r
HH†
2
+
2( det H + detH†)
+
d 1 2 T r
M(
HH† H + H† HH†)
+
d 2 2 T r
M(
H + H†)
T
r
HH†
where H = ξΣξ,
ξ =
exp
i Φ
2f
,
Φ = λ
aφ a,
Σ = λ
bσ b. The v.e.v.
f
the neutral s ala rs a re dened as v i = Σ ii where i = u, d, s , and satisfy the follo wing gap equations: M 2 v i − 2λ 1 v 3 i − λ 2 v 2 v i + v u v d v s v i
=
0. HSQCD 2014 Axial hemi al p
tential
23
SLIDE 29Ee tive s ala r/pseudosala r meson theo ry with µ 5 Generalized Σ mo del F
r
further purp
ses
w e need the non-strange meson se to r and η s
Φ = η
q + π
√
2π+
√
2π−
η
q − π
√
2η s
, Σ =
v u + σ + a
√
2a+
√
2a− v d + σ − a v s
η
q
η
s
=
s ψ
sin ψ
−
sin ψ
s ψ
η η′
F
r µ
5 = 0, w e assume v u = v d = v s = v 0 ≡ fπ ≈ 92 Me V. The
upling
nstants
(in units
f
Me V) a re tted to phenomenology assuming isospin symmetry via χ 2 minimization (MINUIT): b = − 3510100/ m, M 2 = 1255600, = 1252. 2, λ 1 = 67. 007,
λ
2 = 9. 3126, d 1 = − 1051. 7/ m, d 2 = 523. 21/ m, where m ≡ m q = ( m u + m d)/ 2 and m/ m s ≃ 1/ 25. HSQCD 2014 Axial hemi al p
tential
24
SLIDE 30Ee tive s ala r/pseudosala r meson theo ry with µ 5 Generalized Σ mo del V a uum: fo r non-vanishing isosinglet µ 5 w e imp
se
ur
solutions to b e v u = v d = v q = v s . HSQCD 2014 Axial hemi al p
tential
25
SLIDE 31Ee tive s ala r/pseudosala r meson theo ry with µ 5 New eigenstates
f
strong intera tions with LPB (isotriplet) W e p resent a simple ase
f
mixing due to LPB in the isotriplet se to r with π and a . The kineti and mixing terms in the Lagrangian a re given b y
L =
1 2(∂ a 0) 2 + 1 2(∂π) 2 − 1 2 m 2 1 a 2 0 − 1 2 m 2 2π 2 − 4µ 5 a 0 ˙
π,
where m 2 1 = − 2[ M 2 − 2( 3λ 1 + λ 2) v 2 q − λ 2 v 2 s
−
v s + 2( 3d 1 + 2d 2) mv q + 2d 2 m s v s + 2µ 2 5] m 2 2 = 2m v q
b + (
d 1 + 2d 2) v 2 q + d 2 v 2 s
HSQCD
2014 Axial hemi al p
tential
26
SLIDE 32Ee tive s ala r/pseudosala r meson theo ry with µ 5 New eigenstates
f
strong intera tions with LPB (isotriplet) After diagonalization in the momentum rep resentation, the new (momentum-dep endent) eigenstates a re dened ˜
π
and ˜ a . HSQCD 2014 Axial hemi al p
tential
27
SLIDE 33Ee tive s ala r/pseudosala r meson theo ry with µ 5 New eigenstates
f
strong intera tions with LPB (isotriplet) F
r
high energies k 0, | k| > m 1 m 2/( 4µ 5) ≡ k
˜ π
, in-medium ˜
π
go es ta hy
ni .
Nevertheless, energies a re alw a ys p
sitive
(no va uum instabilities). ˜ a mass sho ws an enhan ement, but µ 5 has to b e understo
d
as a p erturbatively small pa rameter. A b etter treatment
f ˜
a w
uld
require heavier degrees
f
freedom. HSQCD 2014 Axial hemi al p
tential
28
SLIDE 34Ee tive s ala r/pseudosala r meson theo ry with µ 5 New eigenstates
f
strong intera tions with LPB (isosinglet) In the isosinglet se to r, w e sho w the mixing
f η
, σ and η′ . The kineti and mixing terms in the Lagrangian a re given b y
L =
1 2[(∂σ) 2 + (∂η q) 2 + (∂η s) 2] − 1 2 m 2 3σ 2 − 1 2 m 2 4η 2 q − 1 2 m 2 5η 2 s
−
4µ 5σ ˙
η
q − 2
√
2 v qη qη s, where m 2 3 = − 2( M 2 − 6(λ 1 + λ 2) v 2 q − λ 2 v 2 s + v s
+
6( d 1 + 2d 2) mv q + 2d 2 m s v s + 2µ 2 5), m 2 4 = 2m v q
b + (
d 1 + 2d 2) v 2 q + d 2 v 2 s
+
2 v s, m 2 5 = 2m s v s
[
b + 2d 2 v 2 q + ( d 1 + d 2) v 2 s ] + v 2 q v s
.
HSQCD 2014 Axial hemi al p
tential
29
SLIDE 35Ee tive s ala r/pseudosala r meson theo ry with µ 5 New eigenstates
f
strong intera tions with LPB (isosinglet) After diagonalization, the new eigenstates a re ˜
σ
, ˜
η
and ˜
η′
. HSQCD 2014 Axial hemi al p
tential
30
SLIDE 36Ee tive s ala r/pseudosala r meson theo ry with µ 5 New eigenstates
f
strong intera tions with LPB (isosinglet) Again, fo r high energies k 0, | k| > k
˜ η
with k
˜ η ≡
m 3 4µ 5 m 5
m
2 4 m 2 5 − 8 2 v 2 q , in-medium ˜
η
go es ta hy
ni .
˜ η′
mass also sho ws an enhan ement and a b etter treatment w
uld
require the in lusion
f
heavier degrees
f
freedom. HSQCD 2014 Axial hemi al p
tential
31
SLIDE 37Ee tive s ala r/pseudosala r meson theo ry with µ 5 De a y widths The ubi
uplings
used to al ulate the widths ˜
η, ˜ σ, ˜ η′ → ˜ π˜ π
from the Lagrangian a re given b y
Lσ
aa = 2[( 3d 1 + 2d 2) m − 2( 3λ 1 + λ 2) v q]σ a 2 0,
Lσππ =
1 v 2 q
(∂
π)
2 v q − ( b + 3( d 1 + 2d 2) v 2 q + d 2 v 2 s ) m
π
2
σ, Lη
aπ = 2 v 2 q
a
0[∂η q∂
π
v q − ( b + ( 3d 1 + 2d 2) v 2 q + d 2 v 2 s ) mη q
π], Lσ
aπ = − 4µ 5 v q
σ
a 0 ˙
π,
Lη
aa = − 2µ 5 v q
˙ η
q a 2 0,
Lηππ =
0. After diagonalization,
ne
repla es the initial {η q, η s, σ} to
{˜ η, ˜ σ, ˜ η′}
and {π, a 0} to {˜
π, ˜
a 0} . The widths a re rstly
mputed
at the rest frame
f
the de a ying pa rti le and se ondly with a b
sted
pa rti le. HSQCD 2014 Axial hemi al p
tential
32
SLIDE 38Ee tive s ala r/pseudosala r meson theo ry with µ 5 De a y widths (at rest)
˜ η
exhibits a smo
th
b ehaviour with Γ˜
η ∼
60 Me V ↔ mean free path ∼ 3 fm L reball ∼ 5 ÷ 10 fm. P
ssible
thermalization! Do wn to µ 5 ∼ 100 Me V, ˜
σ
width de reases and b e omes stable. The visible bumps in these t w
hannels
seem to ree t the ta hy
ni
nature
f
the de a ying ˜
π
. ˜
η′
width gro ws up to the Ge V s ale (violation
f
unita rit y). Mo re degrees
f
freedom a re needed. HSQCD 2014 Axial hemi al p
tential
33
SLIDE 39Ee tive s ala r/pseudosala r meson theo ry with µ 5 De a y widths (moving ˜
η
) Small va riations at lo w 3-momenta: 2 initial bumps slo wly sepa rate as
ne
in reases q . Γ˜
η(µ
5, q) exhibits a saddle p
int
at µ∗ 5 ∼ 240 Me V and q∗ ∼ 500 Me V. F
r
la rge 3-momenta, a third intermediate bump app ea rs ( reation
f
2 ta hy
ns)
and gro ws fast as
ne
in reases q b e oming the global maximum when q 700 Me V. HSQCD 2014 Axial hemi al p
