University of Groningen
Axial-Vector Currents and Chiral Symmetry
Sauerwein, Ubbi
DOI:
10.33612/diss.157530914
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Sauerwein, U. (2021). Axial-Vector Currents and Chiral Symmetry. University of Groningen. https://doi.org/10.33612/diss.157530914
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Chapter
4
Flavor-SU(2) Applications
4.1
Motivation
In this Chapter we focus on the most prominent axial-vector form factor, the axial-vector form factor of the nucleon. At momentum transfer t = 0 it is experimentally very well known GN N
A,aπ
µ(t = 0) ≡ GA(0) = 1.2724(23)[49]. We use flavor-SU(2) chiral perturbation theory, which involves the up and down quark only. Therefore, the degrees of freedom are limited to the pion π, the nucleon N , and the ∆-isobar. Like in flavor-SU(3) we employ the isospin limit mu= md≡ m. In accordance with Eq. (2.1) we define the fields:
Φ = π0 √2 π+ √ 2 π− −π0 , N = p n , and ∆111µ = ∆++µ , ∆112µ = ∆+µ/√3 , ∆122µ = ∆0µ/√3 , ∆222µ = ∆−µ . (4.1) Compared to the flavor-SU(3) case the restriction to two quark flavors leads to fewer terms in the chiral Lagrangian, decreasing the amount of LECs which need to be determined. Additionally, due to fewer baryon masses and conse-quently fewer mass differences, the power counting is simplified. Therefore the restriction to flavor-SU(2) is a good application for our framework.
SU(2)-χPT is not necessarily expected to reproduce the experimental value GA(t = 0) to high accuracy because of neglecting relevant particles, which
contain the strange quark. On the other hand, accurate QCD lattice data of the axial-vector form factor with 2 quark flavors became available in the past few years [33, 34, 35, 36]. A comparison of our results to this QCD lattice data provides a good test of the pion-mass dependence and momentum-transfer dependence of the axial-vector form factor.
On the theoretical side, the axial-vector form factor has been calculated using different renormalization schemes in flavor-SU(2). Many works did not include the isobar [75, 76, 77, 78, 79]. If it is explicitly included, the mass difference between the isobar and the nucleon needs special attention. Refs. [19, 37, 80, 81] approximate the baryon masses, which appear in the loop contributions to the axial-vector form factor. The masses in the chiral limit M and M + ∆ are used in Refs. [19, 80, 81] and subsequently an expansion in ∆/M is performed. On the other hand, Ref. [37] uses physical baryon masses in the loops. Using on-shell masses, the axial-vector form factor of the nucleon has only been determined in Ref. [69]. We reconstruct this work in detail in the following Sections.
4.2
Axial-Vector Form Factor of the Nucleon
4.2.1
Framework
We use the same notation as in the determination of the axial-vector form fac-tor of the nucleon in flavor-SU(2) [69]. The part of the chiral SU(2) Lagrangian, which contributes to the axial-vector form factor, is given by:
LSU (2)= − f2tr h UµUµ i + ¯N i /D − M N − trh ¯∆µ· (i /D − (M + ∆))gµν −i (γµDν+ γνDµ) + γµ(i /D + (M + ∆)) γν ∆ ν i + gAN γ¯ µγ5i UµN + 4 gχN γ¯ µγ5χ0i UµN + gR 2 ¯ N γµγ5[Dν, Fµν−] N − 4 gSN U¯ µUµN − gTN i σ¯ µνUµ, Uν N − gV ¯N i γµ{Uµ, Uν} DνN + h.c. + fS ¯ ∆µ· i Uµ N + h.c. + hAtr h ¯ ∆µ· γ5γν∆µ i Uν i − fA(1)tr ∆¯µ· γνγ 5N Uµ, Uν + h.c. − fA(2)tr ¯ ∆µ· γνγ5N Uµ, Uν + h.c.
− fA(3) ∆¯µ· Uν γνγ5UµN + ¯∆µ· Uµ γνγ5UνN + h.c. − fA(4) ∆¯µ· Uν γνγ5UµN − ¯∆µ· Uµ γνγ5UνN + h.c., (4.2) where we adopt our SU(3) notation of Eqs. (2.4), (2.6), (2.11), (2.12):
Uµ= 12u† (∂µei Φ/f) −i aµ, ei Φ/f u†, u = ei Φ/(2 f ), DµN = ∂µN + ΓµN , Γµ =12u†∂µ− i aµ u +12u∂µ+ i aµ u†, (Dµ∆ν)abc= ∂µ∆abcν + Γ a d,µ∆ dbc ν + Γ b d,µ∆ adc ν + Γ c d,µ∆ abd ν , (Φ · ∆µ)a = kl3Φln∆ kna µ , ( ¯∆ µ· Φ) b= kl3∆¯µknbΦnl, ( ¯N · ∆µ)ab = k3bN¯n∆knaµ , ( ¯∆ µ· N )a b = k3a∆¯µ knbN n, ( ¯∆µ· ∆µ)ab = ¯∆ µ bcd∆ acd µ , Fµν± = u†FµνR u ± u FµνL u†, χ0= 2 B0m 1 , (4.3) FµνR = ∂µaν− ∂νaµ− i [aµ, aν] , FµνL = −(∂µaν− ∂νaµ) − i [aµ, aν] .
In Table 4.1 we compare the normalizations of the SU(2)-LECs of Eq. (4.2) and the SU(3)-LECs from Eqs. (2.5), (2.9), and (2.10). We also give the link to the normalization of SU(2)-LECs of Refs. [15] and [82]. The LECs c6 and
c7 from Ref. [82] as well as the LECs b1 and b2from Ref. [21] are expected
to have an impact on the axial-vector form factor of the nucleon, but they are not included in this work.
The use of on-shell masses in the loop contributions requires the knowl-edge of baryon masses for all used lattice ensembles. Therefore, accurate QCD lattice data of the nucleon and the isobar masses MNand M∆is needed
in addition to the QCD lattice data for the axial-vector form factor. The eight coupled equations in Eq. (3.4) for the determination of the baryon masses in SU(3) reduce to two equations in SU(2):
MN = M + ˜ΣN MN, M∆, mπ ,
M∆= M + ∆ + ˜Σ∆ MN, M∆, mπ . (4.4)
In Ref. [69] the nucleon and isobar masses have been determined from QCD lattice data of three different lattice groups, ETMC [14, 83], CLS [34, 84], and RQCD [85]. Here we do not give the details of the calculation, but sum-marize the quark mass m, the nucleon mass MN, and isobar mass M∆ for
different lattice ensembles specified by the pion mass mπand the lattice
SU(2) LEC SU(3) LEC [15], [82] gA F + D gA= gb A fS C fS =b √ 2 cA hA H hA= gb 1 gS 18 2g0(S)+ gF(S)+ gD(S) gS= cb 3 gV 18 2g(V )0 + gF(V )+ gD(V ) gV =b cm2 gT 14 g(T )F + gD(T ) gT = cb 4 fA(1) 14f1(A) fA(2) 1 4f (A) 2 fA(3) 1 4f (A) 3 fA(4) 14f4(A) gχ 14 g(χ)1 + g (χ) 2 + g (χ) 3 + g (χ) 4 + 2g (χ) 7 gχ = db 16 gR 2 gF(R)+ gD(R) gR= db 22
Table 4.1: Dictionary between our SU(2)-LECs from Eq. (4.2) and our SU(3)-LECs from Eqs. (2.5), (2.9), and (2.10). We compare our SU(3)-LECs to SU(2) Lagrangians of Refs. [15] and [82].
I.1. Therewith we are ready to use on-shell masses in the loop contributions to the axial-vector form factor.
In accordance with the definition in flavor-SU(3) (2.16), we define the axial-vector form factor of the nucleon GA(q2) = GA(t)in flavor-SU(2):
hN (¯p)| Aµi(0) |N (p)i = ¯uN(¯p) γµγ5 τi 2GA(q 2) + γ 5 qµ 2MN τi 2GP(q 2)u N(p),(4.5)
where the form factor GI(q2)from Eq. (2.16) is not needed as ¯B = B = N.
The axial-vector form factor GA(q2)defines the axial charge GA(0)and axial
radius hr2
4.2.2
Analytical Results
Following the procedure in SU(3) with ¯p2 = p2 = MN2, we find the following renormalized result for the axial-vector form factor [69]:
GA(q2) = gAZN + 4 gχm2π+ gRq2+ KπI¯π+ X R=N,∆ KπRI¯πR(MN) + X L,R=N,∆ KLπRI¯LπR(q2) + X L,R=N,∆ KLπR0 ∆ ¯ILπR q2 . (4.6)
This is the SU(2)-equivalent to Eq. (3.21). The renormalized integrals ¯Iπ,
¯
IπR(MN), ¯ILπR(q2), and ∆ ¯ILπR(q2)have been defined in Eqs. (3.11), (3.16),
(3.18), and (3.19) with the loop meson Q = π. The kinematic functions Kπ,
KπR, KLπR, and KLπR0 are expanded in the pion-mass mπ, the mass
differ-ence δ, and the momentum transfer t = q2, applying the following counting
rules:
m2π∼ Q2, δ = M∆− MN(1 + ∆/M ) ∼ Q2,
t ∼ Q2, MN ∼ Q0. (4.7)
These power-counting rules are motivated by the SU(3) relations in Eqs. (3.5) and (3.8). Here the nucleon mass MN takes the role of the large scale like
the external mass MB in SU(3), whereas the difference of external masses
vanishes: δB = 0. We define the mass difference between the isobar and
the nucleon mass δ in accordance with δL and δR in Eq. (3.5), such that it
vanishes in the chiral limit.
To leading order, the kinematic functions are given by [69]:
Kπ= −gA+ g3A/4 f2 + 4 gAfS2 9 f2 α01+ 20 hAfS2 81 f2 α02 −100 fSMNf (3) A 27 f2 ∆ M α03− 4 fSMN(2 f (2) A − f (4) A ) 27 f2 ∆ M α04+ O Q 2 , KπN = 2 gAfS2 3 f2 n −5 6 ∆ M α10M 2 N− 5 24 ∆ M α11t − 5 18α12m 2 π− 5 6α13MNδ o +−2 gA+ g 3 A/4 + 8/3 gAMN(gS− 2 gT) f2 m 2 π+ O Q 4 , Kπ∆= 2 gAfS2 3 f2 n −5 6 ∆ M α20M 2 N + 19 18α22m 2 π− 5 6α23MNδ o
−5 hAf 2 S 9 f2 n14 9 ∆ M α30M 2 N − 1 18 ∆ M α31t −1 27α32m 2 π+ − 1 9α33+ 5 3α34 MNδ o −10 fSMNf (3) A 3 f2 n −20 9 ∆2 M2α40M 2 N + 20 9 α42m 2 π− 40 9 ∆ M α43MNδ o −2 fSMN(2 f (2) A − f (4) A ) 15 f2 n −20 9 ∆2 M2α50M 2 N +20 9 α52m 2 π− 40 9 ∆ M α53MNδ o + O Q4 , KN πN = O Q4 , KN π∆= 2 gAfS2MN2 3 f2 n5 6 ∆2 M2α60M 2 N + 5 24 ∆2 M2α61t −2 α62m2π+ 5 3 ∆ M α63MNδ o + O Q4 , KN π∆0 = 2 gAf 2 SMN4 3 f2 n2 3 ∆2 M2α70M 2 N −2 3 ∆2 M2α72m 2 π+ 4 3 ∆ M α73MNδ o + O Q4 , K∆π∆= − 5 hAfS2MN2 9 f2 n −4 3 ∆2 M2α80M 2 N − 7 9 ∆2 M2α81t +4 3α82m 2 π− 8 3 ∆ M α83MNδ o + O Q4 , K∆πN= KN π∆, K∆πN0 = K 0 N π∆, K 0 N πN = K 0 ∆π∆= 0 . (4.8)
All shown results are consistent with our findings in SU(3). The next order of the K-factors is shown in Eq. (I.1) in the Appendix.
It is not a surprise that we recover the dimensionless parameters αijfrom
the SU(3) analysis, which are shown in the Appendix (H.5). They depend on the ration ∆/M only and are normalized to αij → 1 for ∆ → 0.
Analo-gously to the SU(3) calculation, all power-counting violating contributions are subtracted α0i → 0 and αi0 → 0. In a power-counting scheme, where ∆ is
treated as a small scale, good convergence properties can only be achieved, if all relevant αij do not deviate much from 1. In order to examine this, we
list numerical values for αijwith realistic values ∆/M ∈ {0.2, 0.3, 0.4} in Table
devi-ations appear. In Figure 4.1 we visualize all non-subtracted factors αij with
i 6= 0 6= j. The label ij is given for αij’s with a strong deviation from 1. The
analysis of Table H.1 and Figure 4.1 proves that the kinematic functions K are not well described by the approximation αij ≈ 1. This confirms our wish not
to expand in ∆/M and motivates to keep the full, power-counting respecting coefficients αij with i 6= 0 6= j. 12 53 72 83 23 31 32 αij Δ/M 0.50 1.00 1.50 2.00 0.00 0.25 0.50
Figure 4.1: Factors αij for i 6= 0 6= j in the range 0 ≤ ∆/M ≤ 0.5.
It is desirable to compare our results to existing relativistic approaches. This is not always straightforward, as different renormalization procedures and expansions make it difficult to establish a link to our results. We will focus on a comparison with the works of Refs. [19, 37, 77]. This comparison has already been performed in Ref. [69]. For a comparison with Ref. [19] we try to match the results for internal nucleons. For bubble and tadpole diagrams we succeed, but we disagree with Ref. [19] for the triangle diagram with two internal nucleons. We mostly agree with Eq. (21) in Ref. [77], but only if the
replacement 16M4 NI (P P ) πN N(t) → (16M 4 N− 4tM 2 N)I (P P ) πN N(t)is made. Additionally,
we get a different sign of the gR-term. Ref. [37] gives the results for the loop
contributions to the axial charge in Eq. (A4) and the axial radius in Eq. (A5). By using Eqs. (35) and (36) from Appendix C in Ref. [69], we can rewrite triangle integrals in terms of bubble integrals. All bubble contributions in Ref. [37], denoted by B0(m2N, M
2
π, m2∆)and B0(m2∆, M 2
π, m2∆), can then be
repro-duced. All contributions proportional to the tadpole integral A0(Mπ2)differ by
a minus sign from ours. Furthermore, we disagree with the normalization of the axial radius in Eq. (A5) by a factor 36.
4.3
Fit and Discussion
4.3.1
Fit Details
At first we scrutinize the convergence properties of our chiral expansion. We keep the full structure of the renormalized integrals ¯I..., so the only
ex-pansion is performed in the kinematic functions K.... Only order-Q2 terms
of Eq. (4.8) enter the fit. Formally higher-order terms are suppressed by mπ/MN,
√
t/MN, δ/MN. Here we emphasize that MN denotes the nucleon
on-shell masses, which takes the values 945 MeV < MN < 1400MeV for
150MeV < mπ < 550MeV (see Table I.2). This drastically improves the
convergence pattern of our expansion compared to the use of approximate nucleon masses, either the chiral limit mass MN ≈ M ≈ 900 MeV or the
physical mass MN ≈ MNphys≈ 940 MeV. With approximate masses a
descrip-tion of QCD lattice data up to mπ = 550MeV appears impossible.
To be more explicit about the convergence properties of our expansion of the kinematic functions K..., we compare how well the orders Q2(Eq. (4.8))
and Q4 (Eq. (I.1)) describe the full K factors. We find that the deviation
is only 0.1% for physical pion mass mπ = 140MeV and 1% for mπ = 500
MeV. In contrast, if approximated masses are used, we find a deviation of 17% for mπ = 500MeV. Clearly, such good convergence properties are only
possible through an appropriate treatment of the isobar mass M∆, which is
represented in the parameter δ in our expansion (4.7). We show the specific values of δ for all lattice ensembles in the right panel of Figure 4.2. Approx-imate masses result in a constant value for δ, deviating more and more for larger pion masses from our dynamical variable δ. This strong effect in the expansion parameter δ and the use of on-shell masses in the integrals leads to an enormous reduction of the chiral corrections to the axial charge GA(0)
x 0.1 baryon masses
finite box: ETMC finite box: CLS finite box: RQCD physical chiral limit ch ir al c o rr ec ti o n s to GA (0 ) mπ [GeV] 0.00 -0.40 -0.20 -0.60 0.20 0.40 δ = MΔ - MN (1 + Δ / M)
finite box: ETMC finite box: CLS finite box: RQCD physical value chiral limit δ [G eV ] mπ [GeV] 0.00 -0.10 -0.20 0.00 0.20 0.40
Figure 4.2: Comparison between the use of chiral, physical, and on-shell baryon masses: in the right part we show the expansion parameter δ (4.7) and in the left panel the chiral correction terms to the axial charge. Since for the two cases with approximate baryon masses the chiral correction terms are quite large (orange and black triangles), their contributions are scaled down by a factor of 10. A strong effect is seen starting at pion mass mπ ≈ 200 MeV.
as displayed in the left panel of Figure 4.2. As our corrections represented by colorful points are rather small, the use of approximate masses (both physi-cal or chiral limit) leads to such large corrections that they need to be sphysi-caled down by a factor of 10 in order to be comparable to ours. These are convinc-ing first hints for a good convergence of our chiral expansion.
In our fit we confront our result for the axial-vector form factor of the nu-cleon in Eq. (4.6) with QCD lattice data [34, 83, 85]. We take into account the leading order of the kinematic functions K from Eq. (4.8), but not higher-order contributions from Eq. (I.1). This refers to the preferred loop higher-order Q3
as discussed in the Chapter 3.2, see Figures 3.2 - 3.4. We use the evolu-tionary fit algorithm GENEVA 1.9.0-GSI [86]. All QCD lattice data up to pion mass 550 MeV is included into our fit. There is no QCD lattice data for the axial-vector form factor of the largest pion mass mπ = 545MeV, so our results
are seen to be a prediction. Data points up to momentum transfer t = 0.36 GeV2are included. As the QCD lattice data is determined in boxes of finite
finite-volume effects to the loop contributions of the axial-vector form factor have been studied in Ref. [87]. These explicit effects are not considered in this work. We therefore only include lattice ensembles with mπL ≥ 4.0in our
fit, where L describes the size of the lattice. The use of on-shell masses with their volume-dependence introduces a second source of finite-volume effects in the loop contributions. With these masses being determined in Ref. [69], we are able to use the nucleon and isobar masses with and without finite-volume effects from Tables I.2 and I.1.
The fit minimizes the least-squares differences χ2of our expression of the
axial-vector form factor with respect to the lattice data points. All available lattice points, which meet our requirements, contribute with equal weight. In Table 4.2 we show the details of our fit and the details of the global fit [69], which also includes a fit of the nucleon and isobar masses.
GA-fit global fit [69]
Ndata 52 99 χ2 min/Ndata 1.19 1.04 Ndf 42 73 χ2 min/Ndf 1.47 1.40
Table 4.2: The details of our fit expressed through the minimal χ2
min, the
num-ber of data points Ndata, and the degrees of freedom Ndf. The global fit
includes the GA-fit and additionally the mass fits from Ref. [69].
Even without considering the LECs c6, c7, b1, and b2from Refs. [82, 21] our
chiral SU(2)-Lagrangian (4.2) involves 15 LECs. We have explicitly checked that the LECs c6 and c7 only contribute to higher-order terms than the
con-sidered ones. The explicit impact of the LECs b1and b2 is not known at this
point. Considering the limited amount of data points Ndata we set them to 0.
For the LEC hAwe use the large-Ncrelation [88]:
hA= 9 gA− 6 fS. (4.9)
In Eq. (4.8), the contribution proportional to fA(3) dominates the contribution
2 fA(2)− fA(4) = 0. The LECs gV and f (1)
A from Eq. (4.2) do not enter the final
formula for the axial-vector form factor (4.8). Therefore the number of LECs entering our fit is 10. We achieve a good description of the available QCD lattice data. The quality of the global fit [69] is slightly better than the one of the GA-fit.
4.3.2
Fit Results
We show the LECs, determined by our fit, in Table 4.3 with asymmetric error bars. They are based on a one-σ change for the value of χ2
minand determine
the range of the LECs within the region χ2
min+ 1. These errors are based on
the global fit of the baryon masses and axial-vector form factor in Ref. [69]. The parameters f , M , M + ∆, and gA are in the expected range [58]. We
LEC Fit result LEC Fit result
f [MeV] 87.19(+0.24−0.19) gS[GeV−1] 0.433(+0.081−0.057) M [MeV] 884.80(+0.36−0.72) gV[GeV−2] −1.133(+0.117−0.434) M + ∆ [MeV] 1187.09(+0.58−0.30) gT[GeV−1] 1.554(+0.097−0.054) gA 1.1933(+0.0036−0.0032) gR[GeV−2] 0.925(+0.032−0.024) fS 1.9409(+0.0097−0.0145) gχ[GeV−2] −3.597(+0.034−0.083) hA∗ −0.9057(+0.0964−0.0563) fA(3)[GeV−1] −4.139(+0.064−0.033)
Table 4.3: Low-energy constants (LECs) as determined in our fit. The LEC hA
has not been fitted, but fixed by the large-Nc relation (4.9), whereas gV does
not contribute to the axial-vector factor of the nucleon. For completeness its value has been extracted from Ref. [69].
notice that the LEC fS is a bit larger than its SU(3) equivalent C ≈ 1.60 [62].
This leads to a negative value for hA due to the used large-Ncrelation (4.9).
This is in contrast to the positive sign of its SU(3) partner H, but has also been reported in Ref. [89], where hAis denoted as g1. It is a clear hint that
strangeness effects are not negligible. The values for gS, gT, and gV disagree
significantly from previous SU(2) works like Refs. [90, 91]. These works de-termine the constants gS, gT, and gV in a theory without isobars. Ref. [15]
gives an estimate how to translate this to a theory including isobars. However, this does not allow for an accurate comparison. The values of the LECs gR,
gχ, and f (3)
A are poorly known in the literature. In general, our determined
LECs are in the expected range and reasonably small.
Our results for the axial charge GA(0) = 1.3094(+0.0044−0.0042) is close to the
experimental value GA(0) = 1.2724(23) [49]. Our asymmetric error results
from error propagation of the LECs. In Table 4.4 we compare our result of the axial radius hr2
Ai to the ones given in other works. We find that our value is
smaller than the ones from the literature. The differences of the axial charge and radius to the literature are not surprising, because our SU(2)-description includes strangeness effects only partially.
hr2 Ai [fm 2 ] reference hr2 Ai [fm 2 ] reference 0.1666(+0.0032−0.0035) this work 0.263(38) [37] 0.266(17)(7) [35] 0.46(22) [92] 0.360(36)(+80−88) [34] 0.46(24) [93] 0.213(6)(13)(3) [94]
Table 4.4: The axial radius as determined in our fit in comparison with other works.
To visualize the results of our fit of the axial-vector form factor, we scruti-nize GA(t)of all lattice ensembles separately in Figures 4.3 - 4.6. We
com-pare the colored QCD lattice data points with our result, represented by or-ange lines. In the following we give some details about this comparison. The data of different lattice groups can be distinguished by color, whereas the form of the symbols determines the size of the lattice scale. The order is given by ‘diamond - circle - square’ from smallest lattice scale to largest. Whether finite-volume effects are included in our results for the form factor is deter-mined by the color of the lines. Black lines do not involve any of them and refer to Table I.1. The gray error band is based on the errors of the LECs in Table 4.3. If finite-volume effects in the nucleon and isobar mass are included, so if the values of Table I.2 are used, we show orange lines. As mentioned above, lattice ensembles with mπL < 4.0are not included in the fit, because
explicit finite-volume effects in the form factor integrals are expected to turn relevant. To visualize this exclusion and also the restriction to t < 0.36 GeV2, we use dashed lines. Areas with fitted data points are symbolized with straight
orange lines. For the readers convenience we include the values for mπ, L,
and mπLin each plot. Additionally, the mass difference between the isobar
and the nucleon in the infinite volume and in the box is shown.
As the black lines do not describe the QCD lattice data very well, we con-clude that finite-volume effects are relevant. There is a clear difference be-tween results taking and not taking into account those effects. In general we find a good description of the QCD lattice data with our orange straight lines. The difference between the data points and the dashed orange lines is expected to vanish, if finite-volume effects [87] are explicitly included in the axial-vector form factors.
81 GA t [GeV 2] MeV 330 MΔ - MN = MeV 327 MΔbox- MNbox = fm 2.35 L = 5.28 mπ L = MeV 444 mπ = 0.60 0.80 1.00 1.20 MeV 330 MΔ - MN = MeV 327 MΔbox- MNbox = fm 2.69 L = 6.06 mπ L = MeV 444 mπ = MeV 330 MΔ - MN = MeV 328 MΔbox- MNbox = fm 2.23 L = 5.04 mπ L = MeV 445 mπ = incomplete fit in the box prediction MeV 330 MΔ - MN = MeV 338 MΔbox- MNbox = fm 1.88 L = 4.24 mπ L = MeV 446 mπ = 0.60 0.80 1.00 1.20 0.10 0.30 0.50 MeV 292 MΔ - MN = MeV 297 MΔbox- MNbox = fm 1.88 L = 4.81 mπ L = MeV 503 mπ = 0.10 0.30 0.50 MeV 246 MΔ - MN = MeV 247 MΔbox- MNbox = fm 2.35 L = 6.48 mπ L = MeV 545 mπ = 0.10 0.30 0.50 RQCD ETMC CLS (Two-State) CLS (Summation)
Figure 4.3: The t-dependence of the axial-vector form factor GA(t)for
differ-ent lattice ensembles. The colored lattice points are to be compared to our fitted finite-box results, represented by straight orange lines. Dashed orange lines can be found in regions which do not fulfill our fit restrictions (mπL ≥ 4.0
and t < 0.36 GeV2). Our results in the infinite-volume limit are visualized by black lines.
GA t [GeV 2] MeV 318 MΔ - MN = MeV 338 MΔbox- MNbox = fm 2.69 L = 4.67 mπ L = MeV 342 mπ = 0.60 0.80 1.00 1.20 MeV 313 MΔ - MN = MeV 344 MΔbox- MNbox = fm 2.23 L = 4.04 mπ L = MeV 357 mπ = MeV 386 MΔ - MN = MeV 339 MΔbox- MNbox = fm 2.35 L = 4.58 mπ L = MeV 385 mπ = incomplete fit in the box prediction MeV 360 MΔ - MN = MeV 337 MΔbox- MNbox = fm 2.52 L = 5.21 mπ L = MeV 408 mπ = 0.60 0.80 1.00 1.20 MeV 358 MΔ - MN = MeV 339 MΔbox- MNbox = fm 2.25 L = 4.67 mπ L = MeV 409 mπ = MeV 355 MΔ - MN = MeV 339 MΔbox- MNbox = fm 2.23 L = 4.66 mπ L = MeV 411 mπ = MeV 338 MΔ - MN = MeV 333 MΔbox- MNbox = fm 2.24 L = 4.9 mπ L = MeV 431 mπ = 0.60 0.80 1.00 1.20 0.10 0.30 0.50 MeV 336 MΔ - MN = MeV 366 MΔbox- MNbox = fm 1.68 L = 3.71 mπ L = MeV 436 mπ = 0.10 0.30 0.50 MeV 334 MΔ - MN = MeV 342 MΔbox- MNbox = fm 1.88 L = 4.18 mπ L = MeV 438 mπ = 0.10 0.30 0.50 RQCD ETMC CLS (Two-State) CLS (Summation)
Figure 4.4: The axial-vector form factor GA(t) of the nucleon for lattice
en-sembles with smaller pion mass; see the caption of Figure 4.3.
Clearly, finite volume effects have strong implications for the analytic structure of the axial-vector form factor in the region 350 MeV ≤ mπ ≤ 450 MeV. We
will come back to this interesting region in the end of this Chapter.
mπ= 385MeV. The relation
M∆box− Mbox
N < mπ< M∆− MN, (4.10)
is only fulfilled in this case. Furthermore, we find that the gray error band of our black, infinite volume results is much larger than for all other cases.
12 GA t [GeV 2 ] MeV 306 MΔ - MN = MeV 349 MΔbox- MNbox = fm 2.35 L = 3.32 mπ L = MeV 279 mπ = 0.60 0.80 1.00 1.20 MeV 308 MΔ - MN = MeV 340 MΔbox- MNbox = fm 2.98 L = 4.28 mπ L = MeV 284 mπ = MeV 309 MΔ - MN = MeV 349 MΔbox- MNbox = fm 2.23 L = 3.27 mπ L = MeV 289 mπ = incomplete fit in the box prediction MeV 310 MΔ - MN = MeV 334 MΔbox- MNbox = fm 3.37 L = 4.97 mπ L = MeV 291 mπ = 0.60 0.80 1.00 1.20 MeV 312 MΔ - MN = MeV 343 MΔbox- MNbox = fm 2.8 L = 4.19 mπ L = MeV 295 mπ = MeV 312 MΔ - MN = MeV 324 MΔbox- MNbox = fm 4.48 L = 6.71 mπ L = MeV 296 mπ = MeV 312 MΔ - MN = MeV 345 MΔbox- MNbox = fm 2.69 L = 4.05 mπ L = MeV 297 mπ = 0.60 0.80 1.00 1.20 0.10 0.30 0.50 MeV 314 MΔ - MN = MeV 349 MΔbox- MNbox = fm 2.24 L = 3.42 mπ L = MeV 301 mπ = 0.10 0.30 0.50 MeV 321 MΔ - MN = MeV 348 MΔbox- MNbox = fm 2.52 L = 4.02 mπ L = MeV 315 mπ = 0.10 0.30 0.50 RQCD ETMC CLS (Two-State) CLS (Summation)
Figure 4.5: The axial-vector form factor GA(t) of the nucleon for lattice
Eq. (4.10) has an important consequence. The decay ∆ → N π is allowed in the infinite volume, as M∆ > mπ + MN, but not in the finite volume as
M∆box< mπ+ MNbox. GA t [GeV 2 ] MeV 292 MΔ - MN = MeV 273 MΔbox- MNbox = fm 4.48 L = 3.47 mπ L = MeV 153 mπ = 0.60 0.80 1.00 1.20 MeV 292 MΔ - MN = MeV 282 MΔbox- MNbox = fm 4.5 L = 3.95 mπ L = MeV 173 mπ = MeV 298 MΔ - MN = MeV 337 MΔbox- MNbox = fm 2.98 L = 3.74 mπ L = MeV 248 mπ = incomplete fit in the box prediction MeV 298 MΔ - MN = MeV 330 MΔbox- MNbox = fm 3.36 L = 4.22 mπ L = MeV 248 mπ = 0.60 0.80 1.00 1.20 MeV 298 MΔ - MN = MeV 341 MΔbox- MNbox = fm 2.82 L = 3.55 mπ L = MeV 249 mπ = MeV 298 MΔ - MN = MeV 330 MΔbox- MNbox = fm 3.37 L = 4.25 mπ L = MeV 249 mπ = MeV 299 MΔ - MN = MeV 321 MΔbox- MNbox = fm 4.04 L = 5.15 mπ L = MeV 252 mπ = 0.60 0.80 1.00 1.20 0.10 0.30 0.50 MeV 302 MΔ - MN = MeV 342 MΔbox- MNbox = fm 2.83 L = 3.82 mπ L = MeV 267 mπ = 0.10 0.30 0.50 MeV 304 MΔ - MN = MeV 345 MΔbox- MNbox = fm 2.66 L = 3.69 mπ L = MeV 274 mπ = 0.10 0.30 0.50 RQCD ETMC CLS (Two-State) CLS (Summation)
Figure 4.6: The axial-vector form factor GA(t) of the nucleon for lattice
G
A(0
)
m
π[GeV]
0.80 1.05 1.30 0.26 0.42 RQCD CLS ETMC L = 2.23 - 2.52 fm Infinite volumeFigure 4.7: The axial charge as function of the pion mass compared to the QCD lattice data. The orange lines are our results for the indicated range of lattice sizes, the black lines are our predictions in the infinite-volume limit.
In Figure 4.7 we show the pion mass dependence of our results for the axial charge GA(0)and the corresponding lattice points. As in previous plots,
the black line symbolizes the infinite volume limit and the orange lines give a range of our finite volume results for L = 2.23 − 2.52 fm. Lattice points, which are not in this range are not expected to meet this line. As already observed before, there is large difference between finite volume and infinite volume results. We find a non-analytic behavior in the black curve at the pion mass mπ = 373.49
+17.79
−12.30 MeV. In the infinite volume the jump height is much
larger than in the finite volume. This means that QCD lattice calculations are less sensitive to the observation of this jump. The non-analytic behavior originates from the non-linear equations for the baryon masses of Eq. (4.4). It has already been observed in Ref. [69] and propagates to the axial-vector form factor because of the use of on-shell masses in the loop contributions.
G
A(0
)
m
π[GeV]
1.30 1.80 2.30 0.35 0.40 0.45 lattice data (L = 2.23 - 2.52 fm) L = 2.23 - 2.52 fm L = 3.38 fm Infinite volumeFigure 4.8: The axial charge as function of the pion mass, zoomed in to the region where our results show the non-analytic behavior. The orange lines are our results for the indicated range of lattice sizes, the red lines are our results for the indicated larger lattice size, and the black lines are our predictions in the infinite-volume limit.
Figure 4.8 offers a closer look to the area mπ = 350 − 450MeV, where
the non-analytic behavior is observed. The results in the infinite volume are presented with gray error margins, which result from the errors of the LECs of Table 4.3. We notice that all available lattice points are determined on a rather small lattice size L = 2.23 − 2.52 fm, for which the jump height is shown to be small. Additional to our results for this range of lattice sizes, we give our results for a larger value L = 3.38 fm, represented by a red line. In contrast to the orange line, the jump in the red line is larger than the typical error bars of the QCD lattice data and of comparable size to the jump in the infinite volume. That means that the non-analytical behavior in the axial-vector form factor is more likely to be detectable, if it is calculated on a larger lattice. Therefore, more QCD lattice data in the interval 360 MeV ≤ mπ ≤ 390 MeV on a large
