Ⅱ-4. Hadron Physics

RIKEN Accel. Prog. Rep. 53

Update of single and dihadron cross sections for light hadrons in e+ e−√annihilation at s = 10.58 GeV in Belle††∗1cross section ratio
cross section ratio

π+π- (AEMP) / π+π- (std)π+π- (MVH) / π+π- (std)

1.4
1.2

π+π+ (AEMP) / π+π+ (std)π+π+ (MVH) / π+π+ (std)0.30 < z1 < 0.35

0.35 < z1 < 0.40

10.8
0.6
0.4
0.2 0.20 < z1 < 0.25

0.25 < z1 < 0.30

1.4 0.40 < z < 0.451
1.2

0.45 < z1 < 0.50

0.50 < z1 < 0.55

0.55 < z1 < 0.60

0.65 < z1 < 0.70

0.70 < z1 < 0.75

0.75 < z1 < 0.80

0.85 < z1 < 0.90

0.90 < z1 < 0.95

0.95 < z1 < 1.00

10.8
0.6
0.4
0.2

any hemisphere

1.4 0.60 < z < 0.65
11.2
1
0.8
0.6
0.4
0.2

cross section ratio

The fragmentation process describes the formationof ﬁnal-state hadrons from initial high-energetic partons such as quarks and gluons. Due to the strengthof the coupling of the strong interaction, described byquantum chromodynamics, QCD, this process is obtained only experimentally and parameterized in fragmentation functions, FFs. FFs are universal and factorize for most relevant processes and FF types. They
therefore serve as an important tool in the study ofthe spin, momentum and ﬂavor structure of the nucleon or other bound states of QCD. Its universality
can be used to extract them in one process, such as
electron-positron annihilation where no hadrons exist
in the initial state, and apply it to semi-inclusive deepinelastic lepton nucleon scattering SIDIS or hadronhadron scattering.
In the Belle experiment at the KEKB accelerator,a large amount of data exists that can be analyzed toextract various hadronic cross sections as a functionof the fractional energy the hadrons carry relative tothe initial parton energies, as well as their spin andtransverse momentum dependence.
The extraction of the ﬂavor dependence of fragmentation functions from electron-positron annihilation is
not as easy since the cross sections are a sum over allaccessible ﬂavors, at Belle energies up, down, strange
and charm quarks. To gain some sensitivity the extraction of cross sections for two hadrons in the ﬁnal state
have been published a few years ago,1,2) where the ﬂavor dependence of each of two single hadron fragmentation functions can be used together. However, theinterpretation via single hadron fragmentation functions is only possible when both hadrons appear in opposite hemispheres which in turn can only be ensured
via additional quantities. Such variables are diﬃcult
to treat theoretically, so in an update two diﬀerent
fractional-energy deﬁnitions were used and comparedto the conventional one.
Both deﬁnitions contain scalar products between thetwo hadron’s four-momenta and therefore only havesizable fractional energies when both hadrons are far
separated without the need of an explicit hemisphereassignment. The cross section ratios for pion pairs canbe seen in Fig. 1 relative to the default deﬁnition. Theﬁrst deﬁnition is actually the oldest one suggested for
dihadrons3) which is meant to also be able to interpret the dihadron cross sections at higher orders of thestrong coupling. The second deﬁnition is similar, butstresses the transverse momentum in the two-hadroncross section ratio

R. Seidl∗11.4 0.80 < z < 0.85
11.2
10.8
0.6
0.4
0.2
0.3

0.4

0.5

0.6

0.7

0.8

0.9

z2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

z2

Fig. 1. Dipion cross section ratios as a function of fractionalenergy z2 in bins of z1 relative to the conventional fractional energy definitions. The opposite sign pairs aredisplayed in blue squares and green triangles for theAEMP and MVH definitions while the same sign pairs
are displayed by red circles and purple squares, respectively.

system.4) As can be seen, at very high fractional energies all three deﬁnitions are similar as both hadrons
need to be nearly back-to-back and no phase space fortransverse momentum remains. At low fractional momenta the cross sections using the alternative deﬁnitions are far smaller since the same-hemisphere hadronpairs would appear at much smaller fractional energies
due to the scalar product.
In addition to these updated dihadron cross sections,also single hadron cross sections were updated. The
diﬀerence to previously published single hadron crosssections for pions, kaons and protons2,5) (as well as for
dihadrons using the default fractional energy deﬁnitions) is an improved initial state radiation correction
as well as the separation of correlated and uncorrelated
systematic uncertainties.
References

1) R. Seidl et al., Phys. Rev. D 96, 032005 (2017).
2) R. Seidl et al., Phys. Rev. D 92, 092007 (2015).
3) G. Altarelli, R. K. Ellis, G. Martinelli, S. Y. Pi,Nucl. Phys. B 160, 301 (1979). doi:10.1016/05503213(79)90062-2
4) P. J. Mulders, C. Van Hulse, Phys. Rev. D 100, 034011(2019).
5) M. Leitgab, et al., Phys. Rev. Lett. 111, 062002 (2013).

Condensed from Phys. Rev. D 101, 092004 (2020)
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