A New Derivation of the Twist-3 Gluon Fragmentation
Contribution to Polarized Hyperon Production

Riku Ikarashi1, Yuji Koike2, Kenta Yabe1 and Shinsuke Yoshida3,4

1 Graduate School of Science and Technology, Niigata University, Ikarashi,
Niigata 950-2181, Japan

2 Department of Physics, Niigata University, Ikarashi, Niigata 950-2181, Japan

3 Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou 510006, China

4 Guangdong-Hong Kong Joint Laboratory of Quantum Matter, Southern Nuclear
Science Computing Center, South China Normal University, Guangzhou 510006,
China

Abstract

A novel method of formulating the twist-3 gluon fragmentation function
contribution to hyperon polarization in the proton-proton collision is
presented. The method employs a covariant gauge and takes full advantage of
the Ward-Takahashi identities before performing the collinear expansion. It
provides a robust way of constructing the general cross section formula, and
also a clear understanding for the absence of the ghost-like terms in the
twist-3 cross section in the leading order with respect to the QCD coupling
constant.

##  1 Introduction

In our recent paper [1], we presented a formalism for calculating the twist-3
gluon fragmentation function (FF) contribution to the polarized hyperon
production in the proton-proton collision:

$\displaystyle p(p)+p(p^{\prime})\to\Lambda^{\uparrow}(P_{h},S_{\perp})+X,$
(1)

where $p$, $p^{\prime}$ and $P_{h}$ are the momenta of the particles and
$S_{\perp}$ is the transverse spin vector of final $\Lambda^{\uparrow}$. This
contribution is diagrammatically shown in Fig. 1, and the corresponding
twist-3 cross section can be calculated from the following formula: 111To get
a gauge- and frame-independent twist-3 cross section, the $q\bar{q}g$-type FF
contribution shown in Fig. 2 of [1] needs to be added to (2). Since the
calculation of the contribution is straightforward, it is not considered in
this paper. Readers should refer to [1] for that contribution.

$\displaystyle
E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}=\frac{1}{16\pi^{2}S_{E}}\sum_{i,j=q,\bar{q},g}\int^{1}_{0}\frac{dx}{x}f_{i}(x)\int^{1}_{0}\frac{dx^{\prime}}{x^{\prime}}f_{j}(x^{\prime})$
$\displaystyle\times\biggl{[}{\Omega^{\mu}_{\ \alpha}}{\Omega^{\nu}_{\
\beta}}\int^{1}_{0}dz\,{\rm
Tr}\left[\widehat{\Gamma}^{\alpha\beta}(z)S_{\mu\nu}(P_{h}/z)\right]-i\,{\Omega^{\mu}_{\
\alpha}}{\Omega^{\nu}_{\ \beta}}{\Omega^{\lambda}_{\
\gamma}}\int^{1}_{0}dz\,{\rm
Tr}\left[\widehat{\Gamma}_{\partial}^{\alpha\beta\gamma}(z)\left.\frac{\partial
S_{\mu\nu}(k)}{\partial k^{\lambda}}\right|_{\rm c.l.}\right]$
$\displaystyle+{\Re}\Bigl{\\{}i\,{\Omega^{\mu}_{\ \alpha}}{\Omega^{\nu}_{\
\beta}}{\Omega^{\lambda}_{\
\gamma}}\int^{1}_{0}\frac{dz}{z}\int^{\infty}_{z}\frac{dz^{\prime}}{z^{\prime}}\,\left(\frac{1}{1/z-1/z^{\prime}}\right){\rm
Tr}\left[\widehat{\Gamma}_{F\,abc}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)S^{abc}_{L\,\mu\nu\lambda}\left({1\over
z^{\prime}},{1\over z}\right)\right]\Bigr{\\}}\biggr{]},$ (2)

where $f_{i}(x)$ ($i=q,\,\bar{q},\,g$) is the twist-2 unpolarized quark,
antiquark, and gluon distributions in the unpolarized proton with the parton’s
momentum fraction $x$, $S_{E}=(p+p^{\prime})^{2}$ is the center-of-mass energy
squared, and ${\rm Tr}$ indicates the sum over all spinor or Lorentz indices
depending on the channels. The correlation functions
$\widehat{\Gamma}^{\alpha\beta}(z)$,
$\widehat{\Gamma}_{\partial}^{\alpha\beta\gamma}(z)$ and
$\widehat{\Gamma}_{F\,abc}^{\alpha\beta\gamma}\left(\frac{1}{z^{\prime}},\frac{1}{z}\right)$,
respectively, define intrinsic, kinematical and dynamical twist-3 gluon FFs.
(For the precise definition, see section 2.) $S_{\mu\nu}(k)$ and
$S^{abc}_{L\,\mu\nu\lambda}(z^{\prime},z)$ are the partonic hard parts which
are, at the beginning, convoluted with the Fourier transform of the hadronic
matrix elements $\sim\langle 0|A_{b}^{\nu}(0)|hX\rangle\langle
hX|A_{a}^{\mu}(\xi)|0\rangle$ and $\sim\langle
0|A_{b}^{\nu}(0)|hX\rangle\langle
hX|A_{a}^{\mu}(\xi)gA_{c}^{\lambda}(\eta)|0\rangle$, respectively. The symbol
$\Omega^{\mu}_{\ \alpha}$ is defined as $\Omega^{\mu}_{\ \alpha}=g^{\mu}_{\
\alpha}-P_{h}^{\mu}w_{\alpha}$ with another lightlike vector $w$ satisfying
$P_{h}\cdot w=1$, and $\left.\right|_{\rm c.l.}$ implies the collinear limit,
$k\to P_{h}/z$.

Figure 1: Generic diagrams representing the twist-3 gluon FF contribution to
$pp\to\Lambda^{\uparrow}X$ for the quark and antiquark distributions in the
initial unpolarized protons. The gluon distribution functions in the initial
protons also contribute. The diagrams (a), (b) and (c), respectively,
correspond to $W^{\rm(a)}_{g}$, $W^{\rm(b)}_{g}$ and $W^{\rm(c)}_{g}$ in (10).

In [1], the formula (2) was applied to the process (1) and the cross section
was calculated in the leading order (LO) with respect to the QCD coupling
constant. This completed the LO twist-3 cross section for (1) together with
the known results for the contribution from the twist-3 distribution function
[2, 3, 4] and the twist-3 quark fragmentation function [5]. Since the formula
(2) is a very general one, it can be easily adopted for other processes such
as $e^{+}e^{-}\to\Lambda^{\uparrow}X$[6] and $ep\to e\Lambda^{\uparrow}X$[7,
8], etc.

To derive the general formula (2), we applied in [1] the collinear expansion
to the hard parts $S_{\mu\nu}(k)$ and
$S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})$. Using the Ward-Takahashi
identities for the partonic hard parts, we could eventually rewrite the
twist-3 cross sections in terms of the low derivatives of the hard parts and
the gauge invariant correlation functions of the gluon’s field strengths as in
(2). Actual calculation, however, is extremely complicated and lengthy, and is
not easy to see how the correlation functions of the gauge field $A_{a}^{\mu}$
is converted into those of the field strength
$F^{a}_{\mu\nu}=\partial_{\mu}A^{a}_{\nu}-\partial_{\nu}A^{a}_{\mu}+gf^{abc}A^{b}_{\mu}A^{c}_{\nu}$.
Furthermore, vanishing of the ghost-like terms appearing in the Ward-Takahashi
identities is essential to reach (2), which was not clearly shown in [1].
Therefore an easier way of deriving (2) is very useful.

In this paper, we present a much more robust and concise way of deriving (2).
In this method, we use Ward-Takahashi identities from the outset to convert
gauge fields into a part of the field strengths, which results in substantial
saving in actual calculation. This procedure was once adopted for deriving the
twist-3 three-gluon distribution contribution to the single spin asymmetry in
$ep^{\uparrow}\to eDX$, where three-gluon distribution contributes as an only
source for the asymmetry and appears as a ”pole contribution”[9]. For the
present case of the twist-3 gluon FF for (1), three types of FFs contribute as
a ”nonpole contribution”, and hence the situation is much more complicated.
Furthermore, our present method provides a clear proof for the absence of the
ghost-like terms which appear in the Ward-Takahashi identities. This is
crucial to guarantee the gauge invariance of the twist-3 cross section.

The remainder of the paper is organized as follows: In section 2, a brief
summary of the twist-3 gluon FFs which appear in (2) is given. In section 3
and Appendix, we present a novel derivation of (2) and prove the absence of
the ghost-like terms in the LO twist-3 cross section. Section 4 is devoted to
a brief summary.

##  2 Gluon Fragmentation Functions

Here we summarize the twist-3 gluon FFs in our notation which appear in (2)
[1]222See also [10, 6, 11] for earlier references and more details about the
gluon FFs.. The twist-3 intrinsic gluon FFs are defined as

$\displaystyle\widehat{\Gamma}^{\alpha\beta}(z)$
$\displaystyle={1\over{N^{2}-1}}\sum_{X}\\!\int\\!\frac{d\lambda}{2\pi}e^{-i{\lambda\over
z}}\langle 0|\bigl{(}[\infty
w,0]F^{w\beta}(0)\bigr{)}_{a}|h(P_{h},S_{h})X\rangle\langle
h(P_{h},S_{h})X|\bigl{(}F^{w\alpha}(\lambda w)[\lambda w,\infty
w]\bigr{)}_{a}|0\rangle$
$\displaystyle=-g_{\perp}^{\alpha\beta}\widehat{G}(z)-iM_{h}\epsilon^{P_{h}w\alpha\beta}(S_{h}\cdot
w)\Delta\widehat{G}(z)-iM_{h}\epsilon^{P_{h}wS_{\perp}[\alpha}w^{\beta]}\Delta\widehat{G}_{3T}(z)+M_{h}\epsilon^{P_{h}wS_{\perp}\\{\alpha}w^{\beta\\}}\Delta\widehat{G}_{3\bar{T}}(z),$
(3)

where $N=3$ is the number of colors for a quark, $|h(P_{h},S_{h})\rangle$ is
the spin-1/2 hyperon state with the four momentum $P_{h}$
($P_{h}^{2}=M_{h}^{2}$) and the spin vector $S_{h}$ ($S_{h}^{2}=-M_{h}^{2}$),
and $[\lambda w,\infty w]$ is the gauge link in the adjoint representation
connecting $\lambda w$ and $\infty w$. For the transversely polarized baryon,
we use the spin vector $S_{\perp}$ normalized as $S_{\perp}^{2}=-1$. In the
twist-3 accuracy $P_{h}$ can be regarded as lightlike. For a baryon with large
momentum, $P_{h}\simeq(|\vec{P}_{h}|,\vec{P}_{h})$, another lightlike vector
$w$ is defined as $w=1/(2|\vec{P}_{h}|^{2})(|\vec{P}_{h}|,-\vec{P}_{h})$ which
satisfies $P_{h}\cdot w=1$. $\widehat{G}(z)$ and $\Delta\widehat{G}(z)$ are
twist-2, and $\Delta\widehat{G}_{3T}(z)$ and $\Delta\widehat{G}_{3\bar{T}}(z)$
are twist-3. We also note $\Delta\widehat{G}_{3\bar{T}}(z)$ is naively
$T$-odd, contributing to hyperon polarization. Each function in (3) has a
support on $0<z<1$.

The kinematical FFs are defined from the transverse derivative of the
correlation functions of the field strengths:

$\displaystyle\widehat{\Gamma}_{\partial}^{\alpha\beta\gamma}(z)=\frac{1}{N^{2}-1}\sum_{X}\\!\int\\!\\!\frac{d\lambda}{2\pi}\\!e^{-i{\lambda\over
z}}\langle 0|\bigl{(}[\infty
w,0]F^{w\beta}(0)\bigr{)}_{a}|h(P_{h},S_{\perp})X\rangle$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\langle
h(P_{h},S_{\perp})X|\bigl{(}F^{w\alpha}(\lambda w)[\lambda w,\infty
w]\bigr{)}_{a}|0\rangle\overleftarrow{\partial}^{\gamma}$
$\displaystyle\qquad\qquad=-i\frac{M_{h}}{2}g_{\perp}^{\alpha\beta}\epsilon^{P_{h}wS_{\perp}\gamma}\widehat{G}_{T}^{(1)}(z)+\frac{M_{h}}{2}\epsilon^{P_{h}w\alpha\beta}S_{\perp}^{\gamma}\Delta\widehat{G}_{T}^{(1)}(z)$
$\displaystyle\qquad\qquad\qquad\qquad-i\frac{M_{h}}{8}\left(\epsilon^{P_{h}wS_{\perp}\\{\alpha}g_{\perp}^{\beta\\}\gamma}+\epsilon^{P_{h}w\gamma\\{\alpha}S_{\perp}^{\beta\\}}\right)\Delta\widehat{H}_{T}^{(1)}(z),$
(4)

where each function is defined to be real and has a support on $0<z<1$.

The dynamical FFs are defined from the lightcone correlation functions of
three field strengths:

$\displaystyle\widehat{\Gamma}_{F\,abc}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})$
$\displaystyle=\sum_{X}\\!\int\\!\\!\frac{d\lambda}{2\pi}\\!\int\\!\\!\frac{d\mu}{2\pi}e^{-i{\lambda\over
z_{1}}}e^{-i\mu({1\over z_{2}}-{1\over z_{1}})}\langle
0|F^{w\beta}_{b}(0)|h(P_{h},S_{\perp})X\rangle\langle
h(P_{h},S_{\perp})X|F^{w\alpha}_{a}(\lambda w)gF^{w\gamma}_{c}(\mu
w)|0\rangle$
$\displaystyle=\frac{if^{abc}}{N}\widehat{\Gamma}_{FA}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})+d^{abc}\frac{N}{N^{2}-4}\widehat{\Gamma}_{FS}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}}),$
(5)

where the gauge link operators are suppressed for simplicity, and $f^{abc}$
and $d^{abc}$ are the anti-symmetric and symmetric structure constants of
color SU(N). The dynamical FFs can be defined as the decomposition of the two
correlation functions in (5) as

$\displaystyle\widehat{\Gamma}_{FA}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})$
$\displaystyle=\frac{-if_{abc}}{N^{2}-1}\sum_{X}\\!\int\\!\\!\frac{d\lambda}{2\pi}\\!\int\\!\\!\frac{d\mu}{2\pi}e^{-i{\lambda\over
z_{1}}}e^{-i\mu({1\over z_{2}}-{1\over z_{1}})}\langle
0|F^{w\beta}_{b}(0)|h(P_{h},S_{\perp})X\rangle\langle
h(P_{h},S_{\perp})X|F^{w\alpha}_{a}(\lambda w)gF^{w\gamma}_{c}(\mu
w)|0\rangle$
$\displaystyle=-{M_{h}}\biggl{(}\widehat{N}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\beta}+\widehat{N}_{2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\beta\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\alpha}-\widehat{N}_{2}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\beta}_{\perp}\epsilon^{P_{h}wS_{\perp}\gamma}\biggr{)},$
$\displaystyle\widehat{\Gamma}_{FS}^{\alpha\beta\gamma}(\frac{1}{z_{1}},\frac{1}{z_{2}})$
$\displaystyle=\frac{d_{abc}}{N^{2}-1}\sum_{X}\\!\int\\!\\!\frac{d\lambda}{2\pi}\\!\int\\!\\!\frac{d\mu}{2\pi}e^{-i{\lambda\over
z_{1}}}e^{-i\mu({1\over z_{2}}-{1\over z_{1}})}\langle
0|F^{w\beta}_{b}(0)|h(P_{h},S_{\perp})X\rangle\langle
h(P_{h},S_{\perp})X|F^{w\alpha}_{a}(\lambda w)gF^{w\gamma}_{c}(\mu
w)|0\rangle$
$\displaystyle=-{M_{h}}\biggl{(}\widehat{O}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\beta}+\widehat{O}_{2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\beta\gamma}_{\perp}\epsilon^{P_{h}wS_{\perp}\alpha}+\widehat{O}_{2}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right)g^{\alpha\beta}_{\perp}\epsilon^{P_{h}wS_{\perp}\gamma}\biggr{)}.$
(7)

Correlation functions (2) and (7), respectively, define two independent set of
the complex functions $\left\\{\widehat{N}_{1},\widehat{N}_{2}\right\\}$ and
$\left\\{\widehat{O}_{1},\widehat{O}_{2}\right\\}$ due to the exchange
symmetry of the field strengths. Functions $\widehat{N}_{1}$ and
$\widehat{O}_{1}$ satisfy the relations

$\displaystyle\widehat{N}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$
$\displaystyle=$
$\displaystyle-\widehat{N}_{1}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right),$
$\displaystyle\widehat{O}_{1}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$
$\displaystyle=$
$\displaystyle\widehat{O}_{1}\left(\frac{1}{z_{2}}-\frac{1}{z_{1}},\frac{1}{z_{2}}\right).$
(8)

The real parts of these four FFs are $T$-even and the imaginary parts are
$T$-odd, the latter being the sources of SSAs.
$\widehat{N}_{1,2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ and
$\widehat{O}_{1,2}\left(\frac{1}{z_{1}},\frac{1}{z_{2}}\right)$ have a support
on $\frac{1}{z_{2}}>1$ and $\frac{1}{z_{2}}>\frac{1}{z_{1}}>0$.

##  3 Twist-3 gluon fragmentation contribution to $pp\to\Lambda^{\uparrow}X$

In this section we present a robust way to derive the basic formula (2). The
twist-3 gluon FF contribution to (1) can be written as

$\displaystyle
E_{h}\frac{d\sigma(p,p^{\prime},P_{h};S_{\perp})}{d^{3}P_{h}}=\frac{1}{16\pi^{2}S_{E}}\sum_{i,j=q,\bar{q},g}\\!\int\\!\\!\frac{dx}{x}f_{i}(x)\\!\int\frac{dx^{\prime}}{x^{\prime}}f_{j}(x^{\prime})W_{g}(xp,x^{\prime}p^{\prime},P_{h}/z,S_{\perp}),$
(9)

where $W_{g}$ represents the partonic hard scattering followed by the
fragmentation of a gluon into the final $\Lambda^{\uparrow}$. Figure 1 shows
the generic structure of the LO diagrams for this contribution. Corresponding
to Figs. 1 (a), (b), (c), $W_{g}$ consists of three parts:

$\displaystyle W_{g}(xp,x^{\prime}p^{\prime},P_{h};S_{\perp})\equiv
W^{\rm(a)}_{g}+W^{\rm(b)}_{g}+W^{\rm(c)}_{g}$
$\displaystyle\qquad=\int\frac{d^{4}k}{(2\pi)^{4}}\left[S^{ab}_{\mu\nu}(k)\widehat{\Gamma}^{\mu\nu}_{ab}(k)\right]$
$\displaystyle\qquad+{1\over
2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\biggl{[}S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})\widehat{\Gamma}^{\mu\nu\lambda}_{L\,abc}(k,k^{\prime})+S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})\widehat{\Gamma}^{\mu\nu\lambda}_{R\,abc}(k,k^{\prime})\biggr{]},$
(10)

where $\widehat{\Gamma}^{\mu\nu}_{ab}(k)$,
$\widehat{\Gamma}^{\mu\nu\lambda}_{L\,abc}(k,k^{\prime})$ and
$\widehat{\Gamma}^{\mu\nu\lambda}_{R\,abc}(k,k^{\prime})$ are the hadronic
matrix elements of the gauge (gluon) fields with $k$ and $k^{\prime}$ the four
momenta of the gluons fragmenting into the final $\Lambda$, and
$S^{ab}_{\mu\nu}(k)$, $S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})$ and
$S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})$ are the corresponding partonic hard
scattering parts with the color indices $a,\,b,\,c$ and the Lorentz indices
$\mu,\,\nu,\lambda$. In (10), the factor $1/2$ in front of $W_{g}^{\rm(b)}$
and $W_{g}^{\rm(c)}$ takes into account the exchange symmetry of the gluon
fields in the fragmentation matrix elements. Hadronic matrix elements are
defined as

$\displaystyle\widehat{\Gamma}^{\mu\nu}_{ab}(k)=\sum_{X}\int{d^{4}\xi}\,{e^{-ik\xi}}{\langle
0|}A^{\nu}_{b}(0){|hX\rangle}{\langle hX|}A^{\mu}_{a}(\xi){|0\rangle},$ (11)
$\displaystyle\widehat{\Gamma}^{\mu\nu\lambda}_{L\,abc}(k,k^{\prime})=\sum_{X}\int{d^{4}\xi}\int{d^{4}\eta}\,{e^{-ik\xi}}{e^{-i(k^{\prime}-k)\eta}}{\langle
0|}A^{\nu}_{b}(0){|hX\rangle}{\langle
hX|}A^{\mu}_{a}(\xi)gA^{\lambda}_{c}(\eta){|0\rangle},$ (12)
$\displaystyle\widehat{\Gamma}^{\mu\nu\lambda}_{R\,abc}(k,k^{\prime})=\sum_{X}\int{d^{4}\xi}\int{d^{4}\eta}\,{e^{-ik\xi}}{e^{-i(k^{\prime}-k)\eta}}{\langle
0|}A^{\nu}_{b}(0)gA^{\lambda}_{c}(\eta){|hX\rangle}{\langle
hX|}A^{\mu}_{a}(\xi){|0\rangle},$ (13)

where the gauge coupling $g$ associated with the attachment of the extra gluon
line to the hard part is included in
$\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})$ and
$\widehat{\Gamma}_{R\,abc}^{\mu\nu\lambda}(k,k^{\prime})$. Therefore the hard
parts $S_{\mu\nu}^{ab}(k)$, $S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})$ and
$S_{R\,\mu\nu\lambda}^{abc}(k,k^{\prime})$ are of $O(g^{4})$ in the LO
calculation. From hermiticity, one has
$\widehat{\Gamma}^{\mu\nu\lambda}_{L\,abc}(k,k^{\prime})^{\star}=\widehat{\Gamma}^{\nu\mu\lambda}_{R\,bac}(k^{\prime},k)$
and
$S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})^{\star}=S^{bac}_{R\,\nu\mu\lambda}(k^{\prime},k)$,
which guarantees the reality of $W_{g}$. A standard procedure to extract the
twist-3 effect is the collinear expansion of the hard parts with respect to
$k$ and $k^{\prime}$ around $P_{h}$. We followed the method in [1] to get (2).

Here we present an alternative method which leads to (2) more easily. In this
method we fully use the Ward-Takahashi identities for the hard parts to
convert some of the gluon field $A^{a}_{\mu}$ into a part of the field
strength $F^{a}_{\mu\nu}$. Ward-Takahashi identities for the hard part read

$\displaystyle{k}^{\mu}S^{ab}_{\mu\nu}(k)={k}^{\nu}S^{ab}_{\mu\nu}(k)=0,$ (14)
$\displaystyle{(k^{\prime}-k)}^{\lambda}S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{-if^{abc}}{N^{2}-1}S_{\mu\nu}(k^{\prime})+G^{abc}_{\mu\nu}(k,k^{\prime}),$
(15) $\displaystyle{k}^{\mu}S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S_{\lambda\nu}(k^{\prime})+G^{cba}_{\lambda\nu}(k^{\prime}-k,k^{\prime}),$
(16) $\displaystyle{k^{\prime}}^{\nu}S^{abc}_{L\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$ $\displaystyle 0,$ (17)
$\displaystyle{(k^{\prime}-k)}^{\lambda}S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S_{\mu\nu}(k)-\left(G^{bac}_{\nu\mu}(k^{\prime},k)\right)^{\star},$
(18) $\displaystyle{k^{\prime}}^{\nu}S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$
$\displaystyle\frac{if^{abc}}{N^{2}-1}S_{\mu\lambda}(k)+\left(G^{cab}_{\lambda\mu}(k-k^{\prime},k)\right)^{\star},$
(19) $\displaystyle{k}^{\mu}S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle=$ $\displaystyle 0,$ (20)

where $S_{\mu\nu}(k)\equiv S_{\mu\nu}^{ab}(k)\delta_{ab}$. The $G$-terms are
the ghost-like terms which appear due to the off-shellness and the nonphysical
polarization of the gluon lines entering the fragmentation matrix elements.
Actual forms of those ghost-like terms for $pp\to\Lambda^{\uparrow}X$ were
given in Appendix A of [1] in the LO with respect to the QCD coupling. They
are proportional to $f^{abc}$ and satisfy the relation

$\displaystyle
k^{\mu}G_{\mu\nu}^{abc}(k,k^{\prime})=k^{\prime\nu}G_{\mu\nu}^{abc}(k,k^{\prime})=0.$
(21)

We will see that use of the relations (14)-(21) from the outset brings
enormous saving in the actual calculation and clearer understanding on the
absence of the ghost-like terms in the LO twist-3 cross section333Absence of
the ghost term contribution to the twist-3 cross sections was discussed for
the 3-gluon distribution contribution to $\vec{p}p^{\uparrow}\to DX$ [12] and
twist-3 quark FF contribution to $ep^{\uparrow}\to e\pi X$ [13]..

We first consider $W_{g}^{\rm(a)}$. The integration momentum $k$ can be
decomposed as

$\displaystyle k^{\mu}=(k\cdot w)P_{h}^{\mu}+\Omega^{\mu}_{\ \nu}k^{\nu},$
(22)

where $\Omega^{\mu}_{\ \nu}\equiv g^{\mu}_{\ \nu}-P_{h}^{\mu}w_{\nu}$.
Inserting (22) into (14), one gets

$\displaystyle S_{P_{h}\nu}^{ab}(k)={-1\over k\cdot w}\Omega^{\mu}_{\
\kappa}k^{\kappa}S_{\mu\nu}^{ab}(k),$ (23) $\displaystyle S_{\mu
P_{h}}^{ab}(k)={-1\over k\cdot w}\Omega^{\nu}_{\
\tau}k^{\tau}S_{\mu\nu}^{ab}(k).$ (24)

Then we can write

$\displaystyle
S_{\mu\nu}^{ab}(k)\widehat{\Gamma}_{ab}^{\mu\nu}(k)=S_{\mu\nu}^{ab}(k)g^{\mu}_{\
\alpha}g^{\nu}_{\
\beta}\widehat{\Gamma}_{ab}^{\alpha\beta}(k)=S_{\mu\nu}^{ab}(k)(P_{h}^{\mu}w_{\alpha}+\Omega^{\mu}_{\
\alpha})(P_{h}^{\nu}w_{\beta}+\Omega^{\nu}_{\
\beta})\widehat{\Gamma}_{ab}^{\alpha\beta}(k)$ $\displaystyle={1\over(k\cdot
w)^{2}}S_{\mu\nu}^{ab}(k)\Omega^{\mu}_{\ \kappa}\Omega^{\nu}_{\
\tau}(-k^{\kappa}w_{\alpha}+k\cdot wg^{\kappa}_{\
\alpha})(-k^{\tau}w_{\beta}+k\cdot wg^{\tau}_{\
\beta})\widehat{\Gamma}^{\alpha\beta}_{ab}(k),$ (25)

where we have used (23) and (24) in the last equality. Noting that one can
write

$\displaystyle(-k^{\kappa}w_{\alpha}+k\cdot wg^{\kappa}_{\
\alpha})(-k^{\tau}w_{\beta}+k\cdot wg^{\tau}_{\
\beta})\widehat{\Gamma}^{\alpha\beta}_{ab}(k)$
$\displaystyle\qquad\qquad=\sum_{X}\int d^{4}\xi e^{-ik\xi}\langle
0|F^{(0)\,\tau w}_{b}(0)|hX\rangle\langle hX|F_{a}^{(0)\,\kappa
w}(\xi)|0\rangle\equiv\widehat{\Gamma}_{F\,ab}^{\kappa\tau}(k),$ (26)

where
$F^{(0)\,\kappa\sigma}_{a}\equiv\partial^{\kappa}A_{a}^{\sigma}-\partial^{\sigma}A_{a}^{\kappa}$
is the $O(A)$ piece of the gluon’s field strength, one obtains

$\displaystyle\int{d^{4}k\over(2\pi)^{4}}\left[S_{\mu\nu}^{ab}(k)\widehat{\Gamma}_{ab}^{\mu\nu}(k)\right]=\int{d^{4}k\over(2\pi)^{4}}\left[{1\over(k\cdot
w)^{2}}S_{\mu\nu}^{ab}(k)\Omega^{\mu}_{\ \kappa}\Omega^{\nu}_{\
\tau}\widehat{\Gamma}_{F\,ab}^{\kappa\tau}(k)\right].$ (27)

To extract the twist-3 contribution, we apply the collinear expansion to the
RHS of (27). Writing $k\cdot w=1/z$, the contribution up to twist-3 from (27)
can be obtained as

$\displaystyle
W_{g}^{\rm(a)}=\int{d^{4}k\over(2\pi)^{4}}\left[S_{\mu\nu}^{ab}(k)\widehat{\Gamma}_{ab}^{\mu\nu}(k)\right]^{\rm
twist-3}$ $\displaystyle\qquad\qquad=\int d\left({1\over
z}\right)z^{2}\,\Omega^{\mu}_{\ \kappa}\Omega^{\nu}_{\
\tau}\left[S_{\mu\nu}\left(z\right)\widehat{\Gamma}_{F}^{\kappa\tau}(z)+\Omega^{\lambda}_{\
\rho}\left.{\partial S_{\mu\nu}(k)\over\partial
k^{\lambda}}\right|_{k=P_{h}/z}\widehat{\Gamma}_{\partial
F}^{\kappa\tau\rho}(z)\right],$ (28)

where $S_{\mu\nu}(z)\equiv S_{\mu\nu}^{ab}(P_{h}/z)\delta^{ab}$ and

$\displaystyle\widehat{\Gamma}_{F}^{\kappa\tau}(z)={1\over
N^{2}-1}\sum_{X}\int{d\lambda\over 2\pi}\,e^{-i\lambda/z}\langle
0|F^{(0)\,\tau w}_{a}(0)|hX\rangle\langle hX|F_{a}^{(0)\,\kappa w}(\lambda
w)|0\rangle,$ (29) $\displaystyle\widehat{\Gamma}_{\partial
F}^{\kappa\tau\rho}(z)={1\over N^{2}-1}\sum_{X}\int{d\lambda\over
2\pi}\,e^{-i\lambda/z}\langle 0|F^{(0)\,\tau w}_{a}(0)|hX\rangle\langle
hX|(-i)\partial^{\rho}F_{a}^{(0)\,\kappa w}(\lambda w)|0\rangle.$ (30)

Equations (29) and (30) are, respectively, identified as the $O(g^{0})$-parts
of (3) and (4), and (28) represents the lowest order contribution to the first
and second terms in (2).

Next we proceed to analyze $W_{g}^{\rm(b)}$. Using (22) in (16), we have

$\displaystyle(k\cdot
w)S_{L\,P_{h}\nu\lambda}^{abc}(k,k^{\prime})+\Omega^{\mu}_{\
\rho}k^{\rho}S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})={if^{abc}\over
N^{2}-1}S_{\lambda\nu}(k^{\prime})+G_{\lambda\nu}^{cba}(k^{\prime}-k,k^{\prime}),$
(31)

from which we obtain

$\displaystyle S_{L\,P_{h}\nu\lambda}^{abc}(k,k^{\prime})={1\over k\cdot
w}\left(-\Omega^{\mu}_{\
\rho}k^{\rho}S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})+{if^{abc}\over
N^{2}-1}S_{\lambda\nu}(k^{\prime})+G_{\lambda\nu}^{cba}(k^{\prime}-k,k^{\prime})\right).$
(32)

Likewise, from (22) and (17), we have

$\displaystyle S_{L\,\mu P_{h}\lambda}^{abc}(k,k^{\prime})={-1\over
k^{\prime}\cdot w}\Omega^{\nu}_{\
\tau}k^{\prime\tau}S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime}).$ (33)

As in (25), (32) and (33) can be used to rewrite integrand of $W_{g}^{\rm(b)}$
as

$\displaystyle
S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})=S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})g^{\mu}_{\
\kappa}g^{\nu}_{\ \tau}g^{\lambda}_{\
\sigma}\widehat{\Gamma}_{L\,abc}^{\kappa\tau\sigma}(k,k^{\prime})$
$\displaystyle=S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\bigl{(}P_{h}^{\mu}w_{\kappa}+\Omega^{\mu}_{\
\kappa}\bigr{)}\bigl{(}P_{h}^{\nu}w_{\tau}+\Omega^{\nu}_{\
\tau}\bigr{)}\bigl{(}P_{h}^{\lambda}w_{\sigma}+\Omega^{\lambda}_{\
\sigma}\bigr{)}\widehat{\Gamma}_{L\,abc}^{\kappa\tau\sigma}(k,k^{\prime})$
$\displaystyle={1\over k\cdot w}{1\over k^{\prime}\cdot
w}S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\Omega^{\mu}_{\
\alpha}\Omega^{\nu}_{\ \beta}\bigl{(}-k^{\alpha}w_{\kappa}+k\cdot
wg^{\alpha}_{\ \kappa}\bigr{)}\bigl{(}-k^{\prime\beta}w_{\tau}+k^{\prime}\cdot
wg^{\beta}_{\ \tau}\bigr{)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\bigl{(}P_{h}^{\lambda}w_{\sigma}+\Omega^{\lambda}_{\
\sigma}\bigr{)}\widehat{\Gamma}^{\kappa\tau\sigma}_{L\,abc}(k,k^{\prime})$
$\displaystyle\qquad\qquad+{1\over k\cdot w}\left({if^{abc}\over
N^{2}-1}S_{\sigma\tau}(k^{\prime})+G_{\sigma\tau}^{cba}(k^{\prime}-k,k^{\prime})\right)\widehat{\Gamma}^{w\tau\sigma}_{L\,abc}(k,k^{\prime}).$
(34)

Similarly to (26), we have

$\displaystyle\bigl{(}-k^{\alpha}w_{\kappa}+k\cdot wg^{\alpha}_{\
\kappa}\bigr{)}\bigl{(}-k^{\prime\beta}w_{\tau}+k^{\prime}\cdot wg^{\beta}_{\
\tau}\bigr{)}\widehat{\Gamma}^{\kappa\tau\sigma}_{L\,abc}(k,k^{\prime})$
$\displaystyle=\sum_{X}\int d^{4}\xi\int d^{4}\eta
e^{-ik\xi}e^{-i(k^{\prime}-k)\eta}\langle 0|F^{(0)\,\beta
w}_{b}(0)|hX\rangle\langle hX|F^{(0)\,\alpha
w}_{a}(\xi)gA^{\sigma}_{c}(\eta)|0\rangle$
$\displaystyle\equiv\widehat{\Gamma}^{\alpha\beta\sigma}_{LFA\,abc}(k,k^{\prime}).$
(35)

Using this equation, (34) can be rewritten as

$\displaystyle
S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})$
$\displaystyle\qquad=S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime}){1\over k\cdot
w}{1\over k^{\prime}\cdot w}\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\left(P_{h}^{\lambda}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta
w}(k,k^{\prime})+\Omega^{\lambda}_{\
\sigma}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta\sigma}(k,k^{\prime})\right)$
$\displaystyle\qquad\qquad+{1\over k\cdot w}\left({if^{abc}\over
N^{2}-1}S_{\sigma\tau}(k^{\prime})+G_{\sigma\tau}^{cba}(k^{\prime}-k,k^{\prime})\right)\widehat{\Gamma}^{w\tau\sigma}_{L\,abc}(k,k^{\prime}).$
(36)

Since (36) is integrated over $k$ and $k^{\prime}$ in (10), one can change the
integration variable as $k\to k^{\prime}-k$ in the last term of (36)
containing the ghost-like term. Furthermore, because of
$\widehat{\Gamma}^{w\tau\sigma}_{L\,abc}(k^{\prime}-k,k^{\prime})=\widehat{\Gamma}^{\sigma\tau
w}_{L\,cba}(k,k^{\prime})$, one can change this term as

$\displaystyle{1\over k\cdot
w}G_{\sigma\tau}^{cba}(k^{\prime}-k,k^{\prime})\widehat{\Gamma}^{w\tau\sigma}_{L\,abc}(k,k^{\prime})$
$\displaystyle\qquad\longrightarrow{1\over k^{\prime}\cdot w-k\cdot
w}G_{\sigma\tau}^{cba}(k,k^{\prime})\widehat{\Gamma}^{\sigma\tau
w}_{L\,cba}(k,k^{\prime})={1\over k^{\prime}\cdot w-k\cdot
w}G_{\sigma\tau}^{abc}(k,k^{\prime})\widehat{\Gamma}^{\sigma\tau
w}_{L\,abc}(k,k^{\prime}).$ (37)

Using this form for the last term in (36) and applying the collinear expansion
to the first term in (36) up to twist-3, one obtains

$\displaystyle\left[S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})\right]^{\rm
twist-3}$ $\displaystyle\qquad=z\,z^{\prime}\,\Omega^{\mu}_{\
\alpha}\Omega^{\nu}_{\ \beta}\left\\{S_{L\,\mu\nu P_{h}}^{abc}\left({1\over
z},{1\over z^{\prime}}\right)+\Omega^{\lambda}_{\
\gamma}k^{\gamma}\left.{\partial S_{L\,\mu\nu
P_{h}}^{abc}(k,k^{\prime})\over\partial k^{\lambda}}\right|_{\rm c.l.}\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\Omega^{\lambda}_{\
\gamma}k^{\prime\gamma}\left.{\partial S_{L\,\mu\nu
P_{h}}^{abc}(k,k^{\prime})\over\partial k^{\prime\lambda}}\right|_{\rm
c.l.}\right\\}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta w}(k,k^{\prime})$
$\displaystyle\qquad+z\,z^{\prime}\,\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\ \sigma}S_{L\,\mu\nu\lambda}^{abc}\left({1\over
z},{1\over
z^{\prime}}\right)\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta\sigma}(k,k^{\prime})$
$\displaystyle\qquad+z\,{if^{abc}\over
N^{2}-1}S_{\sigma\tau}(k^{\prime})\widehat{\Gamma}^{w\tau\sigma}_{L\,abc}(k,k^{\prime})+{1\over
1/z^{\prime}-1/z}G_{\sigma\tau}^{abc}(k,k^{\prime})\widehat{\Gamma}^{\sigma\tau
w}_{L\,abc}(k,k^{\prime}),$ (38)

where we have set $k\cdot w={1\over z}$ and $k^{\prime}\cdot w={1\over
z^{\prime}}$. The first term of (38) can be further rewritten by the Ward-
Takahashi identity (15). Collinear limit of (15) gives

$\displaystyle S_{L\,\mu\nu P_{h}}^{abc}\left({1\over z},{1\over
z^{\prime}}\right)={1\over 1/z^{\prime}-1/z}\left\\{{-if^{abc}\over
N^{2}-1}S_{\mu\nu}\left({z^{\prime}}\right)+G_{\mu\nu}^{abc}\left({1\over
z},{1\over z^{\prime}}\right)\right\\},$ (39)

Similarly from the collinear limit of the first derivatives of (15) with
respect to $k$ and $k^{\prime}$, one obtains

$\displaystyle\left.{\partial S_{L\,\mu\nu
P_{h}}^{abc}(k,k^{\prime})\over\partial k^{\lambda}}\right|_{\rm c.l.}={1\over
1/z^{\prime}-1/z}\left\\{S_{L\,\mu\nu\lambda}^{abc}\left({1\over z},{1\over
z^{\prime}}\right)+\left.{\partial G_{\mu\nu}^{abc}(k,k^{\prime})\over\partial
k^{\lambda}}\right|_{\rm c.l.}\right\\},$ (40) $\displaystyle\left.{\partial
S_{L\,\mu\nu P_{h}}^{abc}(k,k^{\prime})\over\partial
k^{\prime\lambda}}\right|_{\rm c.l.}={1\over
1/z^{\prime}-1/z}\left\\{-S_{L\,\mu\nu\lambda}^{abc}\left({1\over z},{1\over
z^{\prime}}\right)+\left.{\partial G_{\mu\nu}^{abc}(k,k^{\prime})\over\partial
k^{\prime\lambda}}\right|_{\rm c.l.}-{if^{abc}\over N^{2}-1}\left.{\partial
S_{\mu\nu}(k^{\prime})\over\partial k^{\prime\lambda}}\right|_{\rm
c.l.}\right\\}.$ (41)

In the last term in the RHS of (38), using the relation (21) and following a
similar procedure as (34) and (35), one can rewrite

$\displaystyle G_{\mu\nu}^{abc}(k,k^{\prime})\widehat{\Gamma}^{\mu\nu
w}_{L\,abc}(k,k^{\prime})=z\,z^{\prime}\,G_{\mu\nu}^{abc}(k,k^{\prime})\Omega^{\mu}_{\
\alpha}\Omega^{\nu}_{\ \beta}\widehat{\Gamma}^{\alpha\beta
w}_{LFA\,abc}(k,k^{\prime}).$ (42)

Using this form, one sees the collinear expansion of the last term in (38)
yields the identical terms as the ghost-like terms in (39), (40) and (41).
This way one obtains the ghost-like terms (i.e., terms containg
$G_{\mu\nu}^{abc}$) in (38) as

$\displaystyle\left[S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})\right]^{\rm
ghost}=2\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}{zz^{\prime}\over
1/z^{\prime}-1/z}\left\\{G_{\mu\nu}^{abc}\left({1\over z},{1\over
z^{\prime}}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad+\Omega^{\lambda}_{\
\gamma}\left(\left.k^{\gamma}{\partial
G_{\mu\nu}^{abc}(k,k^{\prime})\over\partial k^{\lambda}}\right|_{\rm
c.l.}+\left.k^{\prime\gamma}{\partial
G_{\mu\nu}^{abc}(k,k^{\prime})\over\partial k^{\prime\lambda}}\right|_{\rm
c.l.}\right)\right\\}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta
w}(k,k^{\prime}).$ (43)

Using the actual forms of the ghost-like terms given in Appendix A of [1], we
will show in Appendix that (43) does not contribute to the LO twist-3 cross
section. Hence we will discard (43) below.

Remaining terms in (38) can be written as

$\displaystyle\left[S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})\right]^{\rm
twist-3}$ $\displaystyle=\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}{1\over
1/z+i\epsilon}{z^{\prime}\over
1/z^{\prime}-1/z+i\epsilon}\left({-if^{abc}\over
N^{2}-1}\right)S_{\mu\nu}\left({z^{\prime}}\right)\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta
w}(k,k^{\prime})$ $\displaystyle+\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\
\gamma}{1\over{1/z}+i\epsilon}{z^{\prime}\over{1/z^{\prime}}-{1/z}+i\epsilon}S_{L\,\mu\nu\lambda}^{abc}\left({1\over
z},{1\over z^{\prime}}\right)$
$\displaystyle\qquad\qquad\qquad\times\left\\{(k-k^{\prime})^{\gamma}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta
w}(k,k^{\prime})+\left({1\over z^{\prime}}-{1\over
z}\right)\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta\gamma}(k,k^{\prime})\right\\}$
$\displaystyle+\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\ \gamma}{1\over 1/z+i\epsilon}{z^{\prime}\over
1/z^{\prime}-1/z+i\epsilon}\left({-if^{abc}\over
N^{2}-1}\right)k^{\prime\gamma}\left.{\partial
S_{\mu\nu}(k^{\prime})\over\partial k^{\prime\lambda}}\right|_{\rm
c.l.}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta w}(k,k^{\prime})$
$\displaystyle+\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}{z^{\prime 2}\over
1/z+i\epsilon}\left\\{S_{\mu\nu}(z^{\prime})+\Omega^{\lambda}_{\
\gamma}k^{\prime\gamma}\left.{\partial S_{\mu\nu}(k^{\prime})\over\partial
k^{\prime\lambda}}\right|_{\rm c.l.}\right\\}$
$\displaystyle\qquad\qquad\qquad\times{if^{abc}\over
N^{2}-1}\left(-k^{\prime\alpha}w_{\sigma}+k^{\prime}\cdot wg^{\alpha}_{\
\sigma}\right)\left(-k^{\prime\beta}w_{\tau}+k^{\prime}\cdot wg^{\beta}_{\
\tau}\right)\widehat{\Gamma}_{L\,abc}^{w\tau\sigma}(k,k^{\prime}),$ (44)

where the third term in (38) was rewritten as the last term by using (14) and
subsequest collinear expansion of $S_{\mu\nu}(k^{\prime})$. We have introduced
$i\epsilon$ in the denominators, which gives rise to the future pointing gauge
links. Integration of (44) over $k$ and $k^{\prime}$ proceeds as follows. We
first write for the last term of (44)

$\displaystyle\bigl{(}-k^{\prime\alpha}w_{\sigma}+k^{\prime}\cdot
wg^{\alpha}_{\ \sigma}\bigr{)}\bigl{(}-k^{\prime\beta}w_{\tau}+k^{\prime}\cdot
wg^{\beta}_{\
\tau}\bigr{)}\widehat{\Gamma}_{L\,abc}^{w\tau\sigma}(k,k^{\prime})$
$\displaystyle=\int d^{4}\xi\int d^{4}\eta
e^{-ik\xi}e^{-i(k^{\prime}-k)\eta}\sum_{X}\langle
0|F_{b}^{(0)\,w\beta}(0)|hX\rangle$ $\displaystyle\times\langle
hX|g(\partial^{w}A_{a}^{w}(\xi))A_{c}^{\alpha}(\eta)-g(\partial^{w}A_{a}^{\alpha}(\xi))A_{c}^{w}(\eta)+gA_{a}^{w}(\xi)F_{c}^{(0)\,w\alpha}(\eta)+gF_{a}^{(0)\,w\alpha}(\xi)A_{c}^{w}(\eta)|0\rangle\qquad$
$\displaystyle\equiv\widehat{\Gamma}_{LF\partial\,abc}^{\alpha\beta}(k,k^{\prime}).$
(45)

Then the contribution from the first term in $\left\\{\quad\right\\}$ of the
last term of (44) reads

$\displaystyle\int{d^{4}k\over(2\pi)^{4}}\int{d^{4}k^{\prime}\over(2\pi)^{4}}\Omega^{\mu}_{\
\alpha}\Omega^{\nu}_{\ \beta}{z^{\prime 2}\over 1/z+i\epsilon}{if^{abc}\over
N^{2}-1}S_{\mu\nu}(z^{\prime})\widehat{\Gamma}_{LF\partial\,abc}^{\alpha\beta}(k,k^{\prime})$
$\displaystyle=\int d\left({1\over z^{\prime}}\right)\int d\left({1\over
z}\right)\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}\,z^{\prime
2}\,{if^{abc}\over N^{2}-1}S_{\mu\nu}(z^{\prime})\times i\int{d\lambda\over
2\pi}\int{d\mu\over
2\pi}e^{-i\lambda/z}e^{-i\mu\left(1/z^{\prime}-1/z\right)}$
$\displaystyle\qquad\times\sum_{X}\langle
0|F_{b}^{(0)\,w\beta}(0)|hX\rangle\langle hX|gA_{a}^{w}(\lambda
w)A_{c}^{\alpha}(\mu w)-gA_{a}^{\alpha}(\lambda w)A_{c}^{w}(\mu w)|0\rangle$
$\displaystyle+\int d\left({1\over z^{\prime}}\right)\int d\left({1\over
z}\right)\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}{z^{\prime 2}\over
1/z+i\epsilon}{if^{abc}\over N^{2}-1}S_{\mu\nu}(z^{\prime})\int{d\lambda\over
2\pi}\int{d\mu\over
2\pi}e^{-i\lambda/z}e^{-i\mu\left(1/z^{\prime}-1/z\right)}$
$\displaystyle\qquad\times\sum_{X}\langle
0|F_{b}^{(0)\,w\beta}(0)|hX\rangle\langle hX|gA_{a}^{w}(\lambda
w)F_{c}^{(0)\,w\alpha}(\mu w)+gF_{a}^{(0)\,w\alpha}(\lambda w)A_{c}^{w}(\mu
w)|0\rangle,$ (46)

where, in the first term, we have performed integration by parts for
$\lambda$-integration, which kills the factor ${1\over 1/z+i\epsilon}$.
Integration over $1/z$ of this equation can be done immediately. The second
term in $\left\\{\quad\right\\}$ of the last term of (44) can be integrated
parallelly. Following this procedure, $W_{g}^{\rm(b)}$ is obtained by the
integral of (44) as

$\displaystyle W_{g}^{\rm(b)}={1\over
2}\int{d^{4}k\over(2\pi)^{4}}\int{d^{4}k^{\prime}\over(2\pi)^{4}}\left[S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})\right]^{\rm
twist-3}$ $\displaystyle=-\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}\int
d\left({1\over z^{\prime}}\right)f^{abc}z^{\prime
2}S_{\mu\nu}\left({z^{\prime}}\right)$ $\displaystyle\qquad\qquad\times{1\over
N^{2}-1}\sum_{X}\int{d\lambda\over 2\pi}e^{-i\lambda/z^{\prime}}\langle
0|F_{b}^{(0)\,w\beta}(0)|hX\rangle\langle hX|gA_{a}^{w}(\lambda
w)A_{c}^{\alpha}(\lambda w)|0\rangle$ $\displaystyle+\Omega^{\mu}_{\
\alpha}\Omega^{\nu}_{\ \beta}\int d\left({1\over
z^{\prime}}\right)f^{abc}z^{\prime 2}S_{\mu\nu}\left({z^{\prime}}\right)$
$\displaystyle\qquad\qquad\qquad\times{1\over
N^{2}-1}\sum_{X}\int{d\lambda\over 2\pi}e^{-i\lambda/z^{\prime}}\langle
0|F_{b}^{(0)\,w\beta}(0)|hX\rangle\langle hX|gF_{a}^{(0)\,w\alpha}(\lambda
w)\int_{\infty}^{\lambda}d\mu A_{c}^{w}(\mu w)|0\rangle$
$\displaystyle+\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\ \gamma}\int d\left({1\over
z^{\prime}}\right)z^{\prime 2}if^{abc}\left.{\partial
S_{\mu\nu}\left(k^{\prime}\right)\over\partial k^{\prime\lambda}}\right|_{\rm
c.l.}$ $\displaystyle\qquad\qquad\qquad\times{1\over
N^{2}-1}\sum_{X}\int{d\lambda\over 2\pi}e^{-i\lambda/z^{\prime}}\langle
0|F_{b}^{(0)\,w\beta}(0)|hX\rangle\langle
hX|\partial^{\gamma}\left\\{gA_{a}^{w}(\lambda w)A_{c}^{\alpha}(\lambda
w)\right\\}|0\rangle$ $\displaystyle-\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\ \gamma}\int d\left({1\over
z^{\prime}}\right)z^{\prime 2}if^{abc}\left.{\partial
S_{\mu\nu}\left(k^{\prime}\right)\over\partial k^{\prime\lambda}}\right|_{\rm
c.l.}{1\over N^{2}-1}\sum_{X}\int{d\lambda\over
2\pi}e^{-i\lambda/z^{\prime}}\langle 0|F_{b}^{(0)\,w\beta}(0)|hX\rangle$
$\displaystyle\qquad\qquad\qquad\times\langle
hX|\int_{\infty}^{\lambda}d\mu\left\\{\left(\partial^{\gamma}F_{a}^{(0)\,w\alpha}(\lambda
w)\right)gA_{c}^{w}(\mu w)+F_{a}^{(0)\,w\alpha}(\lambda
w)g\partial^{\gamma}A_{c}^{w}(\mu w)\right\\}|0\rangle$ $\displaystyle-{i\over
2}\,\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}\Omega^{\lambda}_{\
\gamma}\int_{1}^{\infty}d\left({1\over
z^{\prime}}\right)\int_{0}^{1/z^{\prime}}d\left({1\over
z}\right){zz^{\prime}\over
1/z^{\prime}-1/z}S_{L\,\mu\nu\lambda}^{abc}\left({1\over z},{1\over
z^{\prime}}\right)\widehat{\Gamma}_{LF\,abc}^{\alpha\beta\gamma}\left({1\over
z},{1\over z^{\prime}}\right),$ (47)

where

$\displaystyle\widehat{\Gamma}_{LF\,abc}^{\alpha\beta\gamma}\left({1\over
z},{1\over z^{\prime}}\right)$ $\displaystyle\quad=\sum_{X}\int{d\lambda\over
2\pi}\int{d\mu\over 2\pi}e^{-i\lambda/z}e^{-i\mu(1/z^{\prime}-1/z)}\langle
0|F^{(0)\,w\beta}_{b}(0)|hX\rangle\langle hX|F^{(0)\,w\alpha}_{a}(\lambda
w)gF^{(0)\,w\gamma}_{c}(\mu w)|0\rangle.\qquad\ $ (48)

The first four terms in (47) come from the combination of the first, third and
the last terms in (44), while the last term in (47) is from the second term of
(44). $\widehat{\Gamma}_{LF\,abc}^{\alpha\beta\gamma}\left({1\over z},{1\over
z^{\prime}}\right)$ can be identified as the lowest order part of the
dynamical FF (5).

Calculation of $W_{g}^{\rm(c)}$ can be performed in the same way. The result
reads

$\displaystyle W_{g}^{\rm(c)}={1\over
2}\int\frac{d^{4}k}{(2\pi)^{4}}\int\frac{d^{4}k^{\prime}}{(2\pi)^{4}}\biggl{[}S^{abc}_{R\,\mu\nu\lambda}(k,k^{\prime})\widehat{\Gamma}^{\mu\nu\lambda}_{R\,abc}(k,k^{\prime})\biggr{]}^{\rm
twist-3}$ $\displaystyle=\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\ \beta}\int
d\left({1\over z}\right)f^{abc}z^{2}S_{\mu\nu}\left({z}\right)$
$\displaystyle\qquad\qquad\times{1\over N^{2}-1}\sum_{X}\int{d\lambda\over
2\pi}e^{-i\lambda/z}\langle 0|A_{b}^{w}(0)gA_{c}^{\beta}(0)|hX\rangle\langle
hX|F_{a}^{(0)\,w\alpha}(\lambda w)|0\rangle$ $\displaystyle+\Omega^{\mu}_{\
\alpha}\Omega^{\nu}_{\ \beta}\sum_{X}\int d\left({1\over
z}\right)f^{abc}z^{2}S_{\mu\nu}\left({z}\right)$
$\displaystyle\qquad\times{1\over N^{2}-1}\sum_{X}\int{d\lambda\over
2\pi}e^{-i\lambda/z}\langle 0|\int^{\infty}_{0}d\mu A_{c}^{w}(\mu
w)F_{b}^{(0)\,w\beta}(0)|hX\rangle\langle hX|gF_{a}^{(0)\,w\alpha}(\lambda
w)|0\rangle$ $\displaystyle-\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\ \gamma}\sum\int d\left({1\over
z}\right)z^{2}if^{abc}\left.{\partial S_{\mu\nu}\left(k\right)\over\partial
k^{\lambda}}\right|_{\rm c.l.}$ $\displaystyle\qquad\qquad\qquad\times{1\over
N^{2}-1}\sum_{X}\int{d\lambda\over 2\pi}e^{-i\lambda/z}\langle
0|A_{b}^{w}(0)gA_{c}^{\beta}(0)|hX\rangle\langle
hX|\partial^{\gamma}F_{a}^{(0)\,w\alpha}(\lambda w)|0\rangle$
$\displaystyle-\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\ \gamma}\int d\left({1\over
z}\right)z^{2}if^{abc}\left.{\partial S_{\mu\nu}\left(k\right)\over\partial
k^{\lambda}}\right|_{\rm c.l.}$ $\displaystyle\qquad\times{1\over
N^{2}-1}\sum_{X}\int{d\lambda\over 2\pi}e^{-i\lambda/z}\langle
0|\int^{\infty}_{0}d\mu gA_{c}^{w}(\mu
w)F_{b}^{(0)\,w\beta}(0)|hX\rangle\langle
hX|\partial^{\gamma}F_{a}^{(0)\,w\alpha}(\lambda w)|0\rangle$
$\displaystyle-{i\over 2}\,\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}\Omega^{\lambda}_{\ \gamma}\int_{1}^{\infty}d\left({1\over
z}\right)\int_{0}^{1/z}d\left({1\over z^{\prime}}\right){zz^{\prime}\over
1/z^{\prime}-1/z}S_{R\,\mu\nu\lambda}^{abc}\left({1\over z},{1\over
z^{\prime}}\right)\widehat{\Gamma}_{RF\,abc}^{\alpha\beta\gamma}\left({1\over
z},{1\over z^{\prime}}\right),$ (49)

where

$\displaystyle\widehat{\Gamma}^{\alpha\beta\gamma}_{RF\,abc}\left({1\over
z},{1\over z^{\prime}}\right)$ $\displaystyle=\sum_{X}\int{d\lambda\over
2\pi}\int{d\mu\over 2\pi}e^{-i\lambda/z}e^{-i\mu(1/z^{\prime}-1/z)}\langle
0|F_{b}^{(0)\,w\beta}(0)gF_{c}^{(0)\,w\gamma}(\mu w)|hX\rangle\langle
hX|F_{a}^{(0)\,w\alpha}(\lambda w)|0\rangle.\qquad$ (50)

We can now compare the sum of $W_{g}^{\rm(b)}$ (47) and $W_{g}^{\rm(c)}$ (49)
with (2). We first remind that the gluon’s field strength is
$F_{a}^{\mu\nu}=\partial^{\mu}A_{a}^{\nu}-\partial^{\nu}A_{a}^{\mu}+gf^{abc}A^{\mu}_{b}A^{\nu}_{c}=F^{(0)\mu\nu}_{a}+gf^{abc}A_{b}^{\mu}A_{c}^{\nu}$.
The first terms of (47) and (49) are the $O(g)$-contribution to (2) from the
$O(A^{2})$-term of the field strength in the intrinsic FF (3). The second
terms of (47) and (49) are the $O(g)$-contribution to (2) from the gauge links
in the intrinsic FF. The third terms of (47) and (49) are the
$O(g)$-contribution to (2) from the $O(A^{2})$-term of the field strength in
the kinematical FF (4). The fourth terms of (47) and (49) are the
$O(g)$-contribution to (2) from the gauge links in the kinematical FF. To
identify the fifth terms of (47) and (49), we note the following relations:

$\displaystyle\widehat{\Gamma}_{RF\,abc}^{\alpha\beta\gamma}\left({1\over
z},{1\over
z^{\prime}}\right)=\widehat{\Gamma}_{LF\,bac}^{\beta\alpha\gamma}\left({1\over
z^{\prime}},{1\over z}\right)^{\star},$ (51) $\displaystyle
S^{abc}_{R\,\alpha\beta\gamma}\left({1\over z},{1\over
z^{\prime}}\right)=S^{bac}_{L\,\beta\alpha\gamma}\left({1\over
z^{\prime}},{1\over z}\right)^{\star}.$ (52)

From these relations, the sum of the last terms in (47) and (49) are the
$O(g)$ contribution to (2) from the dynamical FF (5). This way, the basic
formula (2) has been proved in the leading order with respect to the QCD
coupling constant.

The method used here for the twist-3 gluon FF contribution to
$pp\to\Lambda^{\uparrow}X$ can also be applied to the twist-3 gluon
distribution function contribution to the double-spin asymmetry in
$\vec{p}p^{\uparrow}\to DX$, which occurs as a nonpole contribution[12]. Our
method provides a clearer understanding for the absence of the ghost-like
terms in the corresponding LO twist-3 cross section.

##  4 Summary

In this paper we presented a new derivation of the basic formula (2) for the
twist-3 gluon FF contribution to $pp\to\Lambda^{\uparrow}X$. Our method uses
the Ward-Takahashi identities for the partonic hard parts from the outset
before performing the collinear expansion. This method provides a robust
shortcut to convert the correlation functions of the gauge (gluon) fields into
the gauge invariant correlation functions for the gluon’s field strengths.
Furthermore it provides a clear understanding that the ghost-like terms
appearing in the Ward-Takahashi identities do not contribute to the LO twist-3
cross section. Since this method is quite general, it will become a useful
tool to extend the formula in the next-to-leading order calculation.

## Acknowledgments

This work has been supported by the establishment of Niigata university
fellowships towards the creation of science technology innovation (R.I.), the
Grant-in-Aid for Scientific Research from the Japanese Society of Promotion of
Science under Contract Nos. 19K03843 (Y.K.) and 18J11148 (K.Y.), National
Natural Science Foundation in China under grant No. 11950410495, Guangdong
Natural Science Foundation under No. 2020A1515010794 and research startup
funding at South China Normal University (S.Y.).

## Appendix: Absence of the ghost-like contribution at LO twist-3

Here we show that the ghost-like term (43) does not contribute to the twist-3
cross section. Actual forms of $G_{\mu\nu}^{abc}(k,k^{\prime})$ are given in
(A7) for the $q\bar{q}\to gg$ channel, (A9) for the $qg\to gq$ channel, and
(A10)-(A14) for the $gg\to gg$ channel in [1]. In all channels they take the
structure

$\displaystyle
G_{\mu\nu}^{abc}(k,k^{\prime})=(k^{2}g_{\mu\rho}-k_{\mu}k_{\rho})f^{abc}\widetilde{G}^{\rho}_{\
\nu}(k,k^{\prime}),$ (53)

where $\widetilde{G}^{\rho}_{\ \nu}(k,k^{\prime})$ is some function of $k$ and
$k^{\prime}$ (and $xp$ and $x^{\prime}p^{\prime}$). Inserting this form into
(43), one obtains

$\displaystyle\left[S_{L\,\mu\nu\lambda}^{abc}(k,k^{\prime})\widehat{\Gamma}_{L\,abc}^{\mu\nu\lambda}(k,k^{\prime})\right]^{\rm
ghost}$ $\displaystyle=2\Omega^{\mu}_{\ \alpha}\Omega^{\nu}_{\
\beta}{f^{abc}z^{\prime}\over
1/z^{\prime}-1/z}\left\\{-k_{\mu}\widetilde{G}_{P_{h}\nu}\left({1\over
z},{1\over z^{\prime}}\right)+2P_{h}\cdot k\widetilde{G}_{\mu\nu}\left({1\over
z},{1\over z^{\prime}}\right)-\Omega^{\lambda}_{\
\gamma}k^{\gamma}P_{h\mu}\widetilde{G}_{\lambda\nu}\left({1\over z},{1\over
z^{\prime}}\right)\right.$ $\displaystyle\left.\qquad\qquad-{P_{h\mu}\over
z}\left(\left.\Omega^{\lambda}_{\
\gamma}k^{\gamma}{\partial\widetilde{G}_{P_{h}\nu}(k,k^{\prime})\over\partial
k^{\lambda}}\right|_{\rm c.l.}+\left.\Omega^{\lambda}_{\
\gamma}k^{\prime\gamma}{\partial\widetilde{G}_{P_{h}\nu}(k,k^{\prime})\over\partial
k^{\prime\lambda}}\right|_{\rm
c.l.}\right)\right\\}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta
w}(k,k^{\prime}).\qquad$ (54)

From this form one sees that all terms in $\left\\{\cdots\right\\}$ except for
the first one contribute only at twist-4: $\Omega^{\mu}_{\ \alpha}P_{h\mu}$
extracts “$\alpha=-$” component from $\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta
w}(k,k^{\prime})$ which is subleading, and $\Omega^{\lambda}_{\
\gamma}k^{\gamma}$ and $P_{h}\cdot k$, respectively, causes additional one-
and two- power suppressions. For the first term which contributes at twist-3,
we obtain from (35)

$\displaystyle\Omega^{\mu}_{\
\alpha}k_{\mu}\widehat{\Gamma}_{LFA\,abc}^{\alpha\beta w}(k,k^{\prime})$
$\displaystyle\quad=\sum_{X}\int d^{4}\xi\int d^{4}\eta
e^{-ik\xi}e^{-i(k^{\prime}-k)\eta}\langle 0|F^{(0)\,\beta
w}_{b}(0)|hX\rangle\langle
hX|(-i){\partial\over\partial\xi^{\alpha}}F^{(0)\,\alpha
w}_{a}(\xi)gA^{w}_{c}(\eta)|0\rangle.\qquad$ (55)

Here we note that the QCD equation of motion $D_{\alpha}F^{\alpha
w}_{a}+g\bar{\psi}t^{a}\hbox to0.0pt{/\hss}{\mkern-1.0muw}\psi=0$ implies
$\partial_{\alpha}F^{(0)\,\alpha w}_{a}(\xi)$ is of $O(g)$, and hence the
first term in $\left\\{\cdots\right\\}$ of (54) becomes $O(g^{6})$. Consistent
treatment of this term requires the inclusion of all $O(g^{6})$ diagrams,
which is beyond the scope of this work. We thus conclude that it does not
contribute to the LO cross section. This proves that (43) does not contribute
to the LO twist-3 cross section.

## References

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# Thickness optimization of the output power and effective thermoelectric
figure of merit of thin thermoelectric generator

Kazuhiko Seki  GZR, National Institute of Advanced Industrial Science and
Technology (AIST), Onogawa 16-1 AIST West, Ibaraki, 305-8569, Japan
<EMAIL_ADDRESS>Masakazu Mukaida  NMRI, National Institute of Advanced
Industrial Science and Technology (AIST), Higashi 1-1-1 AIST Central 5,
305-8565, Japan  Qingshuo Wei  NMRI, National Institute of Advanced Industrial
Science and Technology (AIST), Higashi 1-1-1 AIST Central 5, Ibaraki,
305-8565, Japan  Takao Ishida  GZR, National Institute of Advanced Industrial
Science and Technology (AIST), Onogawa 16-1 AIST West, Ibaraki, 305-8569,
Japan

###### Abstract

The conventional thermoelectric figure of merit and the power factor are not
sufficient as a measure of thin film quality of thermoelectric materials,
where the power conversion efficiency depends on the film dimensions. By
considering the film size, the effective thermoelectric figure of merit and
effective Seebeck coefficient are introduced to guarantee that the maximum
energy conversion efficiency increases as the effective thermoelectric figure
of merit increases. Similarly, the effective power factor is defined. By
introducing typical material properties for Bi2Te3 and PEDOT, we study the
thickness dependence of the effective figure of merit and the effective power
factor.

  * February 2022

Thermoelectric generators directly convert waste thermal energy into
electrical power. Electric currents can be generated from temperature
gradients by thermoelectric effect; the energy conversion efficiency is known
to be characterized by the dimensionless thermoelectric figure of merit, given
by [1, 2]

$\displaystyle\mbox{zT}_{\rm b}=\frac{\alpha^{2}T_{\rm m}}{\rho\kappa},$ (1)

where $T_{\rm m}$ is the mean temperature between the cold side and the hot
side, $\alpha$ is the Seebeck coefficient, $\rho$ is the electrical
resistivity, and $\kappa$ is the thermal conductivity. The thermoelectric
figure of merit $\mbox{zT}_{\rm b}$ provides a measure of the quality of
thermoelectric materials and can be used for the guide to compare
thermoelectric properties of materials.

The recent demand for thermoelectric generators for Internet of Things (IoT)
and wearable devices has urged to develop thin film thermoelectric devices,
which are flexible and could operate around room temperature. [3, 4, 5, 6, 7,
8, 9] Such devices should work under limited temperature gradients and require
sufficient energy conversion efficiency. Conventionally, the thermoelectric
figure of merit and the power factor have been used to relate the material
properties such as the electrical resistivity and the thermal conductivity
with the maximum energy conversion efficiency and the maximum output power,
respectively. However, for thin film thermoelectric devices, the electrical
contact resistivity and the heat transfer on the surface facing to the
environments need to be considered because of large areal size compared to the
film thickness. [4, 10, 6, 7] Therefore, the energy conversion efficiency
could be influenced by the film dimensions in addition to the thermoelectric
figure of merit $\mbox{zT}_{\rm b}$ of the bulk material. To consider the
influence of film dimensions on the energy conversion efficiency, we define
below $\mbox{zT}_{\rm eff}(d)$, where $d$ indicates the film thickness.

We first introduce the effective electrical resistivity using the intrinsic
material electrical resistivity $\rho$ $\Omega$m and the contact electrical
resistivity $\rho_{\rm c}$ $\Omega\mbox{m}^{2}$. Since they are connected
electrically in series, the effective resistivity can be expressed as
$\rho_{\rm eff}=\rho+\rho_{\rm c}/d$. When the electrical contact resistivity
of one side is given by $\rho_{{\rm c}0}$ and that of the other side is given
by $\rho_{{\rm c}d}$, the total electrical contact resistivity is expressed as
$\rho_{\rm c}=\rho_{{\rm c}0}+\rho_{{\rm c}d}$. Similarly, we define the
effective thermal conductivity $\kappa_{\rm eff}$ by the intrinsic material
thermal conductivity $\kappa$ W/(mK) and the heat transfer coefficient $h$
W/(m2K) as $\kappa_{\rm eff}=(1/\kappa+1/(hd))^{-1}$, where the heat transfer
coefficient is inverse of the contact thermal resistivity; the effective
thermal conductivity is given by the inverse of the sum of the intrinsic
material thermal resistivity and the contact thermal resistivity. When the
heat transfer coefficient of one side is given by $h_{0}$ and that of the
other side is given by $h_{d}$, the total heat transfer coefficient is
expressed as $h=\left(1/h_{0}+1/h_{d}\right)^{-1}$. Previously, the effective
thermal conductivity was given by the sum of the intrinsic material thermal
conductivity and the thermal interface conductance, which indicates parallel
thermal flows of these components. [5]

Figure 1: (Color online) A slab of thermoelectric material (thickness $d$ and
cross section area $S$) under thermal gradient. The environmental temperature
facing at $x=0$ is denoted by $T_{\rm h}$ and that facing at $x=0$ is denoted
by $T_{\rm c}$. Heat flows under the temperature gradient, where the thermal
conductivity and the heat transfer coefficient are denoted by $\kappa$ and
$h$, respectively. Charges flow under the electromotive force generated by the
temperature difference, where the electrical resistivity and the electrical
contact resistivity are denoted by $\rho$ and $\rho_{c}$, respectively.

We introduce the figure of merit by considering the maximum of energy
conversion efficiency by taking into account the film dimensions. For thick
bulk materials, the maximum conversion efficiency is known to be an increasing
function of zTb at the mean temperature of the cold and the hot side.
Therefore, the maximum conversion efficiency is higher as zTb increases. We
needs to define zT${}_{\rm eff}(d)$ for films in such a way that the maximum
conversion efficiency increases by increasing zT${}_{\rm eff}(d)$.

We consider the case that the film of thickness $d$ and the surface area of
$S$ is located between $x=0$ and $d$ in the x coordinate as shown in Fig. 1.
The ambient temperature around $x=0$ is $T_{\rm h}$ and the ambient
temperature around $x=d$ is $T_{\rm c}$; we assume $T_{\rm h}>T_{\rm c}$. We
also assume that systems operate under a constant temperature difference
rather than under constant heat flux. [11] The energy conversion efficiency
can be expressed as [12]

$\displaystyle\eta=\frac{VI}{\alpha T(0)I-RI^{2}/2+K\left[T(0)-T(d)\right]},$
(2)

where $V$ and $I$ denote the voltage across the film and the electrical
current passing through the film, respectively; $R$ and $K$ denote the
electrical resistance and the thermal conductance, respectively. The
temperature at $x=0$ and $x=d$ is denoted by $T(0)$ and $T(d)$, respectively;
$T(d)$ differs from the environmental temperature denoted by $T_{\rm c}$ if
the heat transfer coefficient is low. We calculate $T(0)$ and $T(d)$ when the
heat transfer coefficient at the hot side ($x=0$) is $h_{\rm h}$ and that at
the cold side ($x=d$) is $h_{\rm c}$. $T(0)$ reduces to $T_{\rm h}$ in the
limit of $h_{\rm h}\rightarrow\infty$. As shown in A, we obtain

$\displaystyle T(0)-T(d)=\Delta T/(1+\kappa/(hd)),$ (3)

where $\Delta T=T_{\rm h}-T_{\rm c}$ and
$h=\left(1/h_{0}+1/h_{d}\right)^{-1}$. We also find

$\displaystyle T_{\rm m}=(T(0)+T(d))/2=(T_{\rm h}+T_{\rm c})/2.$ (4)

When the thermoelectric power is generated, the external load with the
resistance $R_{\rm ext}$ is imposed. We introduce $R_{\rm ext}=yR_{\rm eff}$,
where $R_{\rm eff}=(\rho+\rho_{\rm c}/d)d/S$ for the film with the thickness
$d$ and the surface area of $S$ facing to the environments. [1, 13, 14, 15] We
have $I=\alpha[T(0)-T(d)]/[(1+y)R_{\rm eff}]$ and $V=\alpha[T(0)-T(d)]-IR_{\rm
eff}=\alpha[T(0)-T(d)]y/(1+y)$. The power can be expressed as

$\displaystyle P=IV=\frac{(\alpha_{\rm eff}\Delta T)^{2}}{R_{\rm
eff}}\frac{y}{(1+y)^{2}},$ (5)

where $\alpha_{\rm eff}$ is defined by

$\displaystyle\alpha_{\rm eff}=\frac{\alpha}{1+\kappa/(hd)}.$ (6)

By introducing $\alpha_{\rm eff}$, the thermoelectric power can be expressed
in the conventional form as shown in Eq. (5).

Using $y=R_{\rm ext}/R_{\rm eff}$, the energy conversion efficiency given by
(2) can be expressed for films as

$\displaystyle\eta(y)=\frac{y\Delta T}{(1+y)T_{\rm m}+(1+y)^{2}/Z_{\rm
eff}+y\Delta T/2},$ (7)

where $Z_{\rm eff}$ is defined by

$\displaystyle Z_{\rm eff}=\frac{\alpha_{\rm eff}^{2}}{R_{\rm eff}K_{\rm
eff}},$ (8)

and we have $R_{\rm eff}=(\rho+\rho_{\rm c}/d)d/S$ and $K_{\rm
eff}=\left(\kappa^{-1}+1/(hd)\right)^{-1}S/d$. Using $d\eta(y)/dy=0$, [16] we
find $y$ at the maximum conversion efficiency as $y_{\rm max}=\sqrt{1+Z_{\rm
eff}T_{\rm m}}$; the maximum conversion efficiency is obtained as

$\displaystyle\eta_{\rm max}$ $\displaystyle=\frac{\Delta T\sqrt{1+Z_{\rm
eff}T_{\rm m}}}{2(1+\sqrt{1+Z_{\rm eff}T_{\rm m}})/Z_{\rm eff}+2T_{\rm
m}+T_{\rm h}\sqrt{1+Z_{\rm eff}T_{\rm m}}}$ (9) $\displaystyle=\frac{\Delta
T\sqrt{1+Z_{\rm eff}T_{\rm m}}(\sqrt{1+Z_{\rm eff}T_{\rm m}}-1)}{(2T_{\rm
m}-T_{\rm h})\sqrt{1+Z_{\rm eff}T_{\rm m}}+T_{\rm h}(1+Z_{\rm eff}T_{\rm m})}$
$\displaystyle=\left(\frac{\Delta T}{T_{\rm
h}}\right)\frac{\left(\sqrt{1+Z_{\rm eff}T_{\rm m}}-1\right)}{\sqrt{1+Z_{\rm
eff}T_{\rm m}}+(T_{\rm c}/T_{\rm h})},$

where $\Delta T/T_{\rm h}$ is the Carnot efficiency; the maximum efficiency
approaches the Carnot efficiency by increasing $Z_{\rm eff}$. By introducing
$\alpha_{\rm eff}$ defined by Eq. (6), the conventional form of the maximum
conversion efficiency given by Eq. (9) is obtained, where $Z=\mbox{zT}_{\rm
b}$ given by Eq. (1) is redefined by Eq. (8); it is sufficient to study
$Z_{\rm eff}$ in place of $Z=\alpha^{2}/(RK)$ with $R=\rho d/S$ and $K=\kappa
S/d$ for films. The maximum conversion efficiency is higher for the higher
value of $Z_{\rm eff}$.

The temperature difference at both ends of the material is given by
$T(0)-T(d)$ which is related to the ambient temperature difference $\Delta T$
by Eq. (3). The factor $1+\kappa/(hd)$ in Eq. (3) is absorbed in $\alpha_{\rm
eff}$ in Eq. (6). Using $\alpha_{\rm eff}$, $Z_{\rm eff}$ is defined and the
maximum conversion efficiency is a monotonically increasing function of
$Z_{\rm eff}$ for any thickness of the thermoelectric material. If we use
$\alpha$ in stead of $\alpha_{\rm eff}$ in $Z_{\rm eff}$, the correlation to
the maximum conversion efficiency is lost. Therefore, it is essential to
define $Z_{\rm eff}$ in terms of $\alpha_{\rm eff}$ when the heat transfer
coefficient is low for thin film ($hd<\kappa$).

By taking into account the contact electrical resistivity and the heat
transfer coefficient, the effective thermoelectric figure of merit for film
conductors [Eq. (8)] can be expressed as $\mbox{zT}_{\rm eff}(d)=Z_{\rm
eff}T_{\rm m}$ and $\mbox{zT}_{\rm eff}(d)$ is explicitly expressed as

$\displaystyle\mbox{zT}_{\rm eff}(d)$ $\displaystyle=\frac{\alpha_{\rm
eff}^{2}T_{\rm m}}{\rho_{\rm eff}\kappa_{\rm eff}},$ (10)
$\displaystyle=\mbox{zT}_{\rm b}S_{\rm zT}(d),$ (11)

where $\mbox{zT}_{\rm b}$ is the figure of merit of bulk given by Eq. (1) and
the size factor for the effective figure of merit can be expressed as

$\displaystyle S_{\rm zT}(d)=\frac{1}{[1+\rho_{\rm c}/(\rho
d)][1+\kappa/(hd)]}.$ (12)

The thickness dependence of $\mbox{zT}_{\rm eff}(d)=Z_{\rm eff}T_{\rm m}$ can
be studied using $S_{\rm zT}(d)$; the size factor of the effective figure of
merit [Eq. (12)] is a monotonically increasing function of the thickness $d$
of the thermoelectric generating layer.

It should be stressed that the conventional figure of merit defined only by
the material parameters [$\mbox{zT}_{\rm b}$ in Eq. (1)] is not a good index
of the maximum conversion efficiency for thin film conductors; the appropriate
index is given by Eq. (11), where the size factor given by Eq. (12) is taken
into account. The equivalent expression of Eq. (12) was defined to capture the
interface effects in thermoelectric microrefrigerators. [6]

We also study the power factor of thin film by maximizing the electrical
output power. The maximum power is obtained for $y=1$ in Eq. (5) and is given
by

$\displaystyle P_{\rm max}=\frac{(\alpha_{\rm eff}\Delta T)^{2}}{4R_{\rm
eff}}.$ (13)

The condition $y=1$ leads to $R_{\rm ext}=R_{\rm eff}$, where $R_{\rm
eff}=(\rho+\rho_{\rm c}/d)d/S$ involves the contact resistivity. In general,
the external load resistance should match the effective resistance involving
both the contact resistivity and the intrinsic material resistivity. Equation
(13) indicates that the thickness dependence of the maximum electric output
power can be studied by $\alpha_{\rm eff}^{2}/R_{\rm eff}$. Conventionally,
the bulk power factor is defined by

$\displaystyle\mbox{Pf}_{\rm b}=\frac{\alpha^{2}}{\rho},$ (14)

where $P_{\rm max}$ is multiplied by $d$ to eliminate the trivial thickness
dependence originating from $R=\rho d/S$, which remains even for $\rho_{\rm
c}=0$. As a result, the bulk power factor is expressed only by material
parameters such as $\rho$ and $\alpha$. Using Eq. (13), we obtain the power
factor of thin film with thickness $d$ as $\mbox{Pf}_{\rm eff}(d)=\alpha_{\rm
eff}^{2}/\rho_{\rm eff}$. However, it is more appropriate to define an
effective power factor using $\mbox{Pf}_{\rm g}(d)=\alpha_{\rm eff}^{2}/R_{\rm
eff}$ to absorb all terms associated with $d$ for studying the thickness
dependence. We rewrite $\mbox{Pf}_{\rm g}(d)=\mbox{Pf}_{\rm eff}(d)/d$ as

$\displaystyle\mbox{Pf}_{\rm g}(d)=\mbox{Pf}_{\rm b}S_{\rm Pw}(d),$ (15)

where the size factor for the effective power factor is given by

$\displaystyle S_{\rm Pw}(d)=\frac{1}{d[1+\rho_{\rm c}/(\rho
d)][1+\kappa/(hd)]^{2}}.$ (16)

The equivalent expression of Eq. (13) was introduced previously by noticing
the temperature difference between the actual device temperature and the
ambient temperature. [7]

For a given thickness $d$, both $\rho_{\rm c}<\rho d$ (small enough contact
resistance) and $h>\kappa/d$ (high enough heat transfer coefficient) should be
satisfied to associate the maximum conversion efficiency and the maximum power
with the bulk dimensionless thermoelectric figure of merit [Eq. (1)] and the
bulk power factor [Eq. (14)], respectively. For the maximum conversion
efficiency and the maximum power, the effect of finite thickness should be
considered either when $d<\rho_{\rm c}/\rho$ or when $d<\kappa/h$ holds.

Contrary to the size factor of the effective figure of merit [Eq. (12)], which
is a monotonically increasing function of the thickness $d$, the size factor
of the effective power factor [Eq. (16)] has a maximum at the thickness of the
thermoelectric generating layer given by

$\displaystyle d_{\rm
max}=\frac{\kappa/h+\sqrt{\kappa/h\left(\kappa/h+8\rho_{\rm
c}/\rho\right)}}{2}.$ (17)

When $\kappa/h>8\rho_{\rm c}/\rho$ is satisfied, the above equation is
simplified to

$\displaystyle d_{\rm max}\approx\kappa/h.$ (18)

The appearance of the maximum in the output power can be understood in the
following way. When the film thickness is small, the actual temperature
difference at the both ends of the film is smaller than the ambient
temperature difference as shown in Eq. (3). As a result, the Seebeck power is
reduced. By increasing the film thickness, the actual temperature difference
increases and the Seebeck power also increases. Therefore, the output power
increases up to a certain film thickness. When the film thickness exceeds
$d_{\rm max}$, the electric power output decreases with increasing the film
thickness because the electrical resistance [$R_{\rm eff}=(\rho+\rho_{\rm
c}/d)d/S$] increases with $d$.

Table 1: The physical properties of inorganic materials (Bi2Te3) and organic
materials (PEDOT)

Transport coefficient | Bi2Te3 | PEDOT
---|---|---
Electrical resistivity $\rho$ [$\Omega$cm] | $1\times 10^{-3}\mbox{ }^{\rm b}$ | $2.8\times 10^{-2},1.2\times 10^{-3}\mbox{ }^{\rm e}$
Thermal conductivity $\kappa$ [W/(mK)] | $2.0\mbox{ }^{\rm b}$ |  $0.1(0.17),0.9(0.94)$ $\mbox{ }^{\rm f}$
Electrical contact resistivitya $\rho_{\rm c}$ [$\mu\Omega$ cm2] |  $10^{2}$ $\mbox{ }^{\rm c}$ |  $(1-20)\times 10^{4}$ $\mbox{ }^{\rm g}$
Heat transfer coefficienta $h$ [W/(m2K)] |  $1-100$ d |  $1-100$ d

  * a Both the electrical contact resistivity and heat transfer coefficient should be defined at the interface between the material and the environment. Here, typical values are shown without specifying the environment.

  * b [17]

  * c [18]

  * d[19, 7]

  * e Through plane value is $2.8\times 10^{-2}$ and in-plane value is $1.2\times 10^{-3}$. [20]

  * f Through plane value is 0.1 (0.17) and the in-plane value is 0.9 (0.94). [21, 22] (The values in the parenthesis are from ref. [22].)

  * g The values depend on the combination of PEDOT and the contacted material. [23]

We study the size factor of the effective figure of merit [Eq. (12)] and the
size factor of the effective power factor [Eq. (16)] for the benchmark
organic/inorganic thermoelectric materials. Specifically, we use the physical
properties of inorganic materials (Bi2Te3) and organic materials (PEDOT)
summarized in Table 1. The thickness is varied from mm to cm range according
to Ref. [7]. The maxima calculated from Eq. (17) lie in this length scale by
substituting the material properties in Table 1.

As shown in Fig. 2, $S_{\rm zT}(d)$ (the size factor for the effective figure
of merit) monotonically increases by increasing $d$ for both inorganic
materials (Bi2Te3) and organic materials (PEDOT). By considering anisotropy in
intrinsic material electrical resistivity and intrinsic material thermal
conductivity of PEDOT, we show the case that the temperature gradient is
applied in the in-plane direction, and the case that the temperature gradient
is applied in the through-plane direction for PEDOT, where PEDOT would tend to
lie down at the surface. In the same figure, we also show $S_{\rm Pw}(d)$ (the
size factor for the effective power factor). $S_{\rm Pw}(d)$ shows a maximum
as a function of the film thickness for both inorganic materials (Bi2Te3) and
organic materials (PEDOT). Judging from $S_{\rm Pw}(d)$, PEDOT with the
temperature gradient in the through-plane direction (PEDOT-Through) can be
optimized at the thinner film thickness compared to the other materials. The
optimum value of $d$ for $S_{\rm Pw}(d)$ roughly corresponds to the inflection
point for $S_{\rm zT}(d)$, indicating that the decrease of $S_{\rm zT}(d)$ by
decreasing $d$ mainly occurs around the optimum value of $d$. The output power
can be optimized around $1$ cm for Bi2Te3 and could be optimized below $1$ cm
for PEDOT-Through. Though the effective figure of merit is a monotonically
increasing function of $d$, the large part of the figure of merit can be
reduced by reducing the thickness from $10$ to $1$ cm for Bi2Te3 and the
figure of merit can be largely reduced by reducing the thickness around 1 cm
for PEDOT-Through. By employing the finite element methods, the results
similar to Fig. 2 was reported, where the maximum efficiency monotonically
increases by increasing $d$ and the maximum power shows a maximum as a
function of $d$. [19]

Figure 2: (Color online) $S_{\rm Pw}(d)$ (the size factor for the effective
power factor) and $S_{\rm zT}(d)$ (the size factor for the effective figure of
merit) are shown against $d$. The blue lines indicate PEDOT with the
temperature gradient in the in-plane direction, the black lines indicate PEDOT
with the temperature gradient in the through-plane direction, and the red
lines indicate Bi2Te3. (PEDOT-Through, Bi2Te3, and PEDOT-In from top to
bottom.) The material parameters are shown in Table 1, where the values
without parenthesis, $h=100$ [W/(m2K)] for both Bi2Te3 and PEDOT, and
$\rho_{\rm c}=1.0\times 10^{4}$ $\mu\Omega$ cm2 for PEDOT are used. The brown
dashed-and-dotted line indicates $S_{\rm Pw}(d)=1/d$ originating from the
trivial thickness dependence of $R=\rho d/S$, where we assume $\rho_{\rm c}=0$
and take the limit of $h\rightarrow\infty$. $S_{\rm zT}(d)$ in the limit of
$d\rightarrow\infty$ is $1$.

In Fig. 3, we study the film thickness denoted by $d_{\rm max}$ at the maximum
in $S_{\rm Pw}(d)$ as a function of the heat transfer coefficient denoted by
$h$. $d_{\rm max}$ decreases by increasing the heat transfer coefficient
because the temperature gradient could be closer to the imposed temperature
gradient applied through ambient temperature if the heat transfer is higher.
In Fig. 3, $d_{\rm max}$ of PEDOT-Through is smaller than $d_{\rm max}$ of
Bi2Te3, though the difference decreases by increasing $h$.

Figure 3: (Color online) $d_{\rm max}$ is presented as a function of the heat
transfer coefficient ($h$). The blue line indicates PEDOT with the temperature
gradient in the in-plane direction, the black line indicates PEDOT with the
temperature gradient in the through-plane direction, and the red line
indicates Bi2Te3. (PEDOT-In, Bi2Te3, and PEDOT-Through from top to bottom at
$h=100$W/m2K.) The material parameters are shown in Table 1, where the values
without parenthesis, and $\rho_{\rm c}=1.0\times 10^{4}$ $\mu\Omega$ cm2 for
PEDOT are used.

In Fig. 4, we study the film thickness denoted by $d_{\rm max}$ at the maximum
in $S_{\rm Pw}(d)$ as a function of the contact resistivity denoted by $r_{\rm
c}$. $d_{\rm max}$ increases by increasing $r_{\rm c}$. $d_{\rm max}$ of
PEDOT-Through is more than an order of magnitude smaller than $d_{\rm max}$ of
Bi2Te3.

Figure 4: (Color online) $d_{\rm max}$ is presented as a function of the
contact resistivity ($r_{\rm c}$). The blue line indicates PEDOT with the
temperature gradient in the in-plane direction, the black line indicates PEDOT
with the temperature gradient in the through-plane direction, and the red line
indicates Bi2Te3. (Bi2Te3, PEDOT-In, and PEDOT-Through from top to bottom.)
The material parameters are shown in Table 1, where the values without
parenthesis, and $h=100$ [W/(m2K)] are used.

In practice, thermoelectric generators may be composed of series connected
many semiconductors. When the each unit is identical, $S_{\rm Pw}(d)$ and
$S_{\rm zT}(d)$ of the $N$ repeated units are independent of $N$ as shown
below. If the electrical resistivity, the thermal conductivity, the electrical
contact resistivity, and the heat transfer coefficient of the unit are denoted
by $\rho_{\rm u}$, $\kappa_{\rm u}$, $\rho_{\rm c,u}$, and $h_{\rm u}$, those
for the electrically connected units in series are expressed as $N\rho_{\rm
u}$, $N\kappa_{\rm u}$, $N\rho_{\rm c,u}$, and $Nh_{\rm u}$. Because $S_{\rm
Pw}(d)$ and $S_{\rm zT}(d)$ of the $N$ repeated units are expressed in terms
of the ratio such as $\kappa/h$ and $\rho/\rho_{\rm c}$, they are independent
of $N$. However, the dimensionless figure of merit for a single material
should be generalized if thermoelectric power generator consists of series
connected p-type and n-type semiconductors. We introduce the effective Seebeck
coefficient $\alpha_{\rm eff,p}=\alpha_{\rm p}/[1+\kappa_{\rm p}/(h_{\rm
p}d)]$, effective electrical resistivity $\rho_{\rm eff,p}=\rho_{\rm
p}+\rho_{\rm c,p}/d$, and effective thermal conductivity $\kappa_{\rm
eff,p}=(1/\kappa_{\rm p}+1/(h_{\rm p}d))^{-1}$ of the p-type semiconductor,
where the subscript p indicates the property of p-type semiconductor; the
material properties for the n-type semiconductor are defined similarly. When
the p-type semiconductor with the area $S_{\rm p}$ and n-type semiconductor
with the area $S_{\rm n}$ are connected electrically in series and thermally
in parallel, while the thickness $d$ is common for both semiconductors,
$Z_{\rm eff}=(\alpha_{\rm eff}^{2}T_{\rm m})/(R_{\rm eff}K_{\rm eff})$ given
by Eq. (8) still holds, where we have $R_{\rm eff}=\rho_{\rm eff,p}d/S_{\rm
p}+\rho_{\rm eff,n}d/S_{\rm n}$, $K_{\rm eff}=\kappa_{\rm eff,p}S_{\rm
p}/d+\kappa_{\rm eff,n}S_{\rm n}/d$ and $\alpha_{\rm eff}=\alpha_{\rm
eff,p}-\alpha_{\rm eff,n}$; the minus sign in front of $\alpha_{\rm eff,n}$ is
introduced to take into account the opposite direction of electron flow in
n-type semiconductor against the direction of hole flow in p-type
semiconductor. If $R_{\rm eff}K_{\rm eff}$ is minimized by optimizing the
ration $S_{\rm p}/S_{\rm n}$, we find $\mbox{zT}_{\rm
eff}(d)=\left(\alpha_{\rm eff,p}-\alpha_{\rm eff,n}\right)^{2}T_{\rm
m}/[\sqrt{\rho_{\rm eff,p}\kappa_{\rm eff,p}}+\sqrt{\rho_{\rm
eff,n}\kappa_{\rm eff,n}}]^{2}$, where $S_{\rm p}/S_{\rm n}=\sqrt{\rho_{\rm
p}\kappa_{\rm n}/(\rho_{\rm n}\kappa_{\rm p})}$ is obtained by $d(R_{\rm
eff}K_{\rm eff})/dX$ using $X=S_{\rm p}/S_{\rm n}$. In the limit of
$d\rightarrow\infty$, the known result is recovered. [1]

We studied the case that the intrinsic electrical resistivity, thermal
conductivity (regardless of phonon thermal conductivity or electronic thermal
conductivity), and Seebeck coefficient are material properties independent of
the dimensions. On this basis, we focused to illustrate the genuine
contributions of heat transfer coefficient and contact resistivity to the
maximum energy conversion efficiency and thermoelectric power.

In conclusion, the contact resistivity and heat transfer coefficient at the
interface between the thermoelectric material and the environment influence
the energy conversion efficiency and the electric output power if
thermoelectric materials are thin and having a large surface area. The
conventional thermoelectric figure of merit and the power factor are not
sufficient as a measure of the thin film quality of thermoelectric materials.
The effective thermoelectric figure of merit of thin film was introduced in
such a way that the maximum conversion efficiency is a monotonically
increasing function of the effective thermoelectric figure of merit regardless
of the thickness of the thermoelectric material. The effective thermoelectric
figure of merit thus defined can be expressed as a product of the conventional
thermoelectric figure of merit and the size factor. The size factor of the
effective thermoelectric figure of merit is shown to be an increasing function
of the film thickness. Similarly, we introduced the effective power factor and
the corresponding size factor; we showed the existence of the optimal film
thickness. We studied the thickness dependence of the size factor of the
effective thermoelectric figure of merit and the size factor of the effective
power factor using the physical properties of inorganic materials (Bi2Te3) and
organic materials (PEDOT). We showed that PEDOT with the temperature gradient
in the through-plane direction is advantageous over Bi2Te3 as regards to the
thinner optimal film thickness.

##  Appendix A Derivation of Eq. (3)

The boundary condition at $x=0$ is given by

$\displaystyle\left.\kappa\frac{\partial T}{\partial
x}\right|_{x=0}=h_{0}\left[T(0)-T_{\rm h}\right],$ (19)

where the heat flux flowing into the film is considered. The boundary
condition at $x=d$ is given by

$\displaystyle\left.-\kappa\frac{\partial T}{\partial
x}\right|_{x=d}=h_{d}\left[T(d)-T_{\rm c}\right],$ (20)

where the heat flux flowing out from the film is considered. The steady state
solution of the heat equation $\nabla^{2}T(x)=0$ can be expressed as
$T(x)=C_{1}+C_{2}x$. The two unknown constants $C_{1}$ and $C_{2}$ can be
determined from the two boundary conditions [Eq. (19) and Eq. (20)] and we
obtain

$\displaystyle T(0)$ $\displaystyle=\frac{T_{\rm
h}\left[1+\kappa/(h_{d}d)\right]+T_{\rm
c}\kappa/(h_{0}d)}{1+\kappa/(h_{d}d)+\kappa/(h_{0}d)},$ (21) $\displaystyle
T(d)$ $\displaystyle=\frac{T_{\rm c}\kappa/(h_{d}d)+T_{\rm
c}\left[1+\kappa/(h_{0}d)\right]}{1+\kappa/(h_{d}d)+\kappa/(h_{0}d)}.$ (22)

we find Eq. (3) from Eqs. (21) and (22).

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