pi-eta scattering in generalized chiral perturbation theory

7 Pages·0.19 MB·English

by  J. Novotny

πη scattering in generalized chiral
perturbation theory¶
J. Novotny´†and M. Koles´ar‡
3
Institute of Particle and Nuclear Physics,
0
0 Charles University, V Holeˇsoviˇck´ach 2, 180 00 Czech Republic
2

n
a
J
2 Abstract

We calculate the amplitude of πη scattering in generalized chiral per-
2
v turbation theory at the order O(p4) and present a preliminary results for
1 the numerical analysis of the S-wave scattering length, which seems to be
1 particularly sensitive to the deviations from the standard case.
3
2
1 1 Generalized Chiral Perturbation Theory
2
0
/ As it is well known, on the classical level the QCD Lagrangian with N mass-
h f
p less quarks (corresponding to so-called chiral limit of QCD) is invariant w.r.t.
- chiral symmetry (χS) group SU(N ) × SU(N ) . On the quantum level
p f L f R
e there exist strong theoretical (for Nf ≥ 3) and phenomenological arguments
h for spontaneous symmetry breakdown (SSB) of χS according to the pattern
:
v SU(N ) ×SU(N ) → SU(N ) . As a consequence of Goldstone theorem,
f L f R f V
Xi N2 −1 pseudoscalar Goldstone bosons (GB) appear in the particle spectrum
f
r of the theory. These massless pseudoscalars dominate the low energy dynamics
a
of QCD and interact weekly at low energies E << Λ , where Λ ∼ 1GeV
H H
is the hadronic scale corresponding to the masses of the lightest nongoldstone
hadrons. The most important order parameters of this pattern of SSB are the
Goldstone boson decay constant F and the quark condensate1 hq q i .
0 f f 0
QCD
Within the real QCD the quark mass term L breaks χS explicitly.
f,mass
The GB become pseudogoldstone bosons (PGB) with nonzero masses. Nev-
QCD
ertheless, for m << Λ , L can be treated as a small perturbation.
f H f,mass
As a consequence, the PGB masses M can be expanded in the powers (and
P
logarithms) of the quark masses and the interaction of PGB at energy scale
E << Λ continues to be weak. PGB are identified with π0,π± for N = 2
H f
¶Presented by J. N. at Int. Conf. Hadron Structure ’02, Herlany, Slovakia, September
22-27, 2002
†Jiri.Novotny@mff.cuni.cz
‡Marian.Kolesar@mff.cuni.cz
1The parameter F is however more fundamental in the sense that F 6= 0 means both
0 0
necessary and sufficient condition for SSB, while hq q i 6= 0 corresponds to the sufficient
f f 0
condition only.
1
and π0,π±,K0,K0,K±,η for N = 3. Because M < Λ , the QCD dynam-
f P H
ics at E << Λ is still dominated by these particles and can be described in
H
terms of an effective theory known as chiral perturbation theory (χPT). The
Lagrangian of χPT can be constructed on the base of symmetry arguments
only; the unknown information about the nonperturbative properties of QCD
are hidden in the parameters known as low energy constants (LEC)[1].
In order to be able to treat the effective theory as an expansion in powers of
(p/Λ ) (where p are generic external momenta) and (m /Λ ), it is necessary
H f H
to assign to each term L(m,n) = O(∂mmn) of the effective Lagrangian a single
f
parameter called chiral order. To the terms L with chiral order k it is referred
k
as to O(pk) terms. Obviously, ∂ = O(p). The matter of discussion is, however,
the question concerning the chiral power of m . This question is intimately
f
connected to the scenario according which the SSB of χS is realized.
The standard scenario corresponds to the assumption, that the SSB order
parameter hqqi is large in the sense, that the ratio X
0
2B m
0
X = (1)
M2
π
b
(where B = −hqqi /F2 and m = (m + m )/2) is close to one. Because
0 0 0 u d
M2 = O(p2), it is then natural to take m = O(p2), i.e. k = m+2n. This
π f
results into the Standard χPT b(SχPT in what follows)[2]. This scenario has
been experimentally confirmed [3] for N = 2 and it is perfectly compatible
f
with experiment for N = 3 in π, K sector.
f
LetusnotethatatO(p2)thereisnonefreeparameter, becauseatthischiral
order F = F = 93.2MeV, 2B m = M2 = 135MeV and r = m /m ≃ 26.
0 π 0 π s
AlternativetothiswayofchiralpowercountingisGeneralizedχPT (GχPT)
[4] corresponding to the scenariobwith small quark condensate X <b< 1 so that
it is natural to take m = O(p) and B = O(p). I.e. k = m+n and the O(p2)
f 0
Lagrangian is
F2
L = 0 h∂ U+∂µUi+2B hU+M+M+Ui+A h(U+M)2+(M+U)2i
2 µ 0 0
4
+ Z(cid:0)PhU+M−M+Ui2+ZShU+M+M+Ui2 . (2)
0 0
This scenario is still possible for N = 3, as has been dis(cid:1)cussed in [5]2.
f
In the generalized case, there are two free parameters in the O(p2) effective
Lagrangian, the usual choice is (r,ζ = ZS/A ). Within SχPT at O(p2) we get
0 0
2The point is, that providedwe definethe n-flavorcondensate as
hqqi(n) = lim hqqi ,
0 0
mf→0,f≤n
thetwo-flavorcondensaterelevant fortheχPT withN =2isrelated tothethree-flavorone
f
relevant for theχPT with N =3,
f
huui(02) =huui(03)−msZS1 +...
where ZS is the fluctuation parameter measuring a violation of the Zweig rule in the 0++
1
S
channel. That means, that the three-flavor condensate might be small provided Z is large.
1
Recent phenomenological studies suggest possibility of X(3) ∼1/2, cf. [5] and [6].
2
roughly (r,ζ) = (26,0). Note, that ζ measures the violation of Zweig rule in
the 0++ channel, which is, however, not well under control. In what follows,
we therefore prefer to parametrize the deviations from SχPT directly on terms
of X, i.e. our choice of free parameters is (r = m /m, X), cf. (1) (standard
s
O(p2) values are then (26,1)). The O(p2) LEC can be then expressed in terms
of these free parameters. b
In oder to distinguish between the two scenarios of χS SSB, it is necessary
to find observables, which are sensitive to the deviations from the standard
case. It seems, that the πη scattering might offer such observables, though we
left open the question about their experimental accessibility.
Letusnote, thattheamplitudeofthisprocesswasalreadycalculated within
SχPT to O(p4) (and within the extended SχPT with explicit resonance fields)
in the paper [7], where the authors presented prediction for the scattering
lengths and phase shifts of the S, P and D partial waves. We quote here
their O(p4) results for the S-wave scattering length a (in the units of the pion
0
Compton wavelength): aSχPT = 7.3×10−3 and aSχPT+resonances = 4.9×10−3.
0 0
2 General structure of the πη scattering amplitude
Due to isospin conservation and Bose symmetry, the process is described in
terms of one s−u symmetric invariant amplitude A(s,t;u)
hπbη |πaη i= i(2π)4δ(P −P )δabA(s,t;u)
out in f i
Using analyticity, unitarity, crossing symmetry and assuming chiral expansion
in the same way as in [9] we get the following general form of the O(p4) ampli-
tude3
A(s,t;u) = R(3)(t;s,u)+V (t)+W (s)+W (u)
0 0 0
+[(t−u)s+∆2]W (s)+[(t−s)u+∆2]W (u) (3)
1 1
Here R(3)(t;s,u) is the most general s−u symmetric subtraction polynomial
of the third order4
1 2 λ 2 λ
R(3)(t;s,u) = (α M2+β (t− Σ)+ πη(t− Σ)+ πη(s−u)2
3F2 πη π πη 3 F2 3 F2
π π π
κ 2 κ 2 e
+ πη(s−u)2(t− Σ)+ πη(t− Σ)3). (4)
F4 3 F4 3
π π
e
The unitarity corrections V ,W , W start at O(p4) and are determined by
0 0 1
means of the dispersion integrals along the cuts ((M +M )2,∞) or (4M2,∞)
π η π
with the discontinuities given by the right hand cut discontinuities of the S
and P partial waves in the s and t channel. Using partial waves unitarity, it is
possible to proceed iteratively and determine the relevant O(p4) discontinuities
through the O(p2) amplitudes Aηπ→φαφβ, Aφαφβ→ηπ, Aππ→φαφβ and Aηη→φαφβ.
3Here and in what follows, ∆=M2−M2 and Σ=M2+M2 .
η π η π
4Within the generalized chiral expansion, απη = O(1), βπη = O(p), λπη,λπη = O(p2),
V0,W0,W1 = O(p4) and κπη, κπη = O(p4).
e
e
3
These are real and firstorder polynomials in s, t, u, so that we can parametrize
them with (altogether 11) real parameters. Addingtothis the2extra real coef-
ficients of O(p4) partof the subtraction polynomialR(3)(t;s,u), it is possibleto
parametrize the O(p4) amplitude A(s,t;u) in terms of 13 real free parameters5.
As a result of the iterative procedure we get for the O(p4) unitarity corrections
(4)
V , W , W the following formulas: W (s) = 0 and
0 0 1 1
2
1
W(4)(s) = J (s) α M2
0 πη 3F2 πη π
(cid:18) π (cid:19)
3 1 2 1
+J (s) [β (s− Σ− M2)− (2M2 −Σ+α M2)]2
KK 8F4 πηK 3 3 K 3 K πηK π
π
1 4 5
V(4)(s) = J (s) α M2[β (s− M2)+ α M2]
0 ππ 3F4 πη π ππ 3 π 6 ππ π
π
1 4M2
−J (s) α α M4 1− η
ηη 18F4 ηη πη π M2
π π !
1 2 2 2
+J (s) [β (s− M2− M2)+ ((M −M )2+2α M M )]
KK 8F2 πK 3 π 3 K 3 K π πK K π
π
2
×[β (3s−2M2 −2M2)+α (2M2− M2)]
ηK K η ηK η 3 K
where J (s) is the Chew-Mandelstam function, cf.[9]. The role of χPT is
PQ
then reduced to the determination of the above mentioned parameters in terms
of LEC and quark masses. Let us now briefly comment on the results of the
calculations.
3 πη amplitude in GχPT at O(p4)
Let us write the complete O(p4) amplitude A(s,t,u) in the form
A(s,t,u) = A(2)(s,t,u)+A(3)(s,t,u)+A(4)(s,t,u), where A(k)(s,t,u) = O(pk).
A(2) and A(3) contain the contributions from tree graphs with vertices derived
form Lagrangians L and L = O(p3) respectively (see (2) and e.g. [8]); A(4)
2 3
includes tree graphs with vertices from L = O(p4) as well as 1-loop graphs
4
with vertices from L .
2
BecausetheunitaritycorrectionsstartatO(p4),A(2)(s,t,u)andA(3)(s,t,u)
are both polynomials of the form (cf. (4))
1 2
A(k)(s,t,u) = (α(k)M2+β(k)(t− (M2+M2)), k = 1,2.
3F2 πη π πη 3 π η
π
(2)
Moreover, the amplitude must vanish in the chiral limit, therefore β = 0.
πη
From (2) we get
M2
A(2)(s,t,u) = π α(2),
3F2 πη
π
5In the following formulas, the parameters of the O(p2) amplitudes Aηπ→φαφβ and
Aφαφβ→ηπ are απη, βπη, απηK, βπηK for
φαφβ = πη,KK, the parameters of O(p2) amplitudes Aππ→φαφβ and Aηη→φαφβ are αππ,
βππ, αηη, απK, βπK, αηK, βηK for φαφβ =ππ,ηη, KK.
4
(cid:1)(cid:1)(cid:2)(cid:9)(cid:10)(cid:11)(cid:1)(cid:12)(cid:1)(cid:2)(cid:13)(cid:14)(cid:6)(cid:15)(cid:2)(cid:2)(cid:16)(cid:1)(cid:16)(cid:2)(cid:7)(cid:3)
(cid:1)(cid:3) (cid:17)(cid:18)
(cid:15)(cid:2)(cid:1)
(cid:1)(cid:4)(cid:7)(cid:3)
(cid:15)(cid:2)(cid:2)(cid:7)(cid:3)
(cid:1)(cid:2) (cid:15)(cid:2)(cid:2)
(cid:1)(cid:1)(cid:2)(cid:1)(cid:1)(cid:2)(cid:8)(cid:7)(cid:3)
(cid:3)
(cid:4)(cid:7)(cid:3)
(cid:2)
(cid:1)(cid:2) (cid:1)(cid:3) (cid:4)(cid:2) (cid:4)(cid:3) (cid:5)(cid:2)
(cid:6)
(2)
Figure 1: The dependence of the parameter α on r for X = 1, 0.5 and 0.
πη
The thick solid line corresponds to current algebra result.
where, in terms of the free parameters (r,X)
(1+2r) (2 (1−X)+rε(r)) 2∆
α(2) = 1+ − GMO.
πη (2+r) r−1
In this formula
r −r M2 3M2 +M2−4M2
ε(r) = 2 2 , r = 2 K −1, ∆ = η π K ≃ −3.6. (5)
r2−1 2 M2 GMO M2
π π
The standard O(p2) result (corresponding to the current algebra (CA)) corre-
(2) (2)
sponds to α = 1 [7]. The dependence of α on r and X is shown in Fig. 1.
πη πη
The deviation from the standard case might beeven by a factor ten larger than
thestandardvalue, providedthequarkmass ratio r is smallincomparison with
r and the three-flavor ratio X is smaller than one. This result is encouraging
2
enough to calculate the higher order corrections.
TheO(p3) Lagrangian L contains 9 LEC6, namely ξ, ξ and ρ , i = 1,...,7.
3 i
ξ can be expressed in terms of decay constants as follows
e
∆ F2
mξ = F , ∆ = K −1 ≃ 0.5.
r−1 F F2
π
For the NLO correctiobns in terms of X, ε, ∆ , ∆ , ξ and remaining O(p3)
GMO F
LEC we get
e
β(3) = 4mξ(r+2)
πη
2∆ 2(2r+1)2(1−X) (8+r(6r+13))(r−1)ε ∆
α(π3η) = (rb−eF1)(1− 3(r+2) − 6(r+2) − G3MO)
r2−1 ∆
GMO
−4mξ(2r+1) 1−(X −1)+ ε+
2r+1 2r+1
(cid:18) (cid:19)
8 Σ
+ bmeξ(r+2) +...
3 M2
π
6Someofthem,namelyξ, ρ ,ρ , ρ , ρ ,ρ ,violateZweigruleandmightbeneglected(the
be 3 4 5 6 7
only exception is ξ, which could be relevant as the measure of the fluctuations in the 0++
channel, cf. [5]). e
e
5
(cid:8)(cid:1)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:10)(cid:15)(cid:15)(cid:12)(cid:6)(cid:16)(cid:17)(cid:18)(cid:19)(cid:12)(cid:17)(cid:18)(cid:15)(cid:20)(cid:10)(cid:1)(cid:10)(cid:15)(cid:21)(cid:1)(cid:22)(cid:2)(cid:2)(cid:23)(cid:24)(cid:6)(cid:25)(cid:2)(cid:1) (cid:8)(cid:1)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:10)(cid:15)(cid:15)(cid:12)(cid:6)(cid:16)(cid:17)(cid:18)(cid:19)(cid:12)(cid:17)(cid:18)(cid:15)(cid:20)(cid:10)(cid:1)(cid:10)(cid:15)(cid:21)(cid:1)(cid:22)(cid:2)(cid:2)(cid:23)(cid:24)(cid:6)(cid:25)(cid:2)(cid:1)(cid:3)(cid:4)
(cid:28)(cid:29) (cid:2)(cid:7)(cid:2)(cid:5)(cid:3) (cid:28)(cid:29)
(cid:2)(cid:7)(cid:2)(cid:5)
(cid:25)(cid:2)(cid:1)(cid:26)(cid:6)(cid:2)(cid:6)(cid:3)(cid:26)(cid:3)(cid:2)(cid:27)(cid:2) (cid:2)(cid:7)(cid:2)(cid:5) (cid:25)(cid:2)(cid:1)(cid:26)(cid:6)(cid:2)(cid:6)(cid:3)(cid:26)(cid:3)(cid:2)(cid:27)(cid:2)
(cid:2)(cid:7)(cid:2)(cid:4)(cid:3) (cid:25)(cid:2)(cid:1)(cid:26)(cid:6)(cid:2)(cid:6)(cid:3)(cid:26)(cid:3)(cid:2)(cid:27)(cid:1) (cid:2)(cid:7)(cid:2)(cid:4)(cid:3) (cid:25)(cid:2)(cid:1)(cid:26)(cid:6)(cid:2)(cid:6)(cid:3)(cid:26)(cid:3)(cid:2)(cid:27)(cid:1)
(cid:25)(cid:2)(cid:1)(cid:26)(cid:3)(cid:2)(cid:27)(cid:2) (cid:25)(cid:2)(cid:1)(cid:3)(cid:4)(cid:26)(cid:3)(cid:2)(cid:27)(cid:2)
(cid:1)(cid:2)(cid:2)(cid:1)(cid:1)(cid:2)(cid:7)(cid:2)(cid:4) (cid:25)(cid:2)(cid:1)(cid:26) (cid:3)(cid:2)(cid:27)(cid:1) (cid:1)(cid:2)(cid:2)(cid:1)(cid:1)(cid:2)(cid:7)(cid:2)(cid:4) (cid:25)(cid:2)(cid:1)(cid:3)(cid:4)(cid:26)(cid:3)(cid:2)(cid:27)(cid:1)
(cid:2)(cid:7)(cid:2)(cid:1)(cid:3)
(cid:2)(cid:7)(cid:2)(cid:1)(cid:3)
(cid:2)(cid:7)(cid:2)(cid:1)
(cid:2)(cid:7)(cid:2)(cid:1) (cid:2)(cid:7)(cid:2)(cid:2)(cid:3)
(cid:2)
(cid:2)(cid:7)(cid:2)(cid:2)(cid:3)
(cid:1)(cid:2) (cid:1)(cid:3) (cid:6) (cid:4)(cid:2) (cid:4)(cid:3) (cid:5)(cid:2) (cid:1)(cid:2) (cid:1)(cid:3) (cid:6) (cid:4)(cid:2) (cid:4)(cid:3) (cid:5)(cid:2)
Figure 2: The dependence of the S-wave scattering length on r for X = 1 and
X = 0.5 with µ = M and µ = M . The solid lines mimic the SχPT (X = 1
η ρ
and r = r ) at O(p4). The lowest line corresponds to current algebra result.
2
The ellipses stay for the O(m3) terms which includes the unknown LEC ρ .
f i
TheNNLOO(p4)corrections have thegeneralform(cf. (3))withκ = κ = 0
and with V(4) and W(4) given above. The complete formulas for A(4), which
0 i
are rather lengthy, will be published elsewhere. Let us only briefly commeent on
the result.
As usual, 1-loop graphs which contribute to A(4) are generally divergent;
therefore the renormalization procedure is needed. As a result, the amplitude
depends explicitly on the renormalization scale µ. This scale dependence is
compensated by the implicit scale dependence of the renormalized LEC, this
fact we used as a nontrivial check of our calculations.
Let us also note, that the O(p4) Lagrangian is parametrized by means of
40 LEC, most of them are unknown. Influence of these unknown constant (as
well as that of the unknown O(p3) LEC) can be roughly estimated using the
above mentioned explicite µ dependence of the amplitude. The idea behind is
based on the assumption that the variation of LEC with µ is of the same order
as LEC themselves. Setting all the unknown LEC equal to zero and varying
the scale µ is then (up to the sign) equivalent to the variation of the LEC and
gives therefore information on the impact of the unknown LEC.
Theobservable which seems to besensitive to thedeviation from theSχPT
is the S−wave scattering length a , (note, that at O(p2), a ∝ α(2), while
0 0 πη
P−wave scattering length a starts at O(p3)). InFig. 2we showthenumerical
1
resultsfora asafunctionofthequarkmassratiorforX = 1andX = 0.5with
0
µ = M and µ = M .
η ρ
4 Conclusions
We have calculated the πη scattering amplitude in GχPT to the order O(p4)
andevaluatedtheS−wavescatteringlengthasafunctionofthefreeparameters
r andX. Theinfluenceof theother unknownLEC was estimated usingexplicit
dependence of the loops on the renormalization scale µ. Preliminary numerical
results suggest that S−wave scattering length might be sensitive to the values
6
of the quark condensate and quark mass ratio. GχPT allows for systematically
larger values of a in comparison to the standard case [7]. The dependence
0
of the loop corrections on the renormalization scale can be understood as a
signal for relatively strong dependence on the unknown LEC. In order to get
sharperpredictionfurtherestimateswillbenecessary(resonances,sumrules...).
Numerical analysis of other observables, which has not been completed yet,
might be also interesting.
Acknowledgement. This work was supported by the program “Research
Centers”(projectnumberLN00A006) oftheMinistryofEducationoftheCzech
Republic.
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