New approach to the Sivers effect in the collinear twist-3 formalism

TL;DR

This paper introduces a new non-pole calculation method for the Sivers effect within the collinear twist-3 formalism, demonstrating its consistency with traditional pole methods and highlighting its technical advantages.

Contribution

It extends the non-pole technique to the Sivers effect and clarifies its consistency and advantages over the conventional pole calculation method.

Findings

The non-pole method is consistent with pole calculations at $\\mathcal{O}(\alpha_s)$.

The pole calculation implicitly uses equations of motion and Lorentz invariance.

The non-pole method offers technical advantages in evaluating the Sivers effect.

Abstract

The single-transverse spin asymmetry(SSA) for hadron production in the transversely polarized proton scattering receives major contribution from Sivers effect, which can be systematically described within the collinear twist-3 factorization framework in various processes. Conventional method in the evaluation of the Sivers effect known as pole calculation is technically quite different from non-pole calculation which is another method used in evaluating the final state twist-3 effect. In this paper, we extend the non-pole technique to the Sivers effect, and show the consistency with the conventional method through an explicit calculation of $\mathcal{O}(\alpha_s)$ correction in semi-inclusive deep inelastic scattering. As a result, we clarify that the conventional pole calculation is implicitly using the equation of motion and the Lorentz invariant relations whose importance became widely known in the non-pole calculation. We also clarify some technical advantages in using the new non-pole method.

Equations

$$d\sigma =i{T}_{q,F}\otimes D\otimes d\hat{\sigma },$$

$$e(l)+p^{\uparrow }(p,S_{\perp })\to e(l^{\prime })+h(P_h)+X,$$

$$\frac{d^4\Delta \sigma }{d{x}_{B}dyd{z}_{h}d{P}_{h\perp }}=\frac{{\alpha }_{em}^2}{32{\pi }^2{z}_h{x}_B^2{S}_{ep}^2{Q}^2}{L}^{\mu \nu }{W}_{\mu \nu },$$

$$S_{ep}=(p+l{)}^2,x_B=\frac{Q^2}{2p\cdot q},z_h=\frac{p\cdot P_h}{p\cdot q},y=\frac{p\cdot q}{p\cdot l}.$$

$$L^{\mu \nu }=2\left(l^{\mu }{l}^{\prime \nu }+l^{\nu }{l}^{\prime \mu }-\frac{Q^2}{2}{g}^{\mu \nu }\right).$$

$$\frac{d^4\Delta \sigma }{d{x}_{B}dyd{z}_{h}d{P}_{h\perp }}=\frac{{\alpha }_{em}^2}{32{\pi }^2{z}_h{x}_B^2{S}_{ep}^2{Q}^2}\sum _i\int \frac{dz}{z^2}{W}_i{D}_{i\to h}(z),$$

$$W_{\mathrm{pole}}=\int \frac{d^4{k}_1}{(2\pi {)}^4}\int \frac{d^4{k}_2}{(2\pi {)}^4}\int d^4{y}_1\int d^4{y}_2{e}^{i{k}_1\cdot y_1}{e}^{i(k_2-k_1)\cdot y_2}⟨p{S}_{\perp }|{\bar{\psi }}_j(0)g{A}^{\alpha }(y_2){\psi }_i(y_1)|p{S}_{\perp }⟩H_{ji,\alpha }^{\mathrm{pole}}(k_1,k_2).$$

$$\frac{1}{k^2+iϵ}=P\left(\frac{1}{k^2}\right)-i\pi \delta (k^2),$$

$$H_{ji,\alpha }^{\mathrm{pole}}(k_1,k_2)$$

$$(k_2-k_1{)}^{\alpha }{H}_{ji,\alpha }^{\mathrm{pole}}(k_1,k_2)=0.$$

$$\displaystyle(x_{2}-x_{1}){\partial\over\partial k_{1}^{\beta}}H^{{\rm pole}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}$$

$$\displaystyle(x_{2}-x_{1}){\partial\over\partial k_{2}^{\beta}}H^{{\rm pole}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}$$

$$\displaystyle{\partial\over\partial k_{1}^{\beta}}H^{{\rm SFP(HP)}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}$$

$$\displaystyle{\partial\over\partial k_{2}^{\beta}}H^{{\rm SFP(HP)}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}$$

$$\displaystyle-{\partial\over\partial k_{1}^{\beta}}H^{{\rm SFP(HP)}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}={\partial\over\partial k_{2}^{\beta}}H^{{\rm SFP(HP)}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}.$$

$$\displaystyle-{\partial\over\partial k_{1}^{\beta}}H^{{\rm SGP}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}={\partial\over\partial k_{2}^{\beta}}H^{{\rm SGP}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}.$$

$$H_{ji,\rho }^{\mathrm{pole}}(k_1,k_2)$$

$$A^{\alpha }\simeq \frac{A^n}{p^{+}}{p}^{\alpha }+A_{\perp }^{\alpha },$$

$$W_{\mathrm{pole}}$$

$$\displaystyle W_{\rm pole}=i\int{dx_{1}}\int{dx_{2}}{\rm Tr}\left[M_{F}^{\alpha}(x_{1},x_{2}){\partial\over\partial k_{2}^{\alpha}}H^{{\rm pole}}_{p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}\right],$$

$$M_{ij,F}^{\alpha }(x_1,x_2)$$

$$\displaystyle T_{q,F}(x_{1},x_{2})=\Bigl{(}{g\over 2\pi M_{N}}\Bigr{)}\int{dy_{1}^{-}dy_{2}^{-}\over 4\pi}\,e^{ix_{1}p^{+}y_{1}^{-}}e^{i(x_{2}-x_{1})p^{+}y_{2}^{-}}\langle pS_{\perp}|\bar{\psi}(0){\mathchoice{\c@ncel\displaystyle{n}}{\c@ncel\textstyle{n}}{\c@ncel\scriptstyle{n}}{\c@ncel\scriptscriptstyle{n}}}\epsilon^{\alpha\bar{n}nS_{\perp}}F_{\alpha}^{\ n}(y_{2}^{-})\psi(y_{1}^{-})|pS_{\perp}\rangle.$$

$$\displaystyle{\partial\over\partial k_{2}^{\alpha}}H^{{\rm SGP}}_{ji,p}(k_{1},k_{2})\Bigl{|}_{k_{i}=x_{i}p}={1\over 2NC_{F}}\Bigl{[}i\pi\delta(x_{2}-x_{1})\Bigr{]}\left({\partial\over\partial p_{c}^{\alpha}}-{p_{c\alpha}p^{\mu}\over p_{c}\cdot p}{\partial\over\partial p_{c}^{\mu}}\right)H_{ji}(x_{1}p),\hskip 14.22636pt$$

$$\frac{d^4\Delta \sigma }{d{x}_{B}dyd{z}_{h}d{P}_{h\perp }}$$

$${\hat{\sigma }}_D$$

$${\hat{\sigma }}_{ND}$$

$${\hat{\sigma }}_{SFP}$$

$${\hat{\sigma }}_{HP}$$

$$\hat{s}$$

$$W^{(0)}=\int \frac{d^{4}k}{(2\pi {)}^4}\int d^4{y}_1{e}^{ik\cdot y_1}⟨p{S}_{\perp }|{\bar{\psi }}_j(0){\psi }_i(y_1)|p{S}_{\perp }⟩H_{ji}(k).$$

$$\displaystyle H_{ji}(k)\simeq H_{ji}(xp)+{\partial\over\partial k^{\alpha}}H_{ji}(k)\Bigl{|}_{k=xp}k_{\perp}^{\alpha}.$$

$$W^{(0)}$$

$$W^{(1)}=\int \frac{d^4{k}_1}{(2\pi {)}^4}\int \frac{d^4{k}_2}{(2\pi {)}^4}\int d^4{y}_1\int d^4{y}_2{e}^{i{k}_1\cdot y_1}{e}^{i(k_2-k_1)\cdot y_2}⟨p{S}_{\perp }|{\bar{\psi }}_j(0)g{A}^{\alpha }(y_2){\psi }_i(y_1)|p{S}_{\perp }⟩H_{ji,\alpha }(k_1,k_2).$$

Full text

I Introduction

The origin of the single transverse-spin asymmetries(SSAs) in high-energy hadron scatterings has been a long-standing mystery over 40 years since the strikingly large asymmetries were observed in mid-1970s Klem:1976ui ; Bunce:1976yb . RHIC experiment has provided many data of the SSAs for various hadron productions in the last decade [3-7] and motivated a lot of theoretical work on the development of the perturbative QCD framework. Much theoretical effort has been devoted to develop a reliable QCD-based theory in order to deal with the given experimental data. The twist-3 framework in the collinear factorization approach was established as a rigorous framework which can provide a systematic description of the large SSAs.

It is commonly known that there are two major effects which lead to the large SSAs observed in the experiment, i.e., initial state Sivers effect and final state Collins effect. The Sivers effect is essentially twist-3 contribution generated from a transversely polarized hadron in the initial state. Started from the pioneering work by Efremov and Teryaev Efremov:1981sh , more systematic techniques were developed in a series of studies around ’00 [9-12]. A solid theoretical foundation for the calculation of the Sivers effect was finally laid in Ref. Eguchi:2006mc . We will show the calculation technique in detail in next section and here we just give a brief introduction. The Sivers effect of the transversely polarized proton can be expressed by the dynamical twist-3 function defined by a Fourier transformed proton matrix element $T_{q,F}\simeq{\cal F.T.}\langle pS_{\perp}|\bar{\psi}gF^{+-}\psi|pS_{\perp}\rangle$ and the cross section in deep inelastic scattering (DIS) can be derived as

[TABLE]

where $D$ represents the usual twist-2 fragmentation function and $d\hat{\sigma}$ is a hard partonic cross section. Because all the nonperturbative functions are real in this equation, the partonic cross section has to give an imaginary contribution in order to cancel $i$ in the coefficient. This imaginary contribution can be given by the pole part of a propagator in the partonic scattering. In the quantum field theory, the propagator is defined by the time-ordered product of two fields and it has $i\epsilon$-term in the denominator. The imaginary contribution can emerge from a residue of contour integration. This is a basic mechanism of the pole calculation for the Sivers type contribution. Next we turn to the twist-3 fragmentation effect of spin-0 hadron which is known as Collins effect. The cross section formula for the twist-3 fragmentation contribution was completed in $pp$ collision Metz:2012ct and DIS  Kanazawa:2013uia in a formal way. The dynamical twist-3 fragmentation function can be defined as $\hat{D}_{q,F}\simeq{\cal F.T.}\langle 0|gF^{+-}\psi|hX\rangle\langle hX|\bar{\psi}|0\rangle$ and the cross section in DIS is expressed by the same form as Eq. (1) just replacing $T_{q,F}$ with $\hat{D}_{q,F}$ and $D$ with the usual twist-2 parton distribution function respectively. The main difference is that the fragmentation function $\hat{D}_{q,F}$ is complex and therefore it gives the imaginary contribution. The pole contribution from the hard cross section is no longer needed. This is a mechanism of the non-pole contribution from the Collins effect. The origin of generating the imaginary phase results in the main technical difference between the calculations for the Sivers and the Collins effects. The cross section for the pole contribution only depends on the dynamical function, while the result for the non-pole contribution is expressed in terms of three types of nonperturbative functions, the dynamical, intrinsic and kinematical functions. In general, the hard cross sections in the non-pole calculation are not gauge- and Lorentz-invariant, because they are not physical observables, only their sum leads to physical result as measured by experiment. This problem is solved by using two types of the relations among the nonperturbative functions which are called equation of motion relation and Lorentz invariant relation Kanazawa:2015ajw .

As discussed above, the calculations for the pole contribution and the non-pole contribution are technically different from each other. Although the calculation techniques for those contributions are important basics of the higher twist calculation, not so many theorists are familiar with both of them because of the technical differences. In this paper, we revisit the result of the pole calculation from the viewpoint of the non-pole calculation in order to understand two calculations in a unified way. We show that the non-pole calculation has several technical advantages, thus should be extended to more complicated calculations like next-to-leading order calculation and twist-4 calculation.

The remainder of the paper is organized as follows: in Sec. II we introduce the notation and review the conventional pole calculation in detail. In Sec. III we show the non-pole calculation method for the twist-3 contribution in order to reexamine the pole contributions. Finally, in Sec. IV we summarize the achievements in this paper and make some comments on possible applications of the new non-pole method.

II Conventional pole calculation at twist-3

The conventional collinear expansion framework at twist-3 has been developed in Refs. [8-17]. We review here the pole calculation for semi-inclusive deep inelastic scattering (SIDIS) in order to clarify the difference from the new method of non-pole calculation that we will propose in next section. SIDIS is a suitable process to check the consistency between the two methods because the twist-3 cross section has been already completed in Ref. Eguchi:2006mc based on the conventional method. In addition, SIDIS is a relatively easier process than $pp$-collision because of some technical issue.

We consider the process of polarized SIDIS

[TABLE]

where the initial proton is transversely polarized. $l$ and $l^{\prime}$ are, respectively, the momenta of the incoming and outgoing electrons. $p$ and $S_{\perp}$ are the momentum and the transverse spin of the beam proton, $P_{h}$ is the momentum of the final state hadron. In this paper, we focus on one-photon exchange process with the photon invariant mass $q^{2}=(l-l^{\prime})^{2}=-Q^{2}$, the extension to charged current interaction is straightforward. The polarized cross section for SIDIS is given by

[TABLE]

where the standard Lorentz invariant variables in SIDIS are defined as

[TABLE]

The leptonic tensor is defined as follows:

[TABLE]

In order to simplify the discussion, we will mainly consider the metric part $L^{\mu\nu}\to-Q^{2}g^{\mu\nu}$. We will discuss the result with the full leptonic tensor (5) in the end of Sec. III. The SSA in SIDIS can be generated by both the initial state and final state twist-3 contributions. In this paper, we focus on the contribution from initial state twist-3 distribution functions of the transversely polarized proton, then the polarized differential cross section can be written as

[TABLE]

where $D_{i\to h}(z)$ is the twist-2 unpolarized fragmentation function. The hadronic part $W_{i}$ describes a scattering of the virtual photon on the transversely polarized proton, with the leptonic metric part contracted. We will make the subscript $i$ implicit in the rest part of this paper for simplicity.

In the conventional pole calculation, one needs to consider diagrams as shown in Fig. 1, in which the hadronic part reads

[TABLE]

The twist-3 contribution is generated by the pole term, which comes from the imaginary part of the quark propagator

[TABLE]

thus only $-i\pi\delta(k^{2})$ is considered in the conventional pole method. We perform collinear expansion $k_{i}\to x_{i}p$ for the hard part $H^{{\rm pole}}_{ji,\alpha}(k_{1},k_{2})$. There are three types of pole contributions at the leading-order with respect to QCD coupling constant $\alpha_{s}$. : soft-gluon-pole(SGP, $x_{2}-x_{1}=0$), soft-fermion-pole(SFP, $x_{1}=0$ or $x_{2}=0$, $x_{1}\neq x_{2}$) and hard-pole(HP, $x_{1}=x_{B},~{}x_{2}\neq x_{B}$ or $x_{2}=x_{B},x_{1}\neq x_{B}$) contributions. Full diagrams for each pole contribution are shown in Fig. 2a-2c.

It’s known that there is other type of contribution given by diagrams with two quark lines in the same side of the cut Yoshida:2016tfh . This contribution is relatively easier to calculate, thus will not be discussed in this paper.

One can factor out the $\delta$-functions in $H^{{\rm pole}}_{ji,\alpha}(k_{1},k_{2})$ for the three pole contributions

[TABLE]

where the factor (2\pi)\delta\Bigl{[}(k_{2}+q-{p_{c}})^{2}\Bigr{]} representing the on-shell condition of the unobserved parton, $p_{c}$ is the four-momentum of the final state fragmenting parton.

A systematic way to calculate the pole contributions was developed in Eguchi:2006mc . We confirmed that Ward-Takahashi identity(WTI) shown in Eguchi:2006mc is valid for the diagrams in Fig. 2a-2c as

[TABLE]

Considering $k_{1}$ and $k_{2}$ derivatives, we can derive relations,

[TABLE]

where $H^{{\rm pole}}_{ji,p}(k_{1},k_{2})=p^{\beta}H^{{\rm pole}}_{ji,\beta}(k_{1},k_{2})$. Thus we can derive the following useful relations for SFP and HP

[TABLE]

which lead to

[TABLE]

However, we cannot derive the same relation with Eqs. (12,13) for the SGP diagrams directly from WTI because $H^{{\rm SGP}}_{Lji,\alpha}(x_{1}p,x_{2}p)$ contains $\delta(x_{2}-x_{1})$. So far, the only way to derive the relations is to calculate all the relevant diagrams explicitly, which is annoying in higher order perturbative QCD calculations. In SIDIS at $\mathcal{O}(\alpha_{s})$, the authors of Ref. Eguchi:2006mc have checked explicitly that the above relation also holds true for $H^{{\rm SGP}}_{ji,p}$,

[TABLE]

Now we can perform collinear expansion of the hard part

[TABLE]

where the projection tensor is defined as $\omega^{\alpha}_{\ \beta}=g^{\alpha}_{\ \beta}-\bar{n}^{\alpha}n_{\beta}$ with the unit vectors taken as $\bar{n}=[1,0,0],~{}n=[0,1,0]$. We work in the hadron frame and $p^{\mu}=p^{+}\bar{n}^{\mu}$. We can neglect $k^{-}$-component at the twist-3 accuracy and identify $\omega^{\alpha}_{\ \beta}k_{i}^{\beta}\simeq k^{\alpha}_{i\perp}$. We use the same identification for all vectors projected by $\omega^{\alpha}_{\ \beta}$ below. The next step is to decompose the components of the gluon field $A^{\alpha}$ into longitudinal and transverse as

[TABLE]

where $A^{n}=A\cdot n$ and we neglected $A^{-}$-component. Substitute Eqs. (15,16) into Eq. (7), we can extract the twist-3 contribution

[TABLE]

The last term in Eq. (17) can be eliminated by using the relation Eq. (11), and the first term can be rewritten as

[TABLE]

where $M_{F}^{\alpha}(x_{1},x_{2})$ is the F-type dynamical function which can be further expanded as

[TABLE]

with the nucleon mass $M_{N}$ and the field strength tensor defined as $F^{\alpha n}(y_{2}^{-})=\partial_{\perp}^{\alpha}A^{n}(y_{2}^{-})-\partial^{n}A_{\perp}^{\alpha}(y_{2}^{-})$, notice that the nonlinear gluon term in the field strength tensor has been omitted because it comes from Feynman diagrams with linked gluons more than one, therefore does not show in Eq. (18). $T_{q,F}(x_{1},x_{2})$ is the well-known Qiu-Sterman function, defined as following111We rescaled the function as $T_{q,F}(x_{1},x_{2})\to(g/2\pi M_{N})T_{q,F}(x_{1},x_{2})$ from the original definition in Ref. Qiu:1998ia for convenience. Our definition of $T_{q,F}(x_{1},x_{2})$ is the same with $F^{q}_{FT}(x_{2},x_{1})$ in Ref. Kanazawa:2015ajw .

[TABLE]

Using Eq. (12) for SFP and HP, one can evaluate the derivative of the hard part with respect to $k_{2}$ in Eq. (18). For SGP, we need to rely on the master formula Koike:2006qv

[TABLE]

where $H_{ji}(x_{1}p)$ is the usual $2\to 2$ $\gamma^{*}q\to qg$ scattering cross section without the extra gluon line attached. Combining the three pole contributions together, we obtain the final result based on the conventional pole calculation

[TABLE]

where all hard cross sections are listed below,

[TABLE]

with the standard Mandelstam variables defined as

[TABLE]

In the next section, we show that the new non-pole method can reproduce these hard cross sections. We would like to make a comment on the relation Eq. (14) for the SGP diagrams, this relation is required to construct the gauge-invariant matrix for the dynamical function. However, there is no simple way to prove this relation. We have to check if it’s correct diagram by diagram. This is a frustrating point of the conventional pole calculation. We will show that the new method can avoid such complexity and thus a more flexible calculation technique.

III The new non-pole calculation for Sivers effect

We introduce the new non-pole calculation method in this section. The main difference between the pole and the non-pole methods is on the decomposition of the propagator shown in Eq. (8). In the new method, we directly perform the contour integrations and never carry out the decomposition for any propagators. The new method is expected to remove the mathematical complexity lies in the validity of the decomposition. In this sense, the new method can be regarded as a more flexible approach, and can be easily extended to more complicated cases. Removal of the workload on Eq. (14) is one of important consequences.

III.1 General formalism

In the new method we propose here, the hadronic part should be written as a sum of all the diagrams, i.e., $W=\sum_{i}W^{(i)}$, where $i$ denotes the number of gluon attachment. Let’s start with the diagram in Fig. 3 without any gluon attached, the hadronic part is given by

[TABLE]

The twist-3 contribution from the diagrams without gluon attached can be obtained by performing collinear expansion of the hard part

[TABLE]

Then Eq. (25) can be expanded into two parts

[TABLE]

In general, the first term could give the twist-3 contribution when the hard part gives transverse component $H_{ji}(xp)\sim(\gamma^{\perp})_{ji}$. Next we consider the diagrams with one gluon attachment as shown in Fig. 1 which were also considered in the conventional method. Here we need to consider a set of diagrams $H_{ji,\alpha}(k_{1},k_{2})$ shown in Fig. 4 and their complex conjugate.

We call them non-pole diagrams because we don’t separate the pole term for any propagators. The non-pole contribution to the hadronic part reads

[TABLE]

Similar to the strategy in dealing with the diagrams without gluon attachment, the first step to extract the twist-3 contribution from one gluon attached diagrams is to perform collinear expansion of the hard part

[TABLE]

One also need to decompose the gluon field $A^{\alpha}$ into longitudinal and transverse components as in Eq. (16). Then Eq.(28) can be expanded as follows:

[TABLE]

notice that other terms in the combination of Eqs. (16) and (29) contribute to higher twist. The hard part shown in above equation can be further simplified by using the WTI relations. It is straightforward to derive the counterpart of Eq. (10) for the non-pole diagrams Kanazawa:2013uia

[TABLE]

The non-pole hard part doesn’t have the delta function $\delta(x_{2}-x_{1})$, therefore, we can derive the following useful relations.

[TABLE]

where the sign of $i\epsilon$ was determined by the fact that only the final state interaction exists in SIDIS. If a process has both the initial and the final interactions as in the case of $pp$-collision, the r.h.s of Eq. (32) could have both $\pm i\epsilon$ and the sign can’t be uniquely determined from the WTI (31). We need more consideration on this point when we apply the same technique to $pp$-collision.

We would like to emphasize the validity of WTI to higher order diagrams. The WTI is a consequence of the gauge invariance in QCD and therefore we can use the same Eqs. (31,32) to higher order diagrams as long as we use an appropriate regularization scheme like the dimensional regularization. By using these useful relations derived from WTI, the hard part terms $H_{ji,\alpha}(k_{1},k_{2})$ and $H_{ji,p}(k_{1},k_{2})$ contained in Eq. (30) are respectively given by

[TABLE]

We iteratively used the third relation in Eq. (32) for the first equation. Substituting Eq. (33) into Eq. (30), we obtain the final result

[TABLE]

Summing over all twist-3 contributions in the diagrams in Figs. 1 and 3, represented by Eqs. (27, 34), respectively, we can construct the gauge-invariant expression

[TABLE]

where the matrices are given by

[TABLE]

where operator definition of $f_{1T}^{\perp(1)}(x)$ is

[TABLE]

The definition of $M_{F}^{\alpha}(x_{1},x_{2})$ and its decomposition are already introduced in Eq. (19). In the present case, the first term in Eq. (35) can’t give a twist-3 contribution because the spin projection $\gamma_{\alpha}\epsilon^{\alpha\bar{n}nS_{\perp}}$ is forbidden by $PT$-invariance. Therefore, we can eliminate the first term in Eq. (35) and rewrite the twist-3 hadronic part as

[TABLE]

In the new method presented above, we only needed the well-defined relations Eq. (32) to construct the gauge-invariant matrix elements. We find that the difficulty associated with the relation Eq. (14) in the conventional calculation was removed. This is one of the advantages in the new method. Another advantage is that, by using Eq. (39) and the discussion in Appendix. B, we don’t need to calculate the derivative of the hard part over the momentum $k(k_{i})$, this will significantly reduce the complexity of twist-3 calculation, in particularly for higher order calculations.

III.2 SIDIS at $\mathcal{O}(\alpha_{S})$

In this subsection, we show in detail the calculation of hadronic part for SIDIS at $\mathcal{O}(\alpha_{s})$. We factor out the on-shell $\delta$-function from the hard partonic part,

[TABLE]

where $\bar{H}^{L}_{\alpha}(x_{1}p,x_{2}p)$ is given by a sum of 12 diagrams in Fig. 4 and $\bar{H}^{R}_{\alpha}(x_{1}p,x_{2}p)$ is its complex conjugate. The derivative of $H(k)$ over $k$ can be converted to that over the standard Mandelstam variables $\hat{s},\hat{t},\hat{u}$. For details, see Appendix B. Then we can calculate Eq. (39) as

[TABLE]

where $\hat{\sigma}(\hat{s},\hat{t},\hat{u})$ is the $2\to 2$ partonic cross section in SIDIS. For $q\gamma^{*}\to qg$ channel, it reads

[TABLE]

Notice that, for convenience, we have changed the notation $x_{1}\to x^{\prime},~{}x_{2}\to x$ in $\bar{H}^{L}(x^{\prime}p,xp)$, and $x_{1}\to x,~{}x_{2}\to x^{\prime}$ in $\bar{H}^{R}(x^{\prime}p,xp)$ in order to factor out the common delta function \delta\Bigl{[}(xp+q-p_{c})^{2}\Bigr{]}. We discuss the gauge- and Lorentz-invariances of the hard cross sections associated with $f_{1T}^{\perp(1)}(x)$. The hard cross section with the nonderivative function $f_{1T}^{\perp(1)}(x)$ is not apparently gauge-invariant because of the term $\epsilon^{\alpha pnS_{\perp}}{\rm Tr}[\gamma_{\alpha}\bar{H}(xp)]$. The gauge-invariance requires the unpolarized spin projection $x{\mathchoice{\c@ncel\displaystyle{p}}{\c@ncel\textstyle{p}}{\c@ncel\scriptstyle{p}}{\c@ncel\scriptscriptstyle{p}}}$ with $\bar{H}(xp)$ like Eq. (43). On the other hand, the hard cross section associated with the derivative function ${d\over dx}f_{1T}^{\perp(1)}(x)$ is not Lorentz-invariant. The vector $n$ in the parametrization (19) satisfies $\bar{n}\cdot n=1$ and $n^{2}=0$. These conditions are not enough to uniquely determine the form of $n$ and there are two possible choices in SIDIS,

[TABLE]

We can check that the coefficient $(\epsilon^{q\bar{n}nS_{\perp}}-\epsilon^{p_{c}\bar{n}nS_{\perp}})$ of ${d\over dx}f_{1T}^{\perp(1)}(x)$ depends on the choice of $n$. This ambiguity of the cross section is physically interpreted as the frame-dependence because the spatial components of $n$ is determined so that it has the opposite direction of the momentum $p$ as $\vec{n}=-\vec{p}/p^{+}$. From the requirement of the frame-independence, the cross section has to be proportional to the factor $\left[(\hat{s}+Q^{2})\epsilon^{p_{c}\bar{n}nS_{\perp}}+\hat{t}\epsilon^{q\bar{n}nS_{\perp}}\right]$ as already shown in the cross section (22) derived by the conventional pole method. We will show later that the gauge- and Lorenz-invariances of the cross section are guaranteed by using the relations

[TABLE]

which enable us to express the cross section only in terms of $T_{q,F}(x^{\prime},x)$ as in the case of the conventional calculation. One can find the derivation of these relations in Appendix A.

Now we show how to calculate the hard partonic part $\bar{H}^{L}(x^{\prime}p,xp)$. There are four types of $x^{\prime}$-dependences in Feynman gauge. Fig. 5 shows typical diagrams including $x^{\prime}$-dependent propagators. Each propagator can be calculated as follows:

[TABLE]

where $V_{\alpha\rho\tau}$ comes from the 3-gluon vertex. We can find that all $x^{\prime}$-dependence appear only in the denominators, $x-x^{\prime}-i\epsilon$, $x^{\prime}-i\epsilon$ and $x^{\prime}-x_{B}+i\epsilon$. Products of two denominators can be disentangled as

[TABLE]

From the above discussion, we can conclude that the part of the cross section with $\bar{H}^{L}_{\alpha}(x^{\prime}p,xp)$ is given by

[TABLE]

All the hard parts $H_{Fi}$ are independent of $x^{\prime}$. We can repeat the same discussion on $\bar{H}^{R}_{\alpha}(xp,x^{\prime}p)$. Then we can calculate each hard partonic cross section and obtain the following result for the hadronic part.

[TABLE]

where the hard cross sections are given by

[TABLE]

$\hat{\sigma}_{ND},\hat{\sigma}_{D},\hat{\sigma}_{SFP},~{}\hat{\sigma}_{HP}$ can be found in Eq. (23). Since $H_{Fi}$ are all independent of $x^{\prime}$, the $x^{\prime}$-integration only involves $T_{q,F}(x^{\prime},x)$ and the propagators. Then we can perform $x^{\prime}$-integration in Eq. (50) as

[TABLE]

where we have used the symmetric property of Qiu-Sterman function $T_{q,F}(x^{\prime},x)=T_{q,F}(x,x^{\prime})$ in the integration of the double pole coefficient. Substituting these relations into Eq. (50) and using Eqs. (45,46), we can finally derive the transverse polarized cross section in SIDIS based on the new method as

[TABLE]

This is exactly the same with the result of the conventional calculation (22). We would like to emphasize that the cross section is never gauge- and Lorentz-invariant if the kinematical function $f_{1T}^{\perp(1)}(x)$ and Qiu-Sterman function $T_{q,F}(x,x)$ are independent of each other. The relation between them is needed for the physically acceptable result.

In the end, we make a comment on the generality of our result. We only considered the metric part $L^{\mu\nu}\simeq-Q^{2}g^{\mu\nu}$ in our calculation so that one can easily follow the calculation and clearly see the difference between two calculation methods. It’s a natural question whether the consistency holds when we consider the full leptonic tensor shown in Eq. (5). The conventional way to calculate the cross section in SIDIS is that we expand the hadronic tensor in terms of orthogonal bases. The symmetric part of the tensor $W^{\mu\nu}$ has 10 independent components and one of them is fixed by the condition $q_{\mu}W^{\mu\nu}=0$. Then $W^{\mu\nu}$ can be expanded by 9 independent bases as

[TABLE]

One can find the explicit forms of ${\cal V}_{i\,\rho\sigma}$ and $\tilde{{\cal V}}^{\mu\nu}_{i}$ in Ref. Meng:1991da . Then the contracted form with $L^{\mu\nu}$ is rewritten as

[TABLE]

This equation means that the calculation with the full leptonic tensor $L^{\mu\nu}$ results in the calculation of the hard cross sections $W^{\rho\sigma}\tilde{{\cal V}}_{i\,\rho\sigma}$. Three tensors $\tilde{{\cal V}}^{\mu\nu}_{5,6,7}$ are irrelevant to our study because they are pure imaginary. We verified that the consistency between the two methods holds for all 6 hard cross sections ($i=1,2,3,4,8,9$). This result shows that the consistency holds for the full leptonic tensor and enhances the generality of our result.

IV Summary

We proposed the new nonpole calculation method for the Sivers effect in the twist-3 cross section and confirmed the consistency with the conventional pole calculation. We found out that the relation $f_{1T}^{\perp(1)}(x)=\pi T_{q,F}(x,x)$ is very important to guarantee the gauge- and Lorentz-invariances of the final result. We reproduced this relation without introducing the definition of the TMD Sivers function. The importance of Eq. (45) has been mainly discussed in the context of the matching between the TMD factorization and the collinear twist-3 factorization frameworksScimemi:2018mmi ; Scimemi:2019gge . Our calculation showed that this was also important for the gauge- and Lorentz-invariances of the twist-3 physical observables for the Sivers effect. This result provides a new perspective on the relation. The same technique can be also applied to the gluon Sivers function and the twist-3 gluon distribution functions Koike:2011mb . The relation between them is relatively nontrivial compared to the quark functions. From the requirement of the gauge- and Lorentz-invariances of the twist-3 cross section, we can derive a similar relation with Eq. (45) for the gluon distribution functions.

One of the advantages in the new non-pole calculation method is that we don’t need to prove Eq. (14) for the SGP contribution as required in conventional pole method, which can be only checked through diagram by diagram calculation. It’s known that this relation may not be hold when the description of the fragmentation part is changed to other framework such as NRQCD for heavy quarkonium production. In the new method, we never separate the pole contributions and then no singularity arises from the relation associated with WTI. Our new method will extend the applicability of the collinear twist-3 framework.

In the new method, one does not need to perform derivatives over the initial parton’s transverse momentum in the calculation of Feynman diagrams. We can anticipate that a lot of propagators depend on the initial parton’s momentum in higher order diagrams. The direct operation of the derivatives is highly complicated task. Our method could significantly reduce this complexity as discussed in Sec. III. As mentioned just below Eq. (32), the WTI doesn’t change for the higher order diagrams as long as the gauge invariance is preserved. Most of our results are available without change for the higher order cross section in SIDIS. A set of equations derived in this paper could be useful to derive the first next-leading order cross section for the SSA in $ep$-collision which could be measured at Electron-Ion-Collider in the near future.

We expect the new method presented in this manuscript can be extended to higher-twist calculation, which becomes one of the standard method to investigate the nontrivial nuclear effect in heavy ion collisions Kang:2008us ; Kang:2011bp ; Xing:2012ii ; Kang:2013ufa ; Kang:2014hha . As we don’t need to perform derivatives over the initial parton’s transverse momentum in the new non-pole method, we expect the new approach will be of great use in performing next-to-leading order calculation at higher twist, in which the conventional collinear expansion caused ambiguity in setting up the initial parton’s kinematics Kang:2014ela ; Kang:2016ron , this ambiguity can be resolved in the new non-pole method.

Appendix A Twist-3 quark-gluon correlation functions

A.1 Definition of the twist-3 functions

We introduce the definition of all relevant twist-3 functions for the transversely polarized proton Kanazawa:2015ajw ; Eguchi:2006qz .

D-type dynamical function

[TABLE]

where $D_{\perp}^{\alpha}(y_{2}^{-})=\partial_{\perp}^{\alpha}-igA_{\perp}^{\alpha}(y_{2}^{-})$, $[0,y_{2}^{-}]$ is the Wilson line

[TABLE]

The D-type function $T_{q,D}(x_{1},x_{2})$ is real and antisymmetric $T_{q,D}(x_{1},x_{2})=-T_{q,D}(x_{2},x_{1})$.

Kinematical function

[TABLE]

By using the translation invariance Kanazawa:2015ajw ,

[TABLE]

we can show $M_{\partial}^{*}(x)=-M_{\partial}(x)$ and therefore $f_{1T}^{\perp(1)}(x)$ is real function. The kinematical function $f_{1T}^{\perp(1)}(x)$ has another definition using the quark TMD correlator. Here we recall the definition of the quark Sivers function  Bacchetta:2006tn ,

[TABLE]

We can find a relation between the first moment of $M(x,p_{T})$ and the correlator of the kinematical function $M^{\alpha}_{\partial}(x)$,

[TABLE]

Then $f_{1T}^{\perp(1)}(x)$ can be expressed by the first moment of the quark Sivers function Boer:2003cm ; Kang:2011hk .

[TABLE]

The matching between TMD functions and collinear functions itself is an active research subject in perturbative QCD phenomenology. One can find recent developments in Refs. Scimemi:2018mmi ; Scimemi:2019gge and references therein.

F-type dynamical function

[TABLE]

where the F-type function $T_{q,F}(x_{1},x_{2})$ is real and symmetric $T_{q,F}(x_{1},x_{2})=T_{q,F}(x_{2},x_{1})$.

A.2 Relation among the functions

We can derive an operator identity among the three types of correlators Eguchi:2006qz . In order to derive the relation, we use the identity for the $D_{\perp}^{\alpha}(y_{2}^{-})[y_{2}^{-},y_{1}^{-}]$ in $M^{\alpha}_{D}(x_{1},x_{2})$,

[TABLE]

where we used the step function

[TABLE]

We calculate each term in r.h.s of (64) below.

(1) first term

[TABLE]

(2) second term

[TABLE]

(3) third term

[TABLE]

Combining (66,67,68), we can show $M^{\alpha}_{D}(x_{1},x_{2})={1\over x_{1}-x_{2}+i\epsilon}M^{\alpha}_{F}(x_{1},x_{2})+\delta(x_{2}-x_{1})M^{\alpha}_{\partial}(x_{1})$ and then the relation among the twist-3 functions is given by

[TABLE]

Using the interchange symmetry $x_{1}\leftrightarrow x_{2}$, we can rewrite the above relation as

[TABLE]

From the operator definition (63), one can find that $T_{q,F}(x_{1},x_{2})$ contains the factor $e^{ix_{1}p^{+}(y_{1}^{-}-y_{2}^{-})}$. We can perform $x_{1}$-integration,

[TABLE]

and then

[TABLE]

After the integration of (70) with respect to $x_{1}$, we can derive the relation

[TABLE]

which is nothing but the relation (45). This is well known relation between the first moment of the Sivers function $f_{1T}^{\perp(1)}(x)$ and the Qiu-Sterman function $T_{q,F}(x,x)$ Boer:2003cm ; Ma:2003ut . The same relation can be derived as we performed here in a simple way. One can easily show the relation (46) by the derivative of (73) with respect to $x$.

Appendix B Calculation of the derivative term {\partial\over\partial k^{\alpha}}H(k)\Bigr{|}_{k=xp}

We show how to calculate the hard part {\partial\over\partial k^{\alpha}}H(k)\Bigr{|}_{k=xp} in Eq. (39) without direct operation of the $k$-derivative. We can calculate the part of the kinematical function as

[TABLE]

We focus on the first term in the parenthesis. Because $H(k)$ carries the information about $k$, $q$ and $p_{c}$, it can be written by all possible Lorentz invariant variables,

[TABLE]

where we defined the variables,

[TABLE]

We can set $k^{2}=0$ because {\partial\over\partial k^{\alpha}}k^{2}\Bigr{|}_{k=xp}{\partial\over\partial k^{2}}=2xp^{\alpha}{\partial\over\partial k^{2}} is canceled with $\epsilon^{\alpha\bar{n}nS_{\perp}}$. We find that $\hat{\sigma}(k^{2},\tilde{s},\tilde{t},\hat{u},Q^{2})$ coincides with $\hat{\sigma}(\hat{s},\hat{t},\hat{u})$ in Eq. (43) in the collinear limit $k=xp$. Then the $k$-derivative is converted into $\hat{s}$- and $\hat{t}$- derivatives,

[TABLE]

We calculate the $k$-derivative term in Eq. (74) as

[TABLE]

We can calculate $x$-derivative of $\hat{\sigma}(\hat{s},\hat{t},\hat{u})$ as

[TABLE]

Finally we combine the second term in Eq. (74) and obtain the result in Eq. (42),

[TABLE]

The derivative over the Mandelstam variable can be carried out after the calculation of the diagrams, which is much easier than the direct $k$-derivative of $H(k)$.

Acknowledgments

The authors would like to thank Ivan Vitev and Matthew D. Sievert for fruitful discussion. This research is supported by NSFC of China under Project No. 11435004 and research startup funding at SCNU.