$\displaystyle\quad\times\mbox{exp}\left\\{\frac{1}{4}\,\log{\frac{\zeta}{\mu^{2}}}\,\left[\widetilde{K}(b_{T}^{\star};\,\mu_{b})-\int_{\mu_{b}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\,\gamma_{K}(\alpha_{S}(\mu^{\prime}))\right]\right\\}.$
(157)

The recipe to obtain the previous quantity at NLL accuracy is the following:

  * •

The anomalous dimension $\gamma_{K}$ of the soft kernel is expanded up to
2-loops. The 1-loop coefficient can be found in Eq. (132), while at 2-loops we
have, see Refs. Echevarria:2016scs ; Collins:2017oxh :

$\displaystyle\gamma_{K}^{[2]}=2\,C_{A}\,C_{F}\,\left(\frac{536}{9}-\frac{8\pi^{2}}{3}\right)-\frac{160}{9}\,C_{F}\,n_{f},$
(158)

where $n_{f}$ is the total number of fermion fields that we consider.

  * •

All the other quantities in the exponent are expanded up to 1-loop. Their
expressions are given by, see Refs. Echevarria:2016scs ; Collins:2017oxh :

$\displaystyle\widetilde{K}^{[1]}=0;$ (159)
$\displaystyle\gamma_{D}^{[1]}=6C_{F},$ (160)

where Eq. (159) refers to the 1-loop coefficient of the soft kernel computed
at the reference scale.

Therefore, the $\zeta$-independent part of the Sudakov at NLL is:

$\displaystyle\mbox{exp}\left\\{\frac{1}{4}\,\widetilde{K}(b_{T}^{\star};\,\mu_{b})\,\log{\frac{\mu^{2}}{\mu_{b}^{2}}}+\int_{\mu_{b}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\,\left[\gamma_{D}(\alpha_{S}(\mu^{\prime}),\,1)-\frac{1}{4}\,\gamma_{K}(\alpha_{S}(\mu^{\prime}))\,\log{\frac{\mu^{2}}{\mu^{\prime
2}}}\right]\right\\}\stackrel{{\scriptstyle NLL}}{{=}}$
$\displaystyle\quad\stackrel{{\scriptstyle
NLL}}{{=}}\mbox{exp}\left\\{\log{\frac{\mu}{\mu_{b}}}\,g_{1}(x)+g_{2}(x)\right\\},$
(161)

where:

$\displaystyle x=\frac{\alpha_{S}(\mu)}{4\pi}\,\log{\frac{\mu}{\mu_{b}}},$
(162)

and the functions $g_{1}$ and $g_{2}$ are given by:

$\displaystyle
g_{1}(x)=\frac{\gamma_{K}^{[1]}}{4\beta_{0}}\,\left[1+\frac{\log{(1-2\beta_{0}\,x)}}{2\beta_{0}\,x}\right];$
(163) $\displaystyle
g_{2}(x)=\frac{\gamma_{K}^{[1]}}{4\beta_{0}}\,\frac{\beta_{1}}{\beta_{0}}\,\left[\frac{x}{1-2\beta_{0}\,x}+\frac{1}{2\beta_{0}}\left(\log{(1-2\beta_{0}\,x)}+\frac{1}{2}\log{(1-2\beta_{0}\,x)}^{2}\right)\right]-$
$\displaystyle\quad-\frac{\gamma_{K}^{[2]}}{8\beta_{0}^{2}}\,\left[\frac{2\beta_{0}\,x}{1-2\beta_{0}\,x}+\log{(1-2\beta_{0}\,x)}\right]-\frac{\gamma_{D}^{[1]}}{2\beta_{0}}\,\log{(1-2\beta_{0}\,x)},$
(164)

where $\beta_{0}$ and $\beta_{1}$ are the coefficients of the beta functions
up to $2$ loop:

$\displaystyle\beta_{0}=\frac{11}{3}C_{A}-\frac{2}{3}n_{f},$ (165)
$\displaystyle\beta_{1}=\frac{34}{3}C_{A}^{2}-\frac{11}{3}C_{A}\,n_{f}-2C_{F}n_{f}.$
(166)

This result is in agreement with the Sudakov factor computed e.g. in Ref.
Koike:2006fn . On the other hand, the term depending on the rapidity cut-off
$\zeta$ is given by:

$\displaystyle\mbox{exp}\left\\{\frac{1}{4}\,\log{\frac{\zeta}{\mu^{2}}}\,\left[\widetilde{K}(b_{T}^{\star};\,\mu_{b})-\int_{\mu_{b}}^{\mu}\frac{d\mu^{\prime}}{\mu^{\prime}}\,\gamma_{K}(\alpha_{S}(\mu^{\prime}))\right]\right\\}\stackrel{{\scriptstyle
NLL}}{{=}}$ $\displaystyle\quad\stackrel{{\scriptstyle
NLL}}{{=}}\mbox{exp}\left\\{\frac{1}{4}\,\log{\frac{\zeta}{\mu^{2}}}\,\left[g_{2}^{K}(x)+\frac{1}{\log{\frac{\mu}{\mu_{b}}}}\,g_{3}^{K}(x)\right]\right\\},$
(167)

where $x$ has been defined in Eq. (162), while the functions $g_{2}^{K}$ and
$g_{3}^{K}$ are:

$\displaystyle
g_{2}^{K}(x)=\frac{\gamma_{K}^{[1]}}{2\beta_{0}}\,\log{(1-2\beta_{0}\,x)};$
(168) $\displaystyle
g_{3}^{K}(x)=\frac{x^{2}}{1-2\beta_{0}\,x}\,\left[\gamma_{K}^{[1]}\,\frac{\beta_{1}}{\beta_{0}}-\gamma_{K}^{[2]}+\frac{1}{x}\,\widetilde{K}^{[1]}\right].$
(169)

###  3.3 Non-Perturbative content

The last term in Eq. (148) encodes the whole non-perturbative content of the
TMD fragmentation function, $D_{1}$. Clearly, this cannot be predicted by
perturbative QCD and hence it has to be extracted from experimental data,
through a phenomenological analysis. It involves two functions. The first is
$g_{K}$, which describes the long-distance behavior of the soft kernel
$\widetilde{K}$, defined as:

$\displaystyle
g_{K}(b_{T})=\widetilde{K}(b_{T}^{\star};\,\mu)-\widetilde{K}(b_{T};\,\mu).$
(170)

In most phenomenological applications it is assumed to behave quadratically:

$\displaystyle g_{K}(b_{T})=a\,b_{T}^{2}.$ (171)

with $a\sim 0.01\div 0.1$ GeV2.

The other non-perturbative function is the model $M_{D}$, which represents the
very heart of the TMD, since it identifies $D_{1}$ uniquely. Besides $b_{T}$,
it may depend on the flavor of the fragmenting parton, on its light-cone
momentum fraction $z$ and also on the detected hadron $H$ (in the following,
however, we will assume that it is just a function of $b_{T}$, dropping any
other dependence). Its functional form is arbitrary, as long as it goes to $1$
as $b_{T}\to 0$ and goes to zero in the large-$b_{T}$-limit, $b_{T}\to\infty$.
It is commonly modelled as a Gaussian or as a (second kind) Bessel function,
which in the Fourier conjugate momentum space becomes a power-law, reminding
of a squared propagator for an on-shell fermion. This will be discussed in
more detail in Section 5.

##  4 Final Cross Section

We are now ready to assemble the expression for the final cross section of
$e^{+}e^{-}\to H\,X$ in the $2$-jet limit, as sketched in Eq. (4) in its
general factorized form. It is differential in the fractional energy of the
detected hadron $z_{h}$, defined in Eq. (1), in the transverse momentum
$P_{T}$ of the detected hadron with respect to the thrust axis, and also in
thrust, $T$, as defined in Eq. (3), The factorization procedure allows to
separate out the short-distance contributions, encoded in the partonic cross
section (Section 2), from the long-distance behavior, described by the TMD FFs
(Section 3). Referring to the very general expression presented in Eq. (4), we
should keep in mind that:

  * •

The contribution of the fragmenting gluon is suppressed by orders of
$\mathcal{O}\left({\lambda^{2}}/{Q^{2}}\right)$ in the $2$-jet limit (see
Section 2.1.2). Therefore, the sum over parton types, $j$, runs only over the
flavors $f$ of the fragmenting fermions.

  * •

The final cross section is RG-invariant. In fact, the anomalous dimension of
the partonic cross section, Eq. (127), is exactly equal and opposite to the
anomalous dimension of the TMD FF associated to a fragmenting fermion, Eq.
(143). Therefore, we are allowed to set $\mu=Q$.

  * •

The final cross section depends on two cut-offs, $\lambda$ and $\zeta$, so far
considered as being independent. However, they have been introduced for
intimately related reasons: they both ensure that the partonic cross section
describes all particles involved in the process, _except_ those that are
_collinear_ i.e. with low transverse momentum ($k_{T}\leq\lambda$) and a very
large rapidity ($y\geq y_{1}$), which are included in the TMD FF. In fact, a
neat and consistent separation of the partonic cross section from the TMD
fragmentation function guarantees the successful factorization of the cross
section. Furthermore, both cut-offs have to be intended in the limit in which
they vanish. On one hand, $\zeta\to 0$ (i.e. $y_{1}\to+\infty$, according to
Eq. (145)) is the limit in which the factorization procedure of Ref.
Collins:2011zzd  can be applied. On the other hand, $\lambda\to 0$ corresponds
to the $2$-jet limit for the topology of the final state of the process,
according to the relation ${\lambda^{2}}/{Q^{2}}=\tau_{\mbox{\tiny meas.}}$,
as we discussed at the end of Section 2.1.2. The above considerations suggest
that $\lambda$ and $\zeta$ are closely related. The subtraction mechanism
presented in Section 2.2.2 was built in such a way that the range of
transverse momentum of the collinear emission covered by the unsubtracted hard
part of Eq. (108) would exactly match the $k_{T}$-range covered by the
subtraction term of Eq. (119). In fact, in both cases the collinear gluon can
have a transverse momentum $k_{T}$ no larger than $\lambda$. As far as
rapidity is concerned, however, the subtraction mechanism needs to be applied
with special care. Let’s consider the 1-loop computation of the partonic cross
section associated to a fragmenting fermion. On one hand, the subtraction term
describes a collinear gluon that, when $\tau=0$, has a rapidity larger than
$y_{1}$, linked to $\zeta$ by the definition of Eq. (145). On the other hand,
the rapidity of the collinear gluon described by the unsubtracted hard part
depends on $k_{T}$ and $\tau$, and when $\tau=0$ it cannot be smaller than
$y_{\lambda}$, defined exactly as $y_{1}$ but with $\zeta=\lambda^{2}$.
Therefore, in order to cover the same range of rapidities in both terms, we
have to set $y_{1}=y_{\lambda}$, which means $\zeta=\lambda^{2}$. Any
different value of $\zeta$ would lead to an incorrect separation of the
collinear and the hard contributions in the final cross section. In fact, if
$\zeta>\lambda^{2}$, then $y_{1}<y_{\lambda}$ and the hard part would also
include the contribution of the collinear particles with a rapidity in the
range $y_{1}\leq y\leq y_{\lambda}$, which is already covered by the TMD FFs.
Conversely, if $\zeta<\lambda^{2}$, then $y_{1}>y_{\lambda}$ and the
contribution of the collinear particles with rapidity in the range
$y_{\lambda}\leq y\leq y_{1}$ would not be present at all in the final result.
Consequently, the only possible choice is $\zeta=\lambda^{2}$.

Such considerations lead us to the following formula:

$\displaystyle\frac{d\sigma}{dz_{h}\,dT_{\mbox{\tiny
meas.}}\,dP_{T}^{2}}=\pi\int\,\frac{d^{2}\vec{b}_{T}}{(2\pi)^{2}}\times$
$\displaystyle\quad\times\sum_{f}\,\int_{z_{h}}^{1}\,\frac{dz}{z}\,\frac{d\widehat{\sigma}_{f}(Q,\,\lambda,\,\lambda^{2})}{d{z_{h}}/{z}\,dT_{\mbox{\tiny
meas.}}}\,\widetilde{D}_{1,\,H/f}(z,\,b_{T},\,Q,\,\lambda^{2})\,e^{i\frac{\vec{P}_{T}}{z}\cdot\vec{b}_{T}}\,\left[1+\mathcal{O}\left(\frac{M_{H}^{2}}{Q^{2}}\right)\right],$
(172)

where $\lambda$ is related to the measured value of thrust, $T_{\mbox{\tiny
meas.}}$. Alternatively, the final cross section can be written in terms of
the soft and collinear contributions encoded in the partonic cross section, as
in Eq. (126). Therefore, by using the final, all-order, results of Section
2.2.2 we have:

$\displaystyle\frac{d\sigma}{dz_{h}\,dT_{\mbox{\tiny
meas.}}\,dP_{T}^{2}}=\pi\,\sigma_{B}\,N_{C}\,V\int
d\tau_{1}\,d\tau_{2}\,d\tau_{S}\,J_{B}(\tau_{2})\,S(\tau_{S})\,\times$
$\displaystyle\quad\times\int_{z_{h}}^{1}\,dz\left.J_{q/q}^{(\lambda)}(\tau_{1},\,{z_{h}}/{z},\,\lambda^{2})\right\rvert_{\mbox{\small
sub,
R}}\,\sum_{f}\,e_{f}^{2}\,\int\,\frac{d^{2}\vec{b}_{T}}{(2\pi)^{2}}\widetilde{D}_{1,\,H/f}(z,\,b_{T},\,Q,\,\lambda^{2})\,e^{i\frac{\vec{P}_{T}}{z}\cdot\vec{b}_{T}}\times$
$\displaystyle\quad\times\,\delta(1-T_{\mbox{\tiny
meas.}}-\tau_{1}-\tau_{2}-\tau_{S})\,\left[1+\mathcal{O}\left(\frac{M_{H}^{2}}{Q^{2}}\right)\right],$
(173)

As we pointed out at the end of Section 2.2.2, the structure of this result is
similar to the well-known thrust distribution associated to $e^{+}e^{-}$
events, with the exception of the contributions associated to the radiation of
partons collinear to the thrust axis, which give the non-trivial dependence on
$z_{h}$ and $P_{T}$, encoded in the second line of Eq. (173). Notice that here
we explicitly indicated that the thrust refers to its measured value. This
does not exactly coincide with the variable $T=1-\tau$ that we used in Section
2 in computing the partonic cross section, since the thrust distribution
calculated at partonic level does not encode the information about the non-
perturbative hadronization process. According to Ref. Catani:1991kz , the
observed distribution of “hadronic” thrust, $T_{\mbox{\tiny meas.}}$, is
related to its partonic counterpart, $T$, by a correlation function
$C(T_{\mbox{\tiny meas.}},\,T)$. Therefore, the partonic cross section written
in terms of $T_{\mbox{\tiny meas.}}$ will be obtained as:

$\displaystyle\frac{d\widehat{\sigma}_{f}(Q,\,\lambda,\,\lambda^{2})}{dz\,dT_{\mbox{\tiny
meas.}}}=\int_{0}^{1}\,dT\,C(T_{\mbox{\tiny
meas.}},\,T)\,\frac{d\widehat{\sigma}_{f}(Q,\,\lambda,\,\lambda^{2})}{dz\,dT}=$
$\displaystyle=\mbox{exp}\Bigg{\\{}\frac{1}{4}\,\widehat{K}\left(\alpha_{S}(Q),\,1\right)\,\log{\frac{\lambda^{2}}{Q^{2}}}-\frac{1}{2}\,\int_{\lambda^{2}}^{Q^{2}}\,\frac{d(\lambda^{\prime})^{2}}{(\lambda^{\prime})^{2}}\,G\left(\alpha_{S}(Q),\,1,\,{\lambda^{2}}/{Q^{2}},\,{Q^{2}}/{(\lambda^{\prime})^{2}}\right)\Bigg{\\}}\times$
$\displaystyle\quad\times\,\int_{0}^{1}\,dT\,C(T_{\mbox{\tiny
meas.}},\,T)\,\left.\frac{d\widehat{\sigma}_{f}}{dz\,dT}\right\rvert_{\begin{subarray}{c}\text{ref.}\end{subarray}},$
(174)

having used the expression for the partonic cross section as obtained in Eq.
(137). Monte Carlo simulations show that the correlation function is sharply
peaked around $T_{\mbox{\tiny meas.}}\sim T$ and that it is not strongly
sensitive to the parton distribution Catani:1991kz . Therefore, a formula
suitable for phenomenology (that clearly requires an expression written in
terms of functions of $T_{\mbox{\tiny meas.}}$) can be obtained by
approximating the correlation function with a (smooth) delta function and then
neglecting all the contributions proportional to $C(T_{\mbox{\tiny
meas.}},\,1)$, associated to strictly pencil-like events. In other words, the
partonic cross section written in terms of $\tau_{\mbox{\tiny meas.}}$ can be
obtained from the result of the fixed order computation, expressed in terms of
distributions of $\tau$, by simply neglecting all contributions proportional
to $\delta(\tau)$, by removing all the “plus” that label the plus-
distributions and finally by replacing $\tau$ with its hadronic counterpart.
At NLO this operation gives:

$\displaystyle\frac{d\widehat{\sigma}^{NLO}_{f}(Q,\,\lambda,\,\lambda^{2})}{dz\,dT_{\mbox{\tiny
meas.}}}=\int_{0}^{1}\,dT\,C(T_{\mbox{\tiny
meas.}},\,T)\,\frac{d\widehat{\sigma}^{NLO}_{f}(Q,\,\lambda,\,\lambda^{2})}{dz\,dT}=$
$\displaystyle=-\sigma_{B}\,e_{f}^{2}\,N_{C}\,\frac{\alpha_{S}(Q)}{4\pi}\,C_{F}\,\delta(1-z)\,\frac{3+8\log{\tau_{\mbox{\tiny
meas.}}}}{\tau_{\mbox{\tiny
meas.}}}\,\mbox{exp}\left\\{-\frac{\alpha_{S}(Q)}{4\pi}\,3C_{F}\,\left(\log{\frac{\lambda^{2}}{Q^{2}}}\right)^{2}\right\\}$
(175)

Of course, this is a rather rough approximation which reduces the potential of
a formula like Eq. (138). In particular, neglecting the $\tau=0$ terms not
only eliminates the contributions of pencil-like events, but it also
drastically modifies the behaviour of the cross section in the region where
$T$ is very close to 1. Furthermore, by applying this approximation we loose
part of the information on the $z_{h}$-dependence of the cross section, since
all $z$-distributions that are coupled to $\delta(\tau)$ are removed. As a
consequence, we expect that the dependence on $z_{h}$ will be compromised,
especially at large values of thrust. The $z_{h}$-dependent neglected part,
associated with the last line of Eq. (138), involves two terms: an integral
from $z_{h}$ to $1$ and a contribution dominated by $\log{(1-z_{h})}^{2}$.
Therefore, the discrepancy due to the brute approximation of Eq. (175) should
be more important at small and large values of $z_{h}$, where these two terms
have a non negligible impact on the final cross section. We will come back to
this point in the next Section.

Finally, we require ${\lambda^{2}}/{Q^{2}}=\tau_{\mbox{\tiny meas.}}$, and we
remove the label “meas”, which has now become redundant. At NLO, NLL accuracy,
the cross section for $e^{+}e^{-}\to H,\,X$ differential in $z_{h}$, $T$ and
$P_{T}$ is then given by:

$\displaystyle\frac{d\sigma^{\text{NLO, NLL}}}{dz_{h}\,dT\,dP_{T}^{2}}=$
$\displaystyle\quad=-\sigma_{B}\,\pi\,N_{C}\,\frac{\alpha_{S}(Q)}{4\pi}\,C_{F}\,\frac{3+8\log{(1-T)}}{1-T}\,\mbox{exp}\left\\{-\frac{\alpha_{S}(Q)}{4\pi}\,3C_{F}\,\left(\log{(1-T)}\right)^{2}\right\\}\,\times$
$\displaystyle\quad\times\,\sum_{f}e_{f}^{2}\,\int\,\frac{d^{2}\vec{b}_{T}}{(2\pi)^{2}}\,e^{i\frac{\vec{P}_{T}}{z_{h}}\cdot\vec{b}_{T}}\,\widetilde{D}^{\text{NLL}}_{1,\,H/f}(z_{h},\,b_{T},\,Q,\,(1-T)\,Q^{2})\,\left[1+\mathcal{O}\left(\frac{M_{H}^{2}}{Q^{2}}\right)\right]$
(176)

where, according to Section 3:

$\displaystyle\widetilde{D}^{\text{NLL}}_{1,\,H/f}(z_{h},\,b_{T},\,Q,\,(1-T)\,Q^{2})=\frac{1}{z_{h}^{2}}\,\bigg{(}d_{H/f}(z,\,\mu_{b})+$
$\displaystyle\quad+\frac{\alpha_{S}(\mu_{b})}{4\pi}\int_{z_{h}}^{1}\,\frac{d\rho}{\rho}\,\left[d_{H/f}({z_{h}}/{\rho},\,\mu_{b})\,\rho^{2}\,\mathcal{C}_{q/q}^{[1]}(\rho)+d_{H/g}({z_{h}}/{\rho},\,\mu_{b})\,\rho^{2}\,\mathcal{C}_{g/q}^{[1]}(\rho)\right]\bigg{)}\times$
$\displaystyle\quad\times\mbox{exp}\left\\{\log{\frac{Q}{\mu_{b}}}\,g_{1}(x)+g_{2}(x)\right\\}\mbox{exp}\left\\{\frac{1}{4}\,\log{(1-T)}\,\left[g_{2}^{K}(x)+\frac{1}{\log{\frac{Q}{\mu_{b}}}}\,g_{3}^{K}(x)\right]\right\\}\times$
$\displaystyle\quad\times M_{D}(b_{T})\mbox{
exp}\left\\{-\frac{1}{4}\,g_{K}(b_{T})\,\log{\left(\frac{z_{h}^{2}\,Q^{2}}{M_{H}^{2}}\,(1-T)\right)}\right\\},$
(177)

with $x$ defined as in Eq. (162). For a convenient and straightforward
application of Eq. (177) we recall that the Wilson coefficients
$\mathcal{C}_{q/q}^{[1]}$ and $\mathcal{C}_{g/q}^{[1]}$ are presented in Eqs.
(154) and (155), the functions $g_{1}$, $g_{2}$ and $g_{2}^{K}$, $g_{3}^{K}$
contributing to the Sudakov are computed in Eqs. (163), (164), (168) and
(169), and the non-perturbative functions $g_{K}$ and $M_{D}$ are defined in
Eqs. (171) and (178). Notice that the cross section in Eq. (176) is _not_
resummed in thrust. A proper resummation in $T$ is beyond the purpose of this
paper. Such resummation must also include a correct treatment of the
dependence on $z_{h}$, by considering the terms that have been neglected in
the approximation used in Eq. (175). Clearly, this is strictly connected to
the difficulties in finding a fully resummed expression for the subtracted,
renormalized function $J_{q/q}^{(\lambda)}$ (see the discussion at the end of
Section 2.2.2) and, ultimately, for the whole second line of Eq. (173).

Very recently, the factorization of the $e^{+}e^{-}\to HX$ cross section, as
measured by the BELLE Collaboration (Ref. Seidl:2019jei ), has been
investigated in two papers, both based on the SCET formalism. In Ref.
Kang:2020yqw , the authors propose to integrate out the thrust $T$ and to
reproduce the experimental cross section by combining all the measured thrust
bins, within the range $[0.5-1.0]$. A cross section differential in $z_{h}$,
$P_{T}$ and $T$, is presented in Ref. Makris:2020ltr ; it results from
matching three different kinematical regions, each associated with a different
factorized expression for the final cross section. The phenomenological
application of this formula is not shown.

##  5 A basic phenomenological application

In this conclusive Section we provide a basic but realistic example of the
strong phenomenological potential of Eq. (176). We stress that this is not a
proper and complete data-analysis of the BELLE measurements, as in what
follows there is no fit of the experimental data and no fine tuning of the
free parameters. However, the agreement of the predictions obtained from Eq.
(176) with the experimental data is astonishing, even without a proper fit.

As anticipated in Section 3.3, to model the non-perturbative part of the TMD
FF we will exploit a parameterization that, in the $b_{T}$ space, corresponds
to a Bessel-K function, normalized in such a way that it is $1$ at $b_{T}=0$:

$M_{D}(b_{T},\,m,\,p)=\frac{2^{2-p}}{\Gamma(p-1)}\,(b_{T}\,m)^{p-1}\,K_{p-1}(b_{T}\,m)$
(178)

where $K_{-1+p}$ is the modified Bessel function of the second kind. This
model was successfully applied in Ref. Boglione:2017jlh  and, more recently,
in Ref. Boglione:2020cwn . It is known as the “power law model”, since its
Fourier transform is given by:

$\displaystyle\mathcal{FT}\\{M_{D}\\}=\frac{\Gamma(p)}{\pi\,\Gamma(p-1)}\frac{m^{2(p-1)}}{\left(k_{T}^{2}+m^{2}\right)^{p}},$
(179)

which mimics the power $p$ of a function closely reminiscent of the propagator
of an on-shell fermion (see dedicated discussion in Ref. Collins:2016hqq ).

As mentioned above, we do not perform a proper fit but simply fix the two non
perturbative parameters of the $k_{T}$-distribution to reasonable and
perfectly standard values, inspired by pure common sense: $m=1$ GeV, to
represent a typical hadronic mass, and $p=2$ which translates, in
$k_{T}$-space, in a $k_{T}$-distribution similar to the square of the Feynman
propagator, see Eq. (179). One additional free parameter, $a$, is embedded in
the non perturbative $g_{K}(b_{T})$ function, as defined in Eq. (170). Its
value depends on the choice of $b_{MAX}$; for $b_{MAX}=1$ GeV-1 it is expected
to be in the range $0.01-0.1$ GeV, while considerably larger values are found
at smaller $b_{MAX}$. Interesting discussions and reference values of $a$ can
be found, for example, in Refs Anselmino:2013lza ; Aidala:2014hva ;
Boglione:2014oea ; Boglione:2018dqd . We set $a=0.05$ GeV-1, as an
intermediate value. There is clearly some degree of correlation among the
three parameters $a$, $m$ and $p$, so that different values of $a$ can easily
be compensated by slight variations of $m$ and $p$. Reassuringly, the final
results are not particularly sensitive to the precise value of $a$, after $m$
and $p$ have been fixed. It is important to point out, here, that we do not
need to introduce any overall normalization parameter.

The collinear fragmentation functions used for the implementation of Eq. (176)
are the NNFF10_NLO reference set Bertone:2017tyb , for $\pi^{+}$ production.

To study the $z_{h}$ and $P_{T}$ behaviour of the cross section at fixed
values of thrust, $T$, we consider the data subset that the BELLE
Collaboration has chosen to present in Fig. 9 of their article, Seidl:2019jei
. The differential cross sections corresponding to pion production are shown
as a function of $P_{T}$ for 15 $z_{h}$-bins, covering the range $[0.10-0.85]$
and one thrust bin, $0.85<T<0.9$ (we fix $T=0.875$). The error bars correspond
to statistical and systematical errors added in quadrature, as reported by the
BELLE Collaboration. For the BELLE experiment, $Q=10.58$ GeV, while the
$P_{T}$ bins span a range from $0.06$ GeV to $2.5$ GeV. We expect our TMD
description to hold at low $P_{T}$’s (more properly where the ratio
${P_{T}}/{z_{h}\,Q}$ is small). In our case $P_{T}$ should certainly be no
larger than $1.2-1.5$ Gev/c, depending on the corresponding value of $z_{h}$.

Fig. 6 shows the comparison between the BELLE experimental measurements and
our predictions, represented by the solid (red) line, obtained by applying Eq.
(176), giving the $e^{+}e^{-}\to HX$ cross section, to 1-loop and NLL
accuracy, in the 2 jet limit. Our results are presented without uncertainty
bands as there is no fitting involved in this calculation. The uncertainty
induced by the error on the collinear FFs is not shown in this plot. The
agreement of our results with experimental data is extraordinary, and gives
strong support to the naturalness with which Eq. (176) works. In particular,
our plots show how the full range of $z_{h}$ values explored by BELLE
measurements can be described to an unprecedented level of quality.

Figure 6: The $e^{+}e^{-}\to hX$ cross section computed to NLO and NLL
accuracy, corresponding to 15 $z_{h}$-bins as measured by the BELLE
Collaboration. The central values of the $z_{h}$-bins is indicated in each
panel. Thrust is fixed to one bin, corresponding to $0.85<T<0.9$. Uncertainty
bands are not shown as no fit is involved in this computation. The collinear
fragmentation functions used for the implementation of Eq. (176) are the
NNFF10_NLO reference set Bertone:2017tyb , for $\pi^{+}$ production.
Uncertainties due to the errors on the collinear fragmentation functions are
not shown.

To analyse the $T$ behaviour of the $e^{+}e^{-}\to HX$ cross sections we will
refer to a different subset of the BELLE experimental data, large enough to
span the whole range of values of $T$, $z_{h}$ and $P_{T}$ where Eq. (176) can
safely be applied. In particular we will consider three bins in $T$,
corresponding to $0.70<T<0.80$, $0.80<T<0.85$ and $0.85<T<0.90$. In each of
these T bins, we will consider three bins in $z_{h}$, corresponding to
$0.35<z_{h}<0.40$, $0.55<z_{h}<0.6$ and $0.75<z_{h}<0.80$.

The results obtained by using Eq. (176) are shown in Fig. 7, where they are
compared to the BELLE experimental data corresponding to the nine $T$ and
$z_{h}$ bins described above (individual plot legends clearly indicate their
values in each panel). As in Fig. 6, error bars corresponds to statistical and
systematical errors added in quadrature. The solid lines correspond to our
prediction for the $e^{+}e^{-}\to HX$ cross section to NLO and NLL accuracy,
in the 2 jet limit: they are presented without uncertainty bands as there is
no fitting involved in this calculation. The uncertainty induced by the error
on the collinear unpolarized FF is not shown. As for the previous plots, the
agreement with the experimental data is extremely good, especially if one
considers that it has been obtained without a proper fit or a fine tuning of
the parameters. This confirms the remarkable success of Eq. (176), which can
reproduce the experimentally measured $e^{+}e^{-}\to H\,X$ cross section in a
perfectly natural way, and with a very mild dependence on the specific values
of the free parameters.

Outside of the considered range of $z_{h}$ and $T$ values, the quality of the
description starts deteriorating slowly. This is more evident in Fig. 8 where
our predictions of the $e^{+}e^{-}\to H\,X$ cross section are plotted as
functions of thrust, $T$, in three $z_{h}$ bins and three representative
$P_{T}$ bins (one very low, one central and one rather large value of $P_{T}$,
within the TMD range). While the last $T$-bin is always greatly overestimated,
the description of data deteriorates at extreme values of $z_{h}$.

Figure 7: $e^{+}e^{-}\to hX$ cross section computed to NLO and NLL accuracy,
corresponding to 9 of the bins measured by the BELLE Collaboration. The value
of $z_{h}$ increases by moving down in each column of panels: the upper row of
panels corresponds to $z_{h}=0.375$, the middle row to $z_{h}=0.575$, the
lowest row to $z_{h}=0.775$. Moving left to right, instead, corresponds to
increasing values of thrust. In the leftmost column of panels $T=0.750$, in
the central column $T=0.825$, in the rightmost column $T=0.875$. Uncertainty
bands are not shown as no fit is involved in this computation. The collinear
fragmentation functions used for the implementation of Eq. (176) are the
NNFF10_NLO reference set Bertone:2017tyb  for $\pi^{+}$
production.Uncertainties due to the errors on the collinear fragmentation
functions are not shown.

This is somehow expected. In fact, as we discussed in Section 4, a combined
resummation of Eq. (176) in $T$ and $z_{h}$ would allow us to take into
account terms which are presently neglected and restore a good agreement down
to the lowest values of $z_{h}$. Such a resummation is a project of its own,
which we might consider undertaking in the future.

The high quality of the description of the BELLE experimental data as a
function of thrust represents our most important achievement. For the first
time after their publication, these valuable measurements can be fully
explained, in their $z_{h}$, $P_{T}$ and $T$ behaviour, within a sound and
well founded scheme of factorization, with minimal bias from non-perturbative
free parameters.

Figure 8: $e^{+}e^{-}\to hX$ cross section computed to NLO and NLL accuracy
plotted as a function of thrust, $T$, in three $z$-bins and three
(representative) $P_{T}$-bins, as indicated in the plot legends. In the upper
row of panels $z=0.775$, in the central row $z=0.575$ and in the lower row of
panels $z=0.375$. Uncertainty bands are not shown as no fit is involved in
this computation. The collinear fragmentation functions used for the
implementation of Eq. (176) are the NNFF10_NLO reference set for $\pi^{+}$
production Bertone:2017tyb .Uncertainties due to the errors on the collinear
fragmentation functions are not shown.

##  6 Conclusions

In this paper we have constructed the factorized $e^{+}e^{-}\to HX$ cross
section in the 2-jet limit, providing its explicit expression to NLO and NLL
accuracy. Working in a CSS-like framework, factorization has been built, step
by step, by means of power counting rules and applying an appropriate
subtraction mechanism which allowed us to neatly separate the short distance,
low-$b_{T}$ behaviour, from the long-distance behaviour. In this way, the
cross section results in the convolution of the partonic cross section, which
in our formalism includes all of the hard contributions, with a TMD
fragmentation function, defined according to the scheme presented in Ref.
Boglione:2020cwn . Here TMDs are purely collinear parts, universal and totally
process independent.

The 2-jet limit strongly constrains the final state topology to an extreme
configuration, in which the available phase space is strongly reduced, as
shown in Figs. 2 and 3. In our treatment, this is realized by introducing in
the computation of the partonic cross section a cut-off $\tau_{\mbox{\tiny
MAX}}$, ultimately related to the measured value of thrust. As a consequence,
the TMD FFs appearing in the final cross section acquire a dependence on the
thrust, through their rapidity cut-offs.

In Section 5 we have performed a basic phenomenological analysis in which we
compare the BELLE experimental data to the results obtained by using of our
final formula, Eq. (176), which is differential in $z_{h}$ and $P_{T}$ as well
as in thrust, $T$. This is the first phenomenological analysis of the thrust
dependence of the $e^{+}e^{-}\to HX$ cross section. The agreement of our
predictions with experimental data is excellent, even without performing a
proper fit. In fact, in our calculation we fix the value of the three free
parameters which regulate the non-perturbative behaviour of the TMD FF (one
for $g_{k}$ and two for the model $M_{D}$) by completely general
considerations, based on pure common sense.

The results obtained are very promising, and will represent the basis on which
we will construct a more refined, global analysis, simultaneously considering
the unpolarized cross sections from SIDIS and $e^{+}e^{-}\to HX$ processes.
This will be possible thanks to the formalism proposed in Ref.
Boglione:2020cwn , that allows to define any TMD parton density in a
completely universal way, separating out the soft and process dependent part,
and finally restoring the possibility to perform global analyses including
data from processes belonging to different hadron-classes.

## Acknowledgements

We are grateful to V. Bertone and E. Nocera for providing us with valuable
information about the NLO Neural Network Fragmentation Function sets for pion
production, and J.O. Gonzalez-Hernandez for useful comments and suggestions.
This project has received funding from the European Union’s Horizon 2020
research and innovation programme under grant agreement No 824093.

##  Appendix A Lowest Order Partonic Cross Section

In this section we will review the computation of the lowest order (LO)
partonic cross section, in the $2$-jet limit. Its final state tensor
corresponds to the $e^{+}e^{-}\to q\bar{q}$ Feynman graph shown in Fig. 9. The
LO subtraction mechanism is trivial. In fact, from Eq. (24) we have:

$\displaystyle\left(\widehat{W}_{f,\,R}^{\mu\,\nu}{}^{\mbox{\small,
sub}}(z,\,\tau)\right)^{[0]}=\left(\widehat{W}_{f}^{\mu\,\nu}{}^{\mbox{\small,
unsub}}(z,\,\tau)\right)^{[0]}.$ (180)

Since here the final state topology can only be pencil-like, i.e. $\tau=0$,
there is no need of introducing a thrust cut-off.

Figure 9: The lowest order Feynman graph contributing to the partonic final
state tensor.

For simplicity, we will assume that the fragmenting parton is a quark. Clearly
a similar treatment would apply to the fragmentation of an antiquark. As
usual, we will perform the computation in the partonic analogue of the hadron
frame, where the outgoing parton has momentum $k_{1}$ where only the plus
component is non-vanishing (as in Eq. (23)). As illustrated in Fig. 9, the
only matrix element we have to consider is given by:

$\displaystyle
M_{f}^{\mu\,\nu}{}^{\;[0]}(\epsilon;\,\mu,\,Q)=\begin{gathered}\includegraphics[width=150.79968pt]{M_lo}\end{gathered}=$
(182)
$\displaystyle\quad=e_{f}^{2}\,\sum_{s_{1},\,c}\,\bar{u}_{c,\,f}(k_{1},\,s_{1})\,\gamma^{\mu}\,\not{k}_{2}\,\gamma^{\nu}\,u_{c,\,f}(k_{1},\,s_{1})=e_{f}^{2}\,N_{C}\,\mbox{Tr}\left\\{\not{k}_{1}\,\gamma^{\mu}\,\not{k}_{2}\,\gamma^{\nu}\right\\},$
(183)

where we sum over all the spins and the colors of the fragmenting quark of
flavour $f$ and fractional electric charge $e_{f}$. The projections onto the
relevant Lorentz structures, as in Eqs. (14) and (15), are:

$\displaystyle-
g_{\mu\nu}M_{f}^{\mu\,\nu}{}^{\;[0]}=8\,e_{f}^{2}\,N_{C}(1-\epsilon)\,k_{1}\cdot
k_{2}=4\,e_{f}^{2}\,N_{C}(1-\epsilon)Q^{2};$ (184)
$\displaystyle\frac{k_{1,\,\mu}\,k_{1,\,\nu}}{Q^{2}}M_{f}^{\mu\,\nu}{}^{\;[0]}=0,$
(185)

where we used momentum conservation $k_{1}+k_{2}=q$. Since the projection of
Eq. (185) is zero, the final state tensor will be written as in Eq. (21) and
the partonic cross section will only have the transverse projection. The phase
space integral only involves $k_{2}$ and it is straightforward:

$\displaystyle-
g_{\mu\nu}\widehat{W}_{f}^{\mu\,\nu}{}^{\;[0]}=\frac{1}{4\pi}\,\int\frac{d^{D}k_{2}}{(2\pi)^{D}}\,\left(-g_{\mu\nu}M_{f}^{\mu\,\nu}{}^{\;[0]}\right)\,(2\pi)^{D}\,\delta^{(D)}\left(q-k_{1}-k_{2}\right)\,\times$
$\displaystyle\quad\times\,(2\pi)\,\delta\left(k_{2}^{2}\right)\,\theta(k_{2}^{+})\,\theta(k_{2}^{-})\,\delta(\tau)=$
$\displaystyle\quad=2\,e_{f}^{2}\,N_{C}(1-\epsilon)\,\delta\left(1-z\right)\,\delta(\tau)=$
$\displaystyle\quad=2\widehat{F}_{1,\,f}^{[0]},$ (186)

where the factor $1/(4\pi)$ is the normalization of the hadronic tensor fixed
according to Ref. Collins:2011zzd  and. Boglione:2020cwn  and the last line is
obtained using Eqs. (14) and (15). Finally:

$\displaystyle\widehat{W}_{f}^{\mu\,\nu}{}^{\;[0]}(z,\,\tau)=H_{T}^{\mu\nu}(z)\,\widehat{F}_{1,\,f}^{[0]}(z,\,\tau),$
(187)

where the transverse tensor $H_{T}^{\mu\nu}$ has been defined in Eq. (22). The
resulting partonic cross section follows from Eq. (16) and is given by:

$\displaystyle\frac{d\widehat{\sigma}_{f}^{[0]}}{dz\,dT}=\sigma_{B}\,z\,\widehat{F}_{1,\,f}^{[0]}(z,\,\tau)=\frac{4\pi\alpha^{2}}{3Q^{2}}\,e_{f}^{2}\,N_{C}\,\delta\left(1-z\right)\,\delta(\tau)$
(188)

where the Born cross section $\sigma_{B}$ has been defined in Eq. (18).

##  Appendix B Thrust Divergent Functions

In $e^{+}e^{-}$ scattering, the topology of the final state is related to the
value of thrust, $T$, defined in Eq. (3). In the following, we will focus on
the quasi $2$-jet configuration, where $T$ can be considered sufficiently
large. In this case, the thrust axis $\vec{n}$ defines two opposite
hemispheres: $S_{n}$, which points along the direction of the thrust axis, and
$S_{\bar{n}}$, which points backwards, along the direction $\vec{\bar{n}}$
opposite to $\vec{n}$. Then, Eq. (3) can be approximated by (see Ref.
Catani:1991kz  and Catani:1992ua ):

$\displaystyle\tau=1-T=\frac{M_{n}^{2}}{Q^{2}}+\frac{M_{\bar{n}}^{2}}{Q^{2}}.$
(189)

where $M_{n}$ and $M_{\bar{n}}$ are the invariant masses of the two almost
back-to-back jets. These invariant masses can be computed explicitly at
partonic level, order by order in perturbation theory, by considering the full
momentum flowing into each hemisphere. Clearly, the r.h.s. of Eq. (189) is
trivially zero only when all partons emitted into each hemisphere are virtual.
All other cases require at least one real emission. Therefore, at 1-loop level
the non-trivial configurations have three real particles in the final state. A
quasi $2$-jet configuration can be achieved either when one of them is soft or
when two of them move collinearly. In the following, we will review the 1-loop
computation of the contribution of each of such final state configurations,
obtained by applying the usual soft/collinear approximations to Feynman graphs
(see Ref. Collins:2011zzd ) and equipping them with a delta function that
fixes the value of $\tau=1-T$ according to Eq. (189). The thrust axis
$\vec{n}$ will be identified with the $z$-axis of the frame used to compute
the partonic cross section, i.e. the partonic analogue of the hadron frame,
where the outgoing parton has zero transverse momentum. All quantities will be
computed in dimensional regularization, with $D=4-2\epsilon$ being the
spacetime dimension.

###  B.1 Soft Thrust Function

The 1-loop soft thrust function $S(\epsilon,\,\tau)$ is obtained as a result
of the application of the soft approximation ($T_{S}$ in Ref. Collins:2011zzd
) to the Feynman graphs corresponding the 1-loop partonic cross section. The
case of the fragmenting gluon is suppressed in this approximation, therefore
it can be neglected. On the other hand, when the fragmenting parton is a
fermion, the emitted gluon can be either virtual or real. The virtual
contribution vanishes in full dimensional regularization, hence the soft
thrust function only originates from the real emission graphs. In order to
compute the r.h.s. of Eq. (189), we will label the momenta of the three final
state particles as in Section 2.2.1, where $k_{1}$ refers to the fragmenting
quark, $k_{2}$ to the antiquark and $k_{3}$ to the gluon. In this case we have
to use their soft approximations:

$\displaystyle k_{1}^{(S)}=\left(q^{+},\,0,\,\vec{0}_{T}\right),$ (190)
$\displaystyle k_{2}^{(S)}=\left(0,\,q^{-},\,\vec{0}_{T}\right),$ (191)
$\displaystyle k_{3}^{(S)}=\left(l^{+},\,l^{-},\,\vec{l}_{T}\right),$ (192)

where $q^{+}=q^{-}={Q}/{\sqrt{2}}$. Notice that the soft approximation leaves
$k_{3}$ unchanged in its functional form, but the size of all its components
are now of order ${\lambda^{2}}/{Q}$, where $\lambda$ is a very low energy
scale according to power counting. Depending on the hemisphere in which the
soft gluon has been emitted, we have two different, equally probable,
configurations:

$\displaystyle\tau=y_{2}^{(S)}=\frac{l^{-}}{q^{-}},\mbox{ if }S_{n}\mbox{
emission, i.e. }l^{+}>l^{-};$ (193)
$\displaystyle\tau=y_{1}^{(S)}=\frac{l^{+}}{q^{+}},\mbox{ if
}S_{\bar{n}}\mbox{ emission, i.e. }l^{-}>l^{+}.$ (194)

Therefore, by defining the usual light-like directions
$w_{1}=(1,\,0,\,\vec{0}_{T})$ and $w_{2}=(0,\,1,\,\vec{0}_{T})$, the 1-loop
soft thrust function is obtained as:

$\displaystyle\frac{\alpha_{S}}{4\pi}S^{[1]}(\epsilon,\,\tau)=\int\frac{d^{D}l}{(2\pi)^{D}}\,\frac{\mbox{Tr}_{C}}{N_{C}}\,\Bigg{[}\begin{gathered}\includegraphics[width=102.43008pt]{soft_thrust}\end{gathered}+h.c.\Bigg{]}\,(2\pi)\delta_{+}\left(l^{2}\right)\,\times$
(196)
$\displaystyle\quad\times\,\left[\delta\left(\tau-\frac{l^{-}}{q^{-}}\right)\,\theta\left(l^{+}-l^{-}\right)+\delta\left(\tau-\frac{l^{+}}{q^{+}}\right)\,\theta\left(l^{-}-l^{+}\right)\right]=$
$\displaystyle\quad=\frac{\alpha_{S}}{4\pi}\,8\,C_{F}\,S_{\epsilon}\,\left(\frac{\mu}{Q}\right)^{2\epsilon}\,\tau^{-1-2\epsilon}\,\frac{1}{\epsilon}.$
(197)

Notice that all the divergences of $S(\epsilon,\,\tau)$ are regulated by
dimensional regularization, and we did not encounter any unregulated rapidity
divergence. This is due to the explicit presence of $\tau$, that acts as a
regulator. In fact, the expression in Eq. (197) is divergent if either
$\epsilon$ or $\tau$ vanish. The $\epsilon$-expansion of Eq. (197) follows
from its integrability property. In fact, $S^{[1]}(\epsilon,\,\tau)$ is
integrable with respect to $\tau$, but its trivial expansion in powers of
$\epsilon$ does not. This problem is solved by considering an expansion in
terms of $\tau$ distributions instead of functions. Then, one can easily prove
that:

$\displaystyle\tau^{-1-\epsilon}=-\frac{1}{\epsilon}\,\delta(\tau)+\left(\frac{1}{\tau}\right)_{+}-\epsilon\,\left(\frac{\log{\tau}}{\tau}\right)_{+}+\mathcal{O}\left(\epsilon^{2}\right),\quad\mbox{Re~{}}{\epsilon}<0.$
(198)

###  B.2 Backward Thrust Function

The 1-loop backward thrust function $J_{B}(\epsilon,\,\tau)$ is obtained by
applying the collinear approximation to the backward emission (analogue to
$T_{B}$ in Ref. Collins:2011zzd ) in the 1-loop Feynman graphs. Here
”backward” refers to the direction of emission, which goes along
$\vec{\bar{n}}$ (negative $z$-axis), opposite to the thrust axis. As for the
previous case, the contribution of the fragmenting gluon is suppressed by
power counting and virtual graphs are zero in full dimensional regularization.
In the following, we will assume that the fragmenting parton is a quark, hence
the approximation concerns the configuration where the gluon is collinear to
the antiquark. The case of a fragmenting antiquark is perfectly analogous. The
backward approximation applied to the momenta (labeled as usual) gives:

$\displaystyle k_{1}^{(B)}=\left(q^{+},\,0,\,\vec{0}_{T}\right),$ (199)
$\displaystyle
k_{2}^{(B)}=\left(\frac{l_{T}^{2}}{2(q^{-}-l^{-})},\,q^{-}-l^{-},\,-\vec{l}_{T}\right),$
(200) $\displaystyle
k_{3}^{(B)}=\left(\frac{l_{T}^{2}}{2l^{-}},\,l^{-},\,\vec{l}_{T}\right),$
(201)

As in the previous case, the backward approximation leaves $k_{3}$ unchanged
in its functional form, but resizes its components as
$l^{+}\sim{\lambda^{2}}/{Q}$, $l^{-}\sim Q$ and $l_{T}\sim\lambda$, where
$\lambda$ is a very low energy scale according to power counting. The emission
of the collinear gluon along the direction of the fragmenting quark does not
give any contribution, therefore we only have to consider the emission in the
antiquark hemisphere:

$\displaystyle\tau=y_{1}^{(B)}=\frac{l_{T}^{2}}{Q^{2}}\,\frac{q^{-}}{l^{-}}\,\frac{1}{1-\frac{l^{-}}{q^{-}}},\mbox{
if }S_{\bar{n}}\mbox{ emission, i.e. }l^{-}>l^{+}.$ (202)

By defining $k=k_{2}^{(B)}+k_{3}^{(B)}$ as the total momentum entering into
the antiquark hemisphere, the 1-loop backward thrust function is:

$\displaystyle\frac{\alpha_{S}}{4\pi}J_{B}^{[1]}(\epsilon,\,\tau)=2\,(2\pi)^{D-1}\,\int\frac{dk^{+}\,d^{D-2}\vec{k}_{T}}{(2\pi)^{D}}\,\int\frac{d^{D}k_{2}}{(2\pi)^{D}}\,\times$
$\displaystyle\quad\times\,\frac{\mbox{Tr}_{C}}{N_{C}}\,\frac{\mbox{Tr}}{4}\,\Bigg{[}\gamma^{-}\,\Bigg{(}\begin{gathered}\includegraphics[width=130.88284pt]{B_thrust_X.png}\end{gathered}+h.c.+\,\begin{gathered}\includegraphics[width=93.89418pt]{B_thrust_Urev.png}\end{gathered}\Bigg{)}\Bigg{]}\,\times$
(205)
$\displaystyle\quad\times\,(2\pi)\delta_{+}\left(k_{2}^{2}\right)\,(2\pi)\delta_{+}\left((k-k_{2})^{2}\right)\,\delta^{(D-2)}(\vec{k}_{T})\,\delta\left(\tau-\frac{k^{+}}{q^{+}}\right)\,\theta\left(k^{-}-k_{2}^{-}\right)=$
$\displaystyle\quad=\frac{\alpha_{S}}{4\pi}\,4\,C_{F}\,S_{\epsilon}\,\left(\frac{\mu}{Q}\right)^{2\epsilon}\,\tau^{-1-\epsilon}\,\left[B(2-\epsilon,\,-\epsilon)+\frac{1-\epsilon}{2}\,B(1-\epsilon,\,2-\epsilon)\right].$
(206)

Notice that in $J_{B}(\epsilon,\,\tau)$ the thrust acts as a regulator for the
rapidity divergences, as in Eq. (197). As a consequence, there is no
overlapping with the soft momentum region and, contrary to the CSS scheme, no
subtraction is required.

###  B.3 Jet Thrust Function

The 1-loop jet thrust function is obtained by applying the collinear
approximation to the emission along the fragmenting parton direction (the
analogue of $T_{A}$ in Ref. Collins:2011zzd ) in all Feynman graphs needed for
the 1-loop partonic cross section. The direction of the fragmenting parton
coincides with the thrust axis $\vec{n}$ (positive $z$-axis). The approximated
momenta (labeled as usual) are:

$\displaystyle k_{1}^{(J)}=\left(zq^{+},\,0,\,\vec{0}_{T}\right),$ (207)
$\displaystyle k_{2}^{(J)}=\left(0,\,q^{-},\,\vec{0}_{T}\right),$ (208)
$\displaystyle
k_{3}^{(J)}=\left((1-z)q^{+},\,\frac{l_{T}^{2}}{2(1-z)q^{+}},\,\vec{l}_{T}\right),$
(209)

where $z$ represents the fractional energy associated to the partonic process:

$\displaystyle z=\frac{2k_{1}\cdot q}{q^{2}}=\frac{k_{1}^{+}}{q^{+}}.$ (210)

The emission in the hemisphere opposite to the direction of the fragmenting
parton does not give any contribution. Therefore, the only relevant
configuration is given by:

$\displaystyle\tau=y_{2}^{(J)}=\frac{z}{1-z}\,\frac{l_{T}^{2}}{Q^{2}},\mbox{
if }S_{n}\mbox{ emission, i.e. }l^{+}>l^{-}.$ (211)

In this case, the contribution of fragmenting gluons is not suppressed and
needs to be computed explicitly. By defining $k=k_{1}^{(J)}+k_{3}^{(J)}$ the
total momentum entering the considered hemisphere, the 1-loop thrust jet
function corresponding to a fragmenting gluon is given by:

$\displaystyle\frac{\alpha_{S}}{4\pi}J_{g/q}^{[1]}(\epsilon;\,\tau,\,z)=\int\frac{dk^{-}}{(2\pi)^{D}}\,d^{2-2\epsilon}\vec{k}_{T}\,\frac{\mbox{Tr}_{C}}{N_{C}}\,\frac{\mbox{Tr}_{D}}{4}\,\Bigg{[}\gamma^{+}\,\begin{gathered}\includegraphics[width=79.6678pt]{A_thrust_gluon.png}\end{gathered}\Bigg{]}\,\times$
(213)
$\displaystyle\quad\times\,(2\pi)\,\delta\left((k-k_{1})^{2}\right)\,\delta\left(\tau-\frac{z}{1-z}\,\frac{k_{T}^{2}}{Q^{2}}\right)\,\theta(1-z)=$
$\displaystyle\quad=\frac{\alpha_{S}}{4\pi}\,2\,C_{F}\,S_{\epsilon}\,\theta(1-z)\,\frac{1+(1-z)^{2}-\epsilon
z^{2}}{z^{2}}\,\times$
$\displaystyle\quad\times\,\frac{\Gamma(1-\epsilon)}{\pi^{1-\epsilon}}\,\mu^{2\epsilon}\,\int
d^{2-2\epsilon}\vec{k}_{T}\,\frac{1}{k_{T}^{2}}\,\delta\left(\tau-\frac{z}{1-z}\,\frac{k_{T}^{2}}{Q^{2}}\right)=$
$\displaystyle\quad=\frac{\alpha_{S}}{4\pi}\,2\,C_{F}\,S_{\epsilon}\,\left(\frac{\mu}{Q}\right)^{2\epsilon}\,\theta(1-z)\,\left(\frac{z}{1-z}\right)^{\epsilon}\,\frac{1+(1-z)^{2}-\epsilon
z^{2}}{z^{2}}\,\tau^{-1-\epsilon}.$ (214)

The case of a fragmenting fermion involves more Feynman graphs. The
contribution of the virtual emission is zero in full dimensional
regularization. The 1-loop jet thrust function associated to a fragmenting
quark is given by:

$\displaystyle\frac{\alpha_{S}}{4\pi}J_{q/q}^{[1]}(\epsilon;\,\tau,\,z)=\int\frac{dk^{-}}{(2\pi)^{D}}\,d^{2-2\epsilon}\vec{k}_{T}\,\times$
$\displaystyle\quad\times\,\frac{\mbox{Tr}_{C}}{N_{C}}\,\frac{\mbox{Tr}_{D}}{4}\,\Bigg{[}\gamma^{+}\,\Bigg{(}\begin{gathered}\includegraphics[width=119.50148pt]{A_thrust_X.png}\end{gathered}+h.c.+\,\begin{gathered}\includegraphics[width=82.51282pt]{A_thrust_U.png}\end{gathered}\Bigg{)}\Bigg{]}\,\times$
(217)
$\displaystyle\quad\times\,\theta(1-z)\,\delta\left(\tau-\frac{z}{1-z}\,\frac{k_{T}^{2}}{Q^{2}}\right)=$
$\displaystyle\quad=\frac{\alpha_{S}}{4\pi}\,2\,C_{F}\,S_{\epsilon}\,\theta(1-z)\,\left[\frac{2}{(1-z)}+\frac{(1-\epsilon)\,(1-z)}{z}\right]\,\times$
$\displaystyle\quad\times\,\frac{\Gamma(1-\epsilon)}{\pi^{1-\epsilon}}\,\mu^{2\epsilon}\,\int
d^{2-2\epsilon}\vec{k}_{T}\frac{1}{k_{T}^{2}}\,\delta\left(\tau-\frac{z}{1-z}\,\frac{k_{T}^{2}}{Q^{2}}\right)=$
$\displaystyle\quad=\frac{\alpha_{S}}{4\pi}\,4\,C_{F}\,S_{\epsilon}\,\left(\frac{\mu}{Q}\right)^{2\epsilon}\,\theta(1-z)\,z^{\epsilon}\,\left[(1-z)^{-1-\epsilon}+\frac{1-\epsilon}{2}\,\frac{(1-z)^{1-\epsilon}}{z}\right]\tau^{-1-\epsilon}.$
(218)

Notice that in this case, the $\epsilon$-expansion of the above expression
follows from Eq. (198) and from the analogous expansion for $z$:

$\displaystyle(1-z)^{-1-\epsilon}=-\frac{1}{\epsilon}\,\delta(1-z)+\left(\frac{1}{1-z}\right)_{+}-\epsilon\,\left(\frac{\log{(1-z)}}{1-z}\right)_{+}+\mathcal{O}\left(\epsilon^{2}\right),\quad\mbox{Re~{}}{\epsilon}<0.$
(219)

## References

  * (1) Belle Collaboration, R. Seidl et al., Transverse momentum dependent production cross sections of charged pions, kaons and protons produced in inclusive $e^{+}e^{-}$ annihilation at $\sqrt{s}=$ 10.58 GeV, Phys. Rev. D99 (2019), no. 11 112006, [arXiv:1902.0155].
  * (2) J. C. Collins, D. E. Soper, and G. F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl.Phys. B250 (1985) 199.
  * (3) J. C. Collins, Sudakov form-factors, Adv. Ser. Direct. High Energy Phys. 5 (1989) 573–614, [hep-ph/0312336].
  * (4) J. C. Collins, D. E. Soper, and G. F. Sterman, Factorization of Hard Processes in QCD, Adv. Ser. Direct. High Energy Phys. 5 (1989) 1–91, [hep-ph/0409313].
  * (5) J. C. Collins, Fragmentation of transversely polarized quarks probed in transverse momentum distributions, Nucl. Phys. B396 (1993) 161–182.
  * (6) J. C. Collins, S. F. Heppelmann, and G. A. Ladinsky, Measuring transversity densities in singly polarized hadron hadron and lepton - hadron collisions, Nucl.Phys. B420 (1994) 565–582, [hep-ph/9305309].
  * (7) J. Collins, Foundations of perturbative QCD.  Cambridge University Press, 2011.
  * (8) M. Boglione and A. Simonelli, Universality-breaking effects in $e^{+}e^{-}$ hadronic production processes, arXiv:2007.1367.
  * (9) D. Neill, I. Scimemi, and W. J. Waalewijn, Jet axes and universal transverse-momentum-dependent fragmentation, JHEP 04 (2017) 020, [arXiv:1612.0481].
  * (10) Z.-B. Kang, X. Liu, F. Ringer, and H. Xing, The transverse momentum distribution of hadrons within jets, JHEP 11 (2017) 068, [arXiv:1705.0844].
  * (11) Z.-B. Kang, F. Ringer, and W. J. Waalewijn, The Energy Distribution of Subjets and the Jet Shape, JHEP 07 (2017) 064, [arXiv:1705.0537].
  * (12) Z.-B. Kang, D. Y. Shao, and F. Zhao, QCD resummation on single hadron transverse momentum distribution with the thrust axis, arXiv:2007.1442.
  * (13) Y. Makris, F. Ringer, and W. J. Waalewijn, Joint thrust and TMD resummation in electron-positron and electron-proton collisions, arXiv:2009.1187.
  * (14) G. Passarino and M. Veltman, One Loop Corrections for $e^{+}e^{-}$ Annihilation Into $\mu^{+}\mu^{-}$ in the Weinberg Model, Nucl. Phys. B 160 (1979) 151–207.
  * (15) S. Catani, G. Turnock, B. R. Webber, and L. Trentadue, Thrust distribution in $e^{+}e^{-}$ annihilation, Phys. Lett. B263 (1991) 491–497.
  * (16) G. F. Sterman, Partons, factorization and resummation, TASI 95, in Theoretical Advanced Study Institute in Elementary Particle Physics (TASI 95): QCD and Beyond, pp. 327–408, 6, 1995.  hep-ph/9606312.
  * (17) A. Gehrmann-De Ridder, T. Gehrmann, E. Glover, and G. Heinrich, NNLO corrections to event shapes in $e^{+}e^{-}$ annihilation, JHEP 12 (2007) 094, [arXiv:0711.4711].
  * (18) T. Becher and M. D. Schwartz, A precise determination of $\alpha_{s}$ from LEP thrust data using effective field theory, JHEP 07 (2008) 034, [arXiv:0803.0342].
  * (19) S. Catani, L. Trentadue, G. Turnock, and B. Webber, Resummation of large logarithms in $e^{+}e^{-}$ event shape distributions, Nucl. Phys. B 407 (1993) 3–42.
  * (20) P. F. Monni, T. Gehrmann, and G. Luisoni, Two-Loop Soft Corrections and Resummation of the Thrust Distribution in the Dijet Region, JHEP 08 (2011) 010, [arXiv:1105.4560].
  * (21) M. D. Schwartz, Quantum Field Theory and the Standard Model.  Cambridge University Press, 3, 2014.
  * (22) M. Dasgupta and G. Salam, Resummation of nonglobal QCD observables, Phys. Lett. B 512 (2001) 323–330, [hep-ph/0104277].
  * (23) A. Banfi, G. Marchesini, and G. Smye, Away from jet energy flow, JHEP 08 (2002) 006, [hep-ph/0206076].
  * (24) A. J. Larkoski, I. Moult, and D. Neill, Non-Global Logarithms, Factorization, and the Soft Substructure of Jets, JHEP 09 (2015) 143, [arXiv:1501.0459].
  * (25) T. Becher, R. Rahn, and D. Y. Shao, Non-global and rapidity logarithms in narrow jet broadening, JHEP 10 (2017) 030, [arXiv:1708.0451].
  * (26) S. M. Aybat and T. C. Rogers, TMD Parton Distribution and Fragmentation Functions with QCD Evolution, Phys.Rev. D83 (2011) 114042, [arXiv:1101.5057].
  * (27) S. M. Aybat, J. C. Collins, J.-W. Qiu, and T. C. Rogers, The QCD Evolution of the Sivers Function, Phys.Rev. D85 (2012) 034043, [arXiv:1110.6428].
  * (28) R. F. del Castillo, M. G. Echevarria, Y. Makris, and I. Scimemi, TMD factorization for di-jet and heavy meson pair in DIS, arXiv:2008.0753.
  * (29) M. G. Echevarria, I. Scimemi, and A. Vladimirov, Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order, JHEP 09 (2016) 004, [arXiv:1604.0786].
  * (30) J. Collins and T. C. Rogers, Connecting Different TMD Factorization Formalisms in QCD, Phys. Rev. D 96 (2017), no. 5 054011, [arXiv:1705.0716].
  * (31) Y. Koike, J. Nagashima, and W. Vogelsang, Resummation for polarized semi-inclusive deep-inelastic scattering at small transverse momentum, Nucl.Phys. B744 (2006) 59–79, [hep-ph/0602188].
  * (32) M. Boglione, J. Gonzalez-Hernandez, and R. Taghavi, Transverse parton momenta in single inclusive hadron production in ${e^{+}}{e^{-}}$ annihilation processes, Phys. Lett. B 772 (2017) 78–86, [arXiv:1704.0888].
  * (33) J. Collins, L. Gamberg, A. Prokudin, T. C. Rogers, N. Sato, and B. Wang, Relating Transverse Momentum Dependent and Collinear Factorization Theorems in a Generalized Formalism, Phys. Rev. D94 (2016), no. 3 034014, [arXiv:1605.0067].
  * (34) M. Anselmino, M. Boglione, J. Gonzalez Hernandez, S. Melis, and A. Prokudin, Unpolarised Transverse Momentum Dependent Distribution and Fragmentation Functions from SIDIS Multiplicities, JHEP 04 (2014) 005, [arXiv:1312.6261].
  * (35) C. Aidala, B. Field, L. Gamberg, and T. Rogers, Limits on TMD Evolution From Semi-Inclusive Deep Inelastic Scattering at Moderate $Q$, Phys.Rev. D89 (2014) 094002, [arXiv:1401.2654].
  * (36) M. Boglione, J. O. Gonzalez Hernandez, S. Melis, and A. Prokudin, A study on the interplay between perturbative QCD and CSS/TMD formalism in SIDIS processes, JHEP 02 (2015) 095, [arXiv:1412.1383].
  * (37) M. Boglione, U. D’Alesio, C. Flore, and J. O. Gonzalez-Hernandez, Assessing signals of TMD physics in SIDIS azimuthal asymmetries and in the extraction of the Sivers function, JHEP 07 (2018) 148, [arXiv:1806.1064].
  * (38) NNPDF Collaboration, V. Bertone, S. Carrazza, N. P. Hartland, E. R. Nocera, and J. Rojo, A determination of the fragmentation functions of pions, kaons, and protons with faithful uncertainties, Eur. Phys. J. C 77 (2017), no. 8 516, [arXiv:1706.0704].


as this corresponds to a holomorphic disk with a flow line from the minimum
attached and such a configuration cannot be rigid unless the disk is constant.

######  Theorem 69.

The maps $\Phi_{i}$ gives an $A_{\infty}$-map $CW^{\ast}(C)\to
CE^{\ast}(\Lambda)$ which is an $A_{\infty}$ quasi-isomorphism.

Sketch of proof: In order to see the $A_{\infty}$-relations we study the
boundary of the moduli space $\mathcal{M}^{\mathrm{ha}}(\mathbf{c})$. As usual
the boundary numbering guarantees that there is no boundary splitting on $C$.
The boundary of the moduli space thus consists of the following
configurations.

  * $(i)$

Two level curves with one level in the cobordism and one in either
symplectization end.

  * $(ii)$

Curves which split at the intersection point $C\cap L$.

Splitting $(i)$ corresponds to the map followed by the operation $d$ in
$CE^{\ast}(\Lambda)$. Splitting $(ii)$ corresponds to the tensor product of
the map followed by the product operation $\cdot$ in $CE^{\ast}(\Lambda)$. The
$A_{\infty}$-relations follow.

To see that $\Phi$ is a quasi-isomorphism we show that $\Phi_{1}$ induces an
isomorphism on homology by constructing algebraically one holomorphic disks
interpolating between a Reeb chord of $\Gamma$ and the corresponding word of
Reeb chords of $\Lambda$. The existence of such disks implies that the map
$\Phi_{1}$ has a triangular matrix with respect to the action filtration and
hence is a chain isomorphism, compare [11, Section 6.2].

To construct the required interpolating disks one starts from unique and
uniformly transversely cut out such disks for single chord words obtained by a
straightforward explicit geometric construction. Gluing such disks at their
Lagrangian intersection punctures in $L\cap C$ and using small action to rule
out all breakings except one, we find that there is algebraically one disk
interpolating between a chord on $\Gamma$ and the corresponding word of chords
of $\Lambda$, see [12]. The map being an $A_{\infty}$-quasi-isomorphism then
follows from the word length spectral sequence.

######  Remark 70.

A complete proof of Theorem 69 has not yet appeared. In cases when splitting
at Reeb orbits can be neglected it is possible to fill in the details in the
above outline with classical techniques. As mentioned above, in the general
case one must employ abstract perturbations to control holomorphic planes of
anchored disks. The details of the perturbation scheme will not affect the
argument.

######  Remark 71.

There is also an “upside-down” perspective on the surgery just described.
Namely, one can start from the contact manifold $Y$ and produce the contact
manifold $Y_{0}$ by doing so called $+1$-surgery on $\Gamma$. In complete
analogy with the above one shows that Reeb chords on $\Lambda$ are in natural
one to one correspondence with words of Reeb chords on $\Gamma$ and one can
construct an upside down surgery map of $A_{\infty}$ co-algebras:

$\mathrm{B}CW^{\ast}(C)\to LC_{\ast}(\Lambda).$

A similar argument also shows that this map is a quasi-isomorphism.
Alternatively, one can prove this from the original surgery map using only
algebra as follows. First write $CE^{\ast}(\Lambda)=\Omega LC_{\ast}(\Lambda)$
then

$\mathrm{B}CW^{\ast}(C)\simeq\mathrm{B}\Omega LC_{\ast}(\Lambda)\simeq
LC_{\ast}(\Lambda),$

since $LC_{\ast}(\Lambda)$ is co-nilpotent, see Section 2.2.2.

###  B.3 Legendrian surgery and stopped wrapping

In this section we outline a surgery approach to the computation of wrapped
Floer homology in a Weinstein manifold $X$ with wrapping stopped by a
Legendrian $\Lambda$ in its boundary. We will use the following model for the
ambient manifold. Fix a tubular neighborhood of $\Lambda$ in the contact
boundary $Y$ of $X$ and attach a disk-bundle neighborhood of the $0$-section
in $T^{\ast}([0,\infty)\times\Lambda)$ along the boundary
$T^{\ast}([0,\infty)\times\Lambda)|_{0\times\Lambda}$ just like in Lagrangian
handle attachment. We use a Liouville vector field on this domain that agrees
with the standard Liouville vector field pointing outwards along fibers in the
cotangent bundle over $[T,\infty)\times\Lambda$ for some $T>0$. Let the
components of $\Lambda$ be denoted $\Lambda_{v}$, $v\in Q_{0}$. Fix a base
point $p_{v}\in\Lambda_{v}$ for each $v$. Let $C^{v;\tau}$ denote the
cotangent fiber $T^{\ast}_{(p_{v},\tau)}([0,\infty)\times\Lambda)$. We will
compute the wrapped Floer cohomology of $C^{\tau}=\bigcup_{v\in
Q_{0}}C^{v;\tau}$ for sufficiently large $\tau$ using a surgery approach.

We first consider the surgery map into the Chekanov-Eliashberg algebra with
loop space coefficients. Consider all components of $\Lambda$ decorated by a
positive sign $\Lambda=\Lambda^{+}$ and consider $CE^{\ast}(\Lambda)$ which
now involves, except for Reeb chords, also chains $C_{-\ast}(\Omega\Lambda)$
on the based loop space. We define an $A_{\infty}$\- map

$\Phi\colon CW^{\ast}(C)\to CE^{\ast}(\Lambda),$

where $A_{\infty}$-structure on the left hand side is the standard DG-algebra
structure induced by concatenation and the Pontryagin product. As in Section
A, consider a disk $D_{i+j+2}$ with two dividing punctures that subdivides the
boundary into two arcs, lower and upper. Let the upper arc contain $i$
boundary punctures and is equipped with a decreasing boundary numbering
$\kappa$ and the lower arc $j$ boundary punctures. Let
$\mathbf{c}^{\prime}=c_{i}\dots c_{1}$ be Reeb chords of $C$ and let
$\mathbf{c}_{0}=c_{0;1}\dots c_{0;j}$ be Reeb chords of $\Lambda$. Let

$\mathbf{c}=c_{0;1}\dots c_{0;j}z^{v}c_{i}\dots c_{1}\dots z^{w}$

and consider $\mathcal{M}^{\mathrm{co}}(\mathbf{c};\kappa)$, again as in
Figure 14.

Theorems 60 and 61 show that this moduli space carries a fundamental chain. We
view this chain as parametrizing chains of paths in $\Lambda$ connecting the
Reeb chord endpoints in $\mathbf{c}_{0}$. We write
$[\mathcal{M}^{\mathrm{co}}(\mathbf{c})]$ for the alternating word of chains
of loops and Reeb chords and view it as an element in $CE^{\ast}(\Lambda)$.
Define the maps

$\Psi_{i}\colon CW^{\ast}(C)^{\otimes_{\mathbf{k}}i}\to CE^{\ast}(\Lambda)$

as

$\Psi_{i}(\mathbf{c}^{\prime})=\sum_{\mathbf{c}_{0}}[\mathcal{M}^{\mathrm{st}}(\mathbf{c})].$

######  Theorem 72.

The map $\Psi\colon CW^{\ast}(C)\to CE^{\ast}(\Lambda)$ is an
$A_{\infty}$-map.

###### Proof.

To see that the $A_{\infty}$-relation holds we look at the boundary of the
moduli space $\mathcal{M}^{\mathrm{co}}(\mathbf{c})$ of dimension $d$. The
codimension one boundary consists of three splittings:

  * $(i)$

A one-dimensional curve splits off in the positive symplectization end.

  * $(ii)$

A one-dimensional curve splits off at the negative end.

  * $(iii)$

Splitting at one of the intersection points $z^{v}$.

In order for splittings of the form $(i)$ to contribute to the codimension one
boundary of the moduli space the part of the holomorphic building in $W$
consists of rigid disks with only positive punctures attached at one puncture
to a negative puncture of the disk in the positive end and a disk of dimension
$d-1$ in $\mathcal{M}^{\mathrm{co}}(\mathbf{b})$ attached at the remaining
negative puncture. Assembling the rigid disks and the one-dimensional disk we
get a partial holomorphic disk building that contributes to the
$A_{\infty}$-operations in $CW^{\ast}(L)$ followed by the map $\Psi$.
Splittings of type $(ii)$ corresponds to the map $\Psi$ followed by the
differential $\mu_{1}$ in $CE^{\ast}(\Lambda)$. Finally splittings of type
$(iii)$ correspond to the map $\Psi$ followed by the product $\mu_{2}$ on
$CE^{\ast}(\Lambda)$. We conclude that the terms contributing to
$A_{\infty}$-relations express the codimension one boundary of
$[\mathcal{M}^{\mathrm{co}}(\mathbf{c})]$ in two different ways and hence
$\Psi$ is an $A_{\infty}$-map. ∎

We will use slight generalizations of the map $\Psi$ below. More precisely, if
$p_{j}$, $j=1,\dots,m$ are points in $[0,\infty)\times\Lambda$ and if $F_{j}$
is the cotangent fiber at $p_{j}$ then we have a similar surgery map

$\Psi^{p_{i}p_{j}}\colon CW^{\ast}(F_{i},F_{j})\to
CE^{\ast}_{p_{i}p_{j}}(\Lambda),$

which counts holomorphic disks with a positive Reeb chord connecting $F_{i}$
to $F_{j}$, two Lagrangian intersection punctures at $p_{i}$ and at $p_{j}$,
and a word of chains of loops in $\Lambda$ and Reeb chords of $\Lambda$ as
output, and where $CE^{\ast}_{ij}$ is directly analogous to $CE^{\ast}$ but
where the first chain of loops is in a word is replaced by a chain of paths
from $p_{i}$ to the base point and the last is replaced by a chain of paths
from the base point to $p_{j}$. In this setup the counterpart of the second
component $\Psi_{2}$ is

(29) $\Psi^{p_{i}p_{j}p_{k}}\colon CW^{\ast}(F_{j},F_{k})\otimes
CW^{\ast}(F_{i},F_{j})\to CE^{\ast}_{p_{i}p_{k}}(\Lambda),$

and counts disks with two positive punctures at Reeb chords and two Lagrangian
intersection punctures at $p_{i}$ and $p_{k}$. The counterpart of the
$A_{\infty}$-equations in this setup is then

(30)
$d\circ\Psi^{p_{i}p_{k}}+\Psi^{p_{j}p_{k}}\cdot\Psi^{p_{i}p_{j}}+\Psi^{p_{i}p_{j}p_{k}}\circ(1\otimes\mu_{1}+\mu_{1}\otimes
1)+\Psi^{p_{i}p_{k}}\circ\mu_{2}=0,$

where $d$ is the differential on $CE^{\ast}_{ik}$ and where $\cdot$ is the
(Pontryagin) product $CE^{\ast}_{p_{j}p_{k}}\otimes CE^{\ast}_{p_{i}p_{j}}\to
CE^{\ast}_{p_{i}p_{k}}$. The proofs of these statements are word by word
repetitions of the proof of Theorem 72.

We next sketch a proof that the map $\Psi$ in Theorem 72 is in fact a quasi-
isomorphism, or, in other words, that its first component $\Psi_{1}$ induces
an isomorphism on homology. We filter $CW^{\ast}(C)$ by action of its Reeb
chord generators. To get a corresponding filtration on $CE^{\ast}(\Lambda)$ we
use action on Reeb chords in combination with the energy on the loops. We
start with a discussion of the energy of loops, following [49].

Equip $\Lambda$ with a Riemannian metric and $\Omega=\Omega(\Lambda)$ denote
the space of based loops in $\Lambda$ with the supremum norm: for two loops
$\gamma,\beta\colon[0,1]\to\Lambda$,

$d^{\ast}(\gamma,\beta)=\sup_{t\in[0,1]}\rho(\gamma(t),\beta(t)),$

where $\rho$ is the metric on $\Lambda$ induced by the Riemannian structure.
Then the metric topology on $\Omega$ agrees with the standard compact open
topology.

Let $\Omega^{\prime}=\Omega^{\prime}(\Lambda)$ denote the space of piecewise
smooth paths with metric

$d(\gamma,\beta)=d^{\ast}(\gamma,\beta)+\int_{0}^{1}\left(|\dot{\gamma}|-|\dot{\beta}|\right)^{2}dt,$

where $\dot{\gamma}$ denotes the derivative of $\gamma$. The natural inclusion
$\Omega^{\prime}\to\Omega$ is a homotopy equivalence, [49, Theorem 17.1]. We
will use finite dimensional approximations to study $\Omega^{\prime}$. The
energy of a piece-wise smooth loop in $\Lambda$ is

$E(\gamma)=\int_{0}^{1}|\dot{\gamma}|^{2}dt.$

For $c>0$, let $\Omega^{c}\subset\Omega^{\prime}$ denote the subset of loops
of energy $E<c$. The space $\Omega^{c}$ can be approximated by piece-wise
geodesic loops. More precisely, fixing a subdivision
$0=t_{0}<t_{1}<\dots<t_{m}=1$ of $[0,1]$ we consider the space $B^{c}$ of
loops of energy $E<c$ that are geodesic on each interval $[t_{i},t_{i+1}]$.
Then [49, Lemma 16.1] shows that for all sufficiently fine subdivisions
$B^{c}$ is a finite dimensional manifold (a submanifold of the product
$\Lambda^{\times m}$ in a natural way). Moreover [49, Theorem 16.2], all
critical points of $E|_{\Omega^{c}}$ lie in $B^{c}$ which is a deformation
retract of $\Omega^{c}$ and for generic metric $E|_{B^{c}}$ is a Morse
function.

With these preliminaries established we turn to the actual proof. The first
step will be to describe the Reeb chords of $C^{\tau}$. Let $g$ be a
Riemannian metric on $\Lambda$ as above and let
$f\colon[0,\infty)\to\mathbf{R}$ be a positive function with $f(0)=1$,
$f^{\prime}(0)=-1$, and $f^{\prime}(t)<0$ monotone increasing. Define the
metric $h$ on $\Lambda\times\mathbf{R}$ by

$h=dt^{2}+f(t)g.$

Then if $x$ and $y$ are points in $\Lambda$ and $\gamma\colon[0,s]\to\Lambda$
is a geodesic with $\gamma(0)=x$ and $\gamma(s)=y$ then there is a unique
geodesic $(\gamma(t),r(t))\in\Lambda\times[0,\infty)$ such that

  * •

$(\gamma(0),0)=(x,0)$ and $(\gamma(s),r(s))=(y,0)$;

  * •

$r(t)$ is Morse with has a unique maximum at an interior point $t=t_{0}$.

Note that the Reeb flow in the unit disk bundle is the natural lift of the
geodesic flow. Assume next as above that the metric $g$ on $\Lambda$ is
generic in the sense that the length functional for curves connecting any two
Reeb chord endpoints in $\Lambda$ has only Morse critical points. Concretely,
this means that the index form of any geodesic connecting two Reeb chord
endpoints is non-degenerate. As in Lemma 67, this allows us to control the
Reeb chords of $C^{\tau}$ below a given action for all sufficiently thin
handles. More precisely, let $\epsilon$ denote the size of the tubular
neighborhood of $\Lambda$ in $Y$ where we attach
$T^{\ast}(\Lambda\times[0,\infty))$ and we introduce the following notion of a
geodesic-Reeb chord word. A _geodesic-Reeb chord word_ is a word

$\gamma_{1}c_{1}\gamma_{2}c_{2}\dots c_{m}\gamma_{m},$

where $\gamma_{1}$ is a geodesic from one of the base points $p_{v}$ to the
start point of the Reeb chord $c_{1}$, where $\gamma_{2}$ is a geodesic from
the endpoint of $c_{1}$ to the start point of $c_{2}$, etc, until finally
$\gamma_{m}$ is a geodesic from the endpoint of the Reeb chord $c_{m}$ to one
of the base points $p_{w}$. We define the action of a geodesic-Reeb chord word
to be the sum of actions of its Reeb chords and the energies of its geodesics.

######  Lemma 73.

For any $A>0$ there exists $\epsilon_{0}>0$ and $\tau_{0}>0$ such that for any
$\epsilon<\epsilon_{0}$ and any $\tau>\tau_{0}$ there is a natural one to one
correspondence between Reeb chords of $C^{\tau}$ of action $<A$ and geodesic-
Reeb chord words of $\Lambda$ of action $<A$.

Sketch of proof: The proof uses the transversality of the Reeb chords and of
the geodesics. The basic observation is that the point in the normal fiber of
$\Lambda$ where the Reeb flow hits determines the direction of the geodesic in
$\Lambda\times[0,\infty)$. After introducing a concrete smoothing of corners
the lemma then follows from the finite dimensional inverse function theorem.

To show that $\Psi_{1}$ is a quasi-isomorphism we will show that it is
represented by a triangular matrix with ones on the diagonal with respect to
the action/energy filtration. To this end we will use the Morse theoretic
finite dimensional model for the chain complex underlying the homology of the
based loop space described above. In order to have
$[\mathcal{M}^{\mathrm{co}}(\mathbf{c})]$ defined as a chain in this model we
need to assure that the paths on the boundary of the holomorphic disk are
sufficiently well behaved. We only sketch the construction. On holomorphic
disk with unstable domains, we fix gauge using small spheres surrounding Reeb
chord endpoint, compare [30, Section A.2]. As in [30, Section A.1] we use a
configuration space for holomorphic curves consisting of maps with two
derivatives in $L^{2}$. This means that the restriction to the boundary has
$3/2$ derivatives in $L^{2}$ and in particular the projection to $\Lambda$ has
bounded energy. Since the action of the positive puncture in a holomorphic
disk contributing to the differential controls the norm of the solution it
follows that we can use configuration spaces of bounded energy to study the
disks in the differential: we approximate the boundary curves uniformly by a
piecewise geodesic curve by introducing a uniformly bounded number of
subdivision points and straight line homotopies in small charts.

######  Conjecture 74.

The chain-map $\Psi_{1}\colon CW^{\ast}(C)\to CE^{\ast}(\Lambda)$ induces an
isomorphism on homology.

Sketch of proof: Consider a word of the form

$\gamma_{0}c_{1}\gamma_{1}c_{2}\dots c_{m}\gamma_{m},$

where $\gamma_{j}$ are geodesics in $\Lambda\times[0,\infty)$ and where
$c_{j}$ are Reeb chords. We aim to construct algebraically one disk connecting
the Reeb chord $a$ of $C$, corresponding to this word (see Lemma 73), to the
word itself. We use an inductive argument and energy filtration. To start the
argument we pick additional fiber disks $F_{c^{+}}$ and $F_{c_{-}}$ in
$T^{\ast}(\Lambda\times[0,\infty))$ at $(c^{+},\epsilon)$ and
$(c_{-},\epsilon)$ for very small $\epsilon>0$ near all Reeb chord end points
$c_{+}$ and $c_{-}$ in $\Lambda$. We use the natural counterparts of the for
mixed wrapped Floer homologies. For example, there is a straightforward
analogue of Lemma 73: Reeb chord generators of $CW^{\ast}(F_{c^{+}},C)$
correspond before the surgery words of the form

$\gamma_{1}c_{1}\gamma_{2}\dots c_{m}\gamma_{m},$

where $\gamma$ is a geodesic connecting the base point of $F_{c^{+}}$ to the
startpoint of $c_{1}$, $\gamma_{2}$ from the endpoint of $c_{1}$ to the start
point of $c_{2}$, etc. To start the argument we note that it is
straightforward to construct holomorphic strips corresponding to the short
geodesics starting at $F_{c^{-}}$ followed by the chord $c$ and then the short
geodesic to $F_{c^{+}}$ and to show that they are unique. This corresponds to
a generator of $CW^{*}(F_{c^{-}},F_{c^{+}})$. Likewise, it is immediate to
construct the holomorphic disk connecting a Reeb chord generator of
$CW^{*}(F_{c^{+}},C)$ corresponding to a geodesic, and show that it is unique,
compare Lemma 69.

We now use these two to construct algebraically one disk from the Reeb chord
generator of $CW^{\ast}(F_{c_{-}},C)$ corresponding to the short geodesic, the
chord $c$, and a geodesic connecting the endpoint of $c$ to the base point of
$C$. To this end we consider the natural map

$\Psi^{p_{v}c^{+}c^{-}}\colon CW^{*}(F_{c^{+}},C)\otimes
CW^{*}(F_{c^{-}},F_{c^{+}})\to CE^{\ast}(\Lambda),$

see (30). For the two Reeb chords $a$ connecting $F_{c^{-}}$ to $F_{c^{+}}$
corresponding to the chord $c$ of $\Lambda$, and $b$ connecting $F_{c^{+}}$ to
$C$ corresponding to the geodesic we then have (with $p_{v}$ denoting the base
point):

$\displaystyle
d(\Psi^{p_{v}c^{+}c^{-}}(b,a))+(\Psi^{p_{v}c^{-}}(b))\cdot(\Psi^{c^{+}c^{-}}(a))$
$\displaystyle+\Psi^{p_{v}c^{-}}(\mathfrak{m}_{2}(b,a))+(-1^{|a|-1}\Psi^{p_{v}c^{+}c^{-}}(\mathfrak{m}_{1}(b),a)+\Psi^{p_{v}c^{+}c^{-}}(b,\mathfrak{m}_{1}(a))=0.$

Here we know that the terms containing $\mathfrak{m}_{1}$ and $d$ involves
non-trivial holomorphic disks or Morse flows in the finite dimensional
approximation and hence lowers action/energy by an amount bounded below by
some $\delta>0$ which we assume is much larger than $\epsilon>0$ above.
Therefore, if we restrict attention to a small action window we find

$(\Psi^{p_{v}c^{-}}(b)\cdot\Psi^{c^{+}c^{-}}(a))+\Psi^{p_{v}c^{-}}(\mathfrak{m}_{2}(b,a))=0.$

Here the first term is simply the Pontryagin product at the common endpoint of
the curves, which is homologous to the word $\epsilon^{\prime}c\gamma$ of the
small geodesic, the Reeb chord and then the longer geodesic, by rounding the
corner at $c^{+}$. It follows that $\mathfrak{m}_{2}(a,b)=r$, where $r$ is a
Reeb chord with action between the sum of the actions of $a$ and $b$ and the
action of $\epsilon^{\prime}c\gamma$ and that $\Psi^{c^{-}p_{v}}(r)$ contains
this word with coefficient $\pm 1$. Noting that there is only one Reeb chord
in the action window studied we find that the desired coefficient equals $\pm
1$. It is now clear how to continue the induction, in each step we add one
more geodesic or Reeb chord to any word. Using already constructed curves and
(30) in a small action window we find that the map $\Psi_{1}$ has a triangular
action matrix with $\pm 1$ on the diagonal, hence it is a quasi-isomorphism.

## References

  * [1] Abouzaid, M., Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Selecta Math. (N.S.) 15 (2009), no. 2, 189–270.
  * [2] Abouzaid, M., On the wrapped Fukaya category and based loops. J. Symplectic Geom. 10 (2012), no. 1, 27–79.
  * [3] Abouzaid, M.; Seidel, P., An open string analogue of Viterbo functoriality. Geom. Topol. 14 (2010), no. 2, 627–718.
  * [4] Adams J. F., On the cobar construction. Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 409–412.
  * [5] Adams, J. F.; Hilton. P. J., On the chain algebra of a loop space. Comment. Math. Helv. 30 (1956), 305–330.
  * [6] Bao E.; Honda K., Semi-global Kuranishi charts and the definition of contact homology. preprint arXiv:1512.00580
  * [7] Barraud, J. F.; Cornea, O., Lagrangian intersections and the Serre spectral sequence. Ann. of Math. (2) 166 (2007), no. 3, 657–722.
  * [8] Beilinson A.; Ginzburg V.; Soergel W., Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9 (1996), 473–527.
  * [9] Boardman J. M., Conditionally convergent spectral sequences. Contemp. Math., 239, Amer. Math. Soc., Providence, RI, 1999.
  * [10] Bourgeois, F. ; Chantraine B., Bilinearized Legendrian contact homology and the augmentation category. J. Symplectic Geom. 12 (2014), no. 3, 553–583.
  * [11] Bourgeois, F.; Ekholm, T.; Eliashberg, Y., Effect of Legendrian surgery. Geom. Topol. 16 (2012), no. 1, 301–389.
  * [12] Bourgeois, F.; Ekholm, T.; Eliashberg, Y., work in progress.
  * [13] Bourgeois, F.; Eliashberg, Y.; Hofer, H.; Wysocki, K.; Zehnder, E., Compactness results in symplectic field theory. Geom. Topol. 7 (2003), 799–888.
  * [14] Brown, Edgar H., Jr., Twisted tensor products. I. Ann. of Math. (2) 69 1959 223–246.
  * [15] Carlsson, G.; Milgram, R. J., Stable homotopy and iterated loop spaces. Handbook of algebraic topology, 505–583, North-Holland, Amsterdam, 1995.
  * [16] Casals R.; Murphy E., Legendrian Fronts for Affine Varieties. preprint arXiv:1610.06977
  * [17] Chen K-T., Iterated path integrals Bull. Amer. Math. Soc. Volume 83, Number 5 (1977), 831–879.
  * [18] Cieliebak, K.; Ekholm, T.; Latschev, J., Compactness for holomorphic curves with switching Lagrangian boundary conditions. J. Symplectic Geom. 8 (2010), no. 3, 267–298.
  * [19] Cieliebak, K.; Latschev, J., The role of string topology in symplectic field theory. New perspectives and challenges in symplectic field theory, 113–146, CRM Proc. Lecture Notes, 49, Amer. Math. Soc., Providence, RI, 2009.
  * [20] Civan, G.; Koprowski, P.; Etnyre, J.; Sabloff J. M.; Walker, A., Product structures for Legendrian contact homology. Math. Proc. Cambridge Philos. Soc. 150 (2011), no. 2, 291–311.
  * [21] Drinfeld V., DG quotients of DG categories. J. of Algebra 272 (2004) 643–691.
  * [22] Eilenberg S.; Moore J. C., Limits and spectral sequences. Topology 1, 1962, 1–23.
  * [23] Ekholm, T., Morse flow trees and Legendrian contact homology in 1-jet spaces. Geom. Topol. 11 (2007), 1083–1224.
  * [24] Ekholm, T., Rational symplectic field theory over $\mathbb{Z}_{2}$ for exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) 10 (2008), no. 3, 641–704.
  * [25] Ekholm, T.; Etnyre, J. B.; Sabloff, J. M., A duality exact sequence for Legendrian contact homology. Duke Math. J. 150 (2009), no. 1, 1–75.
  * [26] Ekholm, T.; Etnyre, J.; Sullivan, M., Orientations in Legendrian contact homology and exact Lagrangian immersions. Internat. J. Math. 16 (2005), no. 5, 453–532.
  * [27] Ekholm, T.; Etnyre, J.; Sullivan, M., Legendrian contact homology in $P\times\mathbf{R}$. Trans. Amer. Math. Soc. 359 (2007), no. 7, 3301–3335
  * [28] Ekholm, T.; Honda, K.; Kalman, T., Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc. (JEMS) 18 (2016), 2627–2689
  * [29] Ekholm, T.; Oancea, A., Symplectic and contact differential graded algebras. preprint arXiv:1506.01078, to appear in Geom. Topol.
  * [30] Ekholm, T.; Smith, I., Exact Lagrangian immersions with one double point revisited. Math. Ann. 358 (2014), no. 1-2, 195–240.
  * [31] Etgü T.; Lekili Y., Koszul duality patterns in Floer theory. preprint arXiv:1502.07922, to appear in Geom. Topol.
  * [32] Félix, Y.; Thomas, J.-C., Extended Adams-Hilton construction. Pac. Jour. of Math. 128 (1987), no. 2, 251–263.
  * [33] Fukaya, K. Application of Floer homology of Lagrangian submanifolds to symplectic topology. Morse theoretic methods in nonlinear analysis and in symplectic topology, 231–276, NATO Sci. Ser. II Math. Phys. Chem., 217, Springer, Dordrecht, 2006.
  * [34] Fukaya, K.; Oh, Y.-G.; Ohta, H.; Ono, K., Lagrangian intersection Floer theory: anomaly and obstruction. Part I. AMS/IP Studies in Advanced Mathematics, 46.1. American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.
  * [35] Goerss, P. G.; Jardine, J. F., Simplicial homotopy theory. Modern Birkhäuser Classics. Birkhäuser Verlag, Basel, 2009. xvi+510 pp.
  * [36] Gromov, M., Pseudoholomorphic curves in symplectic manifolds. Invent. Math. 82 (1985), no. 2, 307–347.
  * [37] Herscovich E., Hochschild (co)homology and Koszul duality. preprint arXiv:1405.2247.
  * [38] Hofer, H., A general Fredholm theory and applications. Current developments in mathematics, 2004, 1–71, Int. Press, Somerville, MA, 2006
  * [39] Holstein J.V.S., Morita cohomology. PhD thesis, University of Cambridge 2014.
  * [40] Holstein J.V.S., Morita cohomology, Math. Proc. Cambridge Philos. Soc. 158 (2015), no. 1, 1–26.
  * [41] Jones, J. D. S., Cyclic homology and equivariant homology. Invent. Math. 87 (1987), no. 2, 403–423.
  * [42] Kan, D. M., Abstract homotopy. IV, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 542–544.
  * [43] Kontsevich, M., Symplectic geometry of homological algebra. available at the authors webpage (2009).
  * [44] Lefèvre-Hasegawa K., Sur les $A_{\infty}$-catégories. PhD thesis, Université Paris 7, 2002.
  * [45] Lekili, Y., Polishchuk, A., Arithmetic mirror symmetry for genus 1 curves with $n$ marked points, A. Sel. Math. New Ser. (2016).
  * [46] Loday J.-L.; Vallette B., Algebraic operads. Grundlehren Math. Wiss. 346, Springer, Heidelberg, 2012.
  * [47] Lu D. M.; Palmieri J. H.; Wu Q. S.; Zhang J. J., Koszul equivalence in $A_{\infty}$ algebras. New York J. Math. 14 (2008), 325–378.
  * [48] McCleary, J., Userś guide to spectral sequences. Mathematics Lecture Series, 12. Publish or Perish, Inc., Wilmington, DE, 1985.
  * [49] Milnor J., Morse Theory. Annals of Mathematics Studies, No. 51 Princeton University Press, Princeton, N.J. 1963
  * [50] Nadler D., Wrapped Microlocal Sheaves on pairs of pants. preprint, arXiv:1604.00114.
  * [51] Neisendorfer, J.A., What is loop multiplication anyhow? J. Homotopy Relat. Struct. (2016).
  * [52] Ng L.; Rutherford D.; Shende V.; Sivek S.; Zaslow E., Augmentations are sheaves. preprint arXiv:1502.04939.
  * [53] Pardon, J., An algebraic approach to virtual fundamental cycles on moduli spaces of pseudo-holomorphic curves. Geom. Topol. 20 (2016), no. 2, 779–1034.
  * [54] Positselski L., Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence. Mem. Amer. Math. Soc. 212 (2011), no. 996.
  * [55] Prouté A., $A_{\infty}$-structures, Modéles minimaux de Baues-Lemaire et Kadeishvili et homologie des fibrations. Repr. Theory Appl. Categ. 21 (2011), 1–99.
  * [56] Schwede, S.; Shipley, B., Equivalences of monoidal model categories. Algebr. Geom. Topol. 3 (2003), 287–334
  * [57] Seidel, P., Fukaya categories and Picard-Lefschetz theory. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2008.
  * [58] Serre, J.-P. Homologie singulière des espaces fibrés, applications, Ann. Math., 54 (1951), 429–505.
  * [59] Sylvan, Z., On partially wrapped Fukaya categories. preprint arXiv:1604.02540
  * [60] Sylvan, Z., Partially wrapped Floer theory. Talk at the Workshop on Homological Mirror Symmetry: Methods and Structures, IAS, November, 2016.
  * [61] Tabuada, G. N., Théorie homotopique des DG-catégories. Thesis, Université Paris-7, 2007. arXiv:0710.4303.
  * [62] Weibel, C., An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, Cambridge, 1994.