Renormalon Effects in Quasi Parton Distributions

We investigate the renormalon ambiguity from bubble-chain diagrams in the isovector unpolarized quasi-parton distribution function (PDF) of a hadron. We confirm the assertion by Braun, Vladimirov and Zhang (arXiv:1810.00048) that the leading IR renormalon ambiguity is formally an $\mathcal{O}({\Lambda}^2_{QCD}/x^2P_z^2)$ effect, with $x$ the parton momentum fraction and $P_z$ the hadron momentum, together with a new $\mathcal{O}(\delta(x)\Lambda^2_{QCD}/P_z^2)$ contribution such that the quark number is conserved. This implies the convergence of the perturbative matching kernel between a quasi-PDF and a PDF would eventually fail for small $x$. However, in both the R-scheme designed to cancel the leading IR renormalon and the typically used RI/MOM scheme in lattice QCD for the same quasi-PDF, we find good convergence in the kernel based on three-loop bubble-chain diagram analyses. These results are encouraging for the quasi-PDF program. However, firm conclusions can only be drawn after the complete higher loop QCD calculations are carried out.


I. INTRODUCTION
Large momentum effective theory (LaMET) enables computations of parton distributions of hadrons on a Euclidean lattice.LaMET relates equal-time spatial correlators (whose Fourier transforms are called quasi-distributions) to lightcone distributions in the infinite hadron momentum limit [2,3].For large but finite momenta accessible on a realistic lattice, LaMET relates quasi-distributions to physical ones through a factorization theorem, which involves a matching coefficient and power corrections that are suppressed by the hadron momentum.The proof of factorization was developed in Refs.[4][5][6].
The quark PDF in a proton is related to a lightcone correlator where the proton momentum P µ is alone the z-direction, P µ = (P 0 , 0, 0, P z ), and ξ ± = (t ± z)/ √ 2 is the lightcone coordinates.x is the parton momentum fractions relative to the proton.A + is a gauge field and ψ is a quark field.The expression in Eq.( 1) is boost invariant alone the z-direction.The corresponding quasi-PDF is related to an equal time correlator [2,3] LaMET relates the flavor non-singlet quark quasi-PDF and PDF of a hadron through the factorization theorem where Λ and Λ are renormalization parameters and PDF is defined in the infinite momentum frame with P z → ∞ [2,3].The (quasi-)PDF for negative momentum fraction corresponds to the (quasi-)PDF of the anti-particle.In particular, Q(−|y|, Λ) = −Q(|y|, Λ) with Q denoting an antiquark PDF.
Let us first discuss the case that both the quasi-PDF and PDF are defined in cut-off schemes with UV cut-off Λ and Λ, respectively.The factorization theorem is based on a large momentum expansion in powers of 1/P z .The O(1/P 2 z ) correction has a similar convolutional structure to the first term: . The scales follow the hierarchy Λ P z Λ Λ QCD , such that the short distance (compared with 1/Λ) physics is encoded in the matching kernels (or Wilson coefficients) Z and Z n while the long distance physics is encoded in the matrix elements Q and Q n with Q of twist-2 and Q n of twist-n.In this series, the long and short distance physics are strictly separated by the scale Λ.
However, the PDF's extracted from experimental data are typically defined in the modified minimal subtraction (MS) scheme.This is because the PDF's convolute with Wilson coefficients to form observables in high energy experiments, and it is technically much easier to compute the Wilson coefficients to higher loops in MS than a hard cut-off scheme.However, there is no strict separation between long and short distance physics in the MS scheme.This means the Wilson coefficients could still contain non-perturbative physics called renormalons [132][133][134] to cause slow convergence in their perturbative expansions.
Mathematically, the renormalon effect could arise when Borel transform is applied to improve the convergence of a perturbation series.Then an inverse Borel transform is applied after the Borel series is summed.However, sometimes the bad convergence of the series can not be tamed by the Borel transform but appears as ambiguities in the inverse Borel transform.When Q is defined in the MS scheme with renormalization scale µ and the quasi-PDF Q is defined in the regularization-independent momentum subtraction (RI/MOM) scheme [13] with scale µ , Eq.( 3) can be rewritten as Since scale separation is not complete in MS and RI/MOM, low energy non-perturbative physics could still exist in Z, which slows down the convergence of the perturbative computation of Z.It can be shown that by resumming a class of bubble-chain diagrams in Z with the help of Borel transform, there are ambiguities in the inverse Borel transform which behave like non-perturbative functions Q n /P n−2 z with n ≥ 4. Since these ambiguities are associated with different choices of the integration paths in the inverse Borel transform, they are not physical.They should be cancelled by the Q n terms.Therefore all the Q n 's need to be defined consistently.An independent determination of a higher twist Q n is not meaningful unless the treatment of renormalon ambiguity in the lower twist Q n is specified.Furthermore, one can estimate the size of Q n by the size of the renormalon ambiguity in Z.It was through this analysis that Braun, Vladimirov and Zhang asserted that the leading IR renormalon ambiguity was an O(Λ 2 QCD /x 2 P 2 z ) effect [1], which was larger than the previous estimation O(Λ 2 QCD /P 2 z ) based on dimensional analysis by a factor 1/x 2 .Therefore, this result has a big impact to the LaMET program.This motivates us to re-investigate the renormalon problem in quasi-PDF.

II. RENORMALON AMBIGUITY IN QUASI-PDF'S
In this section we first briefly review the Borel transform and the possible ambiguity in its inverse transformation.After the discoveries of renormalons [132][133][134] and further conceptual developments [134][135][136][137][138][139], higher order behaviors of perturbation series in the context of renormalon ambiguities became relevant phenomenolically [140][141][142].(See, e.g., Ref. [143] for a review).We will compute the renormalon ambiguity caused by the bubble-chain diagrams to the quasi-PDF.We will compute the Feynman diagrams in momentum space then compare our result with the coordinate space computation obtained in Ref. [1].
We start with writing the matching kernel of Eq.( 4) as a series expansion in the strong coupling constant α s , where we have only kept the ξ = x/y dependence and dropped the other quantities for brevity.In the MS scheme with renormalization scale µ, α s (µ) = 4π where β 0 ≡ 11 3 N c − 2 3 n f , N c being the number of colors, and n f being the number of quark flavors, and Λ QCD is the strong interaction scale.The series is more convergent after the Borel transform After the Borel series is summed, one can perform an inverse Borel transform back to the original series: If the integrant has poles on the real positive w axis, then the above integral will depend on the path of integration on the complex w plane.This uncertainty is called renormalon ambiguity.In an operator product expansion with the MS scheme, the renormalon ambiguities in the lower order Wilson coefficients will be cancelled by the higher order matrix elements.The existence of the renormalon ambiguity is usually demonstrated in the large n f limit with powers of (α s n f ) counted as O(1) and summed to all orders.Then renormalon ambiguity can arise in the α s expansion of these diagrams.However, it is also well known that QCD is not asymptotically free in the large n f limit.Therefore, even though the finite n f contribution can be formally included by an 1/n f expansion, this expansion is not convergent in QCD.However, if we take the QCD β function as a reference, it is probably reasonable to expect that subleading 1/n f corrections are about the same magnitude as the leading order.If this expectation turns out to be true, then the large n f analysis will still be useful.Therefore, we focus on the leading diagrams in the large n f limit to study the renormalon ambiguity in the matching formula between a quasi-PDF and a PDF defined in Eq. (4).
When (α s n f ) is counted as O(1), the bubble chain diagrams that dress the gluon propagator in the second row of Fig. 1 are all O(1).The gray bubble, which is the gluon vacuum polarization, becomes a fermion loop diagram in the large n f limit.We will add the gluon loop contribution to the gluon vacuum polarization as an 1/n f correction as performed in [1,143].This is formally a subleading effect in the 1/n f expansion, but numerically important in QCD-it changes the sign of the QCD beta function from positive to negative.Therefore, the grey bubble in Fig. 1 is the gluon vacuum polarization at one loop [132,133,143,144] where k is the gluon momentum and the 5/3 factor is associated with the MS scheme.Summing the bubble-chains yields a factor modifying the gluon propagator: which has a form of an inverse Borel transform with the Borel series 4π .We will rewrite it as times a factor which is moved to Eq.( 12).Therefore, to compute the diagrams of Fig. 1, one just have to replace the gluon propagator by a dressed one to obtain the Borel series: where Eq.( 6) is used.Below we list the unpolarized isovector PDF and quasi-PDF results of these diagrams.The Borel series of the bubble-chain diagrams in Fig. 1 yields the quark-level quasi-PDF q(x, ρ) (defined similarly to Eq. ( 2) but with the external hadronic state replaced by a single quark state of momentum p µ ): We have computed f up to an integral of the Feynman parameter y, It is easy to see that for positive and finite ρ, if x is different from 0 or 1 and w is different from 1/2 or −1, then the integrant in the above integral will not diverge.For x = 0, 1, the y integration diverges when w is bigger than a certain value.To see the poles in w, one performs the y integration by assuming w is small enough such that the y integration is finite.Then an analytic continuation of w is performed to look for poles in w.After this exercise, no additional pole at x = 0, 1 is found.The w = 1/2 pole is corresponding to the UV divergence of the Wilson line self-energy diagram, which can be subtracted away by the renormalization procedure, such as the RI/MOM scheme used in this work.Therefore we do not consider it further here.The w = −1 pole is not on the path of integration of the inverse Borel transform and hence does not cause any ambiguity.
Setting the external quark on-shell (ρ → 0), Eq.( 14) for the unphysical region (x > 1 and x < 0) becomes where 2 F 1 is a hypergeometric function.There is no IR renormalon in this region, except the contribution near x = 1 which yields the delta function in Eq. (20).The UV renormalon at w = 1/2 can be subtracted away by the renormalization procedure discussed above and will not be discussed further.
In the physical region 0 < x < 1, IR renormalons exist.They correspond to single poles of f at all positive integer values of w: The leading IR renormalon ambiguity in the inverse Borel transform of q is proportional to the residue of the single pole at w = 1.Eqs.( 15) and ( 16) yield When w → 1, the term (1 − w)/|1 − x| w is zero for x = 1 but undetermined for x = 1.It is in fact a delta function since lim w→1 This is consistent with the result of Ref. [145]: Eq. ( 17) yields lim w→1 and the leading IR renormalon ambiguity in the inverse Borel transform of q is thus given by We stress that the IR renormalon is associated with the IR divergence generated when all the particles in the loop become on-shell simultaneously.This is purely an IR effect.Therefore, there is no dependence on the UV regulator in Eq. ( 21).
After discussing the quasi-PDF, we now turn to the PDF.The quark-level PDF q(x, µ 2 −p 2 ) is defined similarly to Eq. ( 1) but with the external hadronic state replaced by a single quark state of momentum p µ .The Borel series of the bubble-chain diagrams in Fig. 1 for quark-level PDF can be obtained by taking p z to infinity before integrating the spacial loop momenta in other directions.The result is where FIG. 2: The leading renormalon ambiguity δ Qren (x, P z ) in the proton isovector quasi-PDF together with the corresponding PDF, Q(x), and the one-loop quasi-PDF Q(x, P z ), all in the MS scheme.δ Qren (x, P z ) is generated with Eq. ( 24), P z = 1.5 GeV and Q(x) from Ref. [146] (CJ12).Q, which is also shown in Fig. 3, is generated by convoluting Q(x) with the one-loop matching kernel.δ Qren (x, P z ) has a large negative contribution near x = 0 such that the area under the curve is zero.This is a consequence of quark number conservation.The size of δ Qren (x, P z ) is bigger than Q(x, P z ) for almost the entire region of −0.
and where the dimensional regularization parameter is kept because we want to perform a w expansion later and this helps to keep the factor Γ(w + ) regulated.The quark-level PDF does not have any IR renormalon ambiguity as there is no pole in the Borel series in the positive w real axis.This result is expected, since the PDF is corresponding to quasi-PDF in the p z → ∞ limit and the leading IR renormalon, which is is proportional to 1/p 2 z as shown in Eq. ( 21), vanishes in this limit.
We can now study how this leading renormalon ambiguity δ Qren affects the quasi-PDF of a hadron, Q, through the matching kernel of Eq.( 4).The kernel can be converted from Eq.( 21) by replacing the quark momentum p z with yP z , P z being the hadron momentum, and replacing x with x/y, such that where the integration limit |x| in the second line can be replaced by 0 since Q(x) vanishes outside of x = [−1, 1] and we have expanded (ξQ(x/ξ) − Q(x))/(1 − ξ) around ξ = 1 before acting it with δ(1 − ξ) in the fourth line.
The plus-function in Eq. ( 24) ensures quark number conservation, such that dx δ Qren = 0.One can show the first two terms in the last line of Eq. ( 24) cancel after integrating over x, and the last two terms are proportional to d dx (Q(x)/x), which yields vanishing boundary terms after integrating over x.Our Eq. ( 24) is the same as the coordinate space calculation in Ref. [1]. 1n Fig. 2, the leading renormalon ambiguity δ Qren (x, P z ) in the proton isovector quasi-PDF, is shown, together with the corresponding PDF, Q(x), and the one-loop quasi-PDF Q(x, P z ), all in the MS scheme.Numerically, δ Qren (x, P z ) is a significant effect.Its size is bigger than Q(x, P z ) for almost the entire region of −0.3 < ∼ x < ∼ 0.1.In the computation, we have used Q from the proton isovector (i.e. the (u − d) quark combination) PDF extracted by the CTEQ-JLab collaboration (CJ12) [146].Q, which is also shown in Fig. 3, is computed with Q(x) convoluted with the one-loop matching kernel.The renormalon ambiguity δ Qren of Q is computed with Eq. ( 24) with P z = 1.5 GeV and Λ QCD = 0.254 GeV.The areas under Q and Q are both one, the isovector charge, while the area under δ Qren is zero, with a large negative contribution near x = 0.This is the result of fermion number conservation.
The singular behavior near x = 0 requires the introduction of an IR regulator When ¯ → 0 is taken at the end of the calculation, the contribution for |x| < |¯ | becomes δ(x) with a divergent prefactor.This illusive contribution could be easily overlooked in a numerical analysis.But it is critical for the demonstration of quark number conservation, which is a property of the bubble-chain diagrams that we considered.Both δ Qren (x, P z ) and which is consistent with Ref. [1].The quasi-PDF Q(x, P z ), however, has support for all values of x.Hence δ Qren has no 1/(1 − x) enhancement compared to Q. Therefore, Since the renormalon ambiguity discussed above should be cancelled by power corrections, one can rewrite Eq.( 4) as where the first power correction has no 1/(1 − x) factor and the second power correction has a divergent prefactor.

III. REMOVING THE LEADING RENORMALON AMBIGUITY
Although the bubble-chain diagrams employed in the analysis above might not be numerically the dominant contribution in QCD, we still expect that they will generate all structures of the renormalon ambiguity [140].Hence Eq.( 28) should be quite robust and is expected to be free from the shortcoming of the bubble-chain diagram analysis.
The R-scheme proposed in Ref. [147] is designed to remove the leading renormalon ambiguity in a general operator product expansion.In our case, the leading renormalon ambiguity in the matching factor Z and the leading power corrections are both proportional to 1/P 2 z in Eq.( 28).These 1/P 2 z terms can be removed in the following combination: The remaining power corrections can in principle be removed in a similar way as well.The logarithmic term comes from the (α s ln P z )/P 2 z correction in the kernel Z which can be recasted to anomalous dimension γ such that the leading renormalon ambiguity in Z scales as 1/P (2+γ) z .This power can be removed by replacing 2 → 2 + γ in the square brackets.Now the remaining power correction is O(1/P 2 z P 2 z ) ∼ O(1/P 4 z ).So the procedure can be applied again to remove this power correction.However, the computation of the anomalous dimensions beyond the bubble-chain diagrams could be a challenge in removing subleading renormalon ambiguities with this method.
Ref. [1] also proposed a method to remove the leading renormalon ambiguity by replacing the γ z of Eq.( 2) with a combination of γ z and γ t .The idea is that PDF can be extracted using either γ z or γ t matrix element.However, the leading renormalon ambiguity contributes differently in the two cases.By choosing a specific combination of the γ z and γ t matrix elements, the leading renormalon ambiguity is largely canceled.However, knowing what combination to use is difficult beyond the bubble-chain approximation.Also, this proposal has another technical issue.On a lattice, the matrix element using γ z will mix with another operator of the same dimension that needs to be removed, while the matrix element using γ t does not have this problem.Therefore, typically γ t is used in lattice computations.However, for the purpose of this work, both of the choices are equivalent since the analysis is performed in the continuum.

IV. BUBBLE-CHAIN CONTRIBUTIONS IN FIXED ORDER PERTURBATIONS
In the previous section, we have seen that the leading IR renormalon ambiguity of Eq.( 24) yields 1/P 2 z power corrections of Eq.( 28) with singular prefactors after summing the bubble-chain diagrams to all orders in α s .These 1/P 2 z terms are canceled in the R-scheme by design in Eq. (29).It is natural to ask, if the cancellation works when the series is summed to all orders in α s , can the cancellation also happen if the series is truncated to a fixed order?It seems the answer should be yes if the cancelation works for the infinite series and for different values of α s (within the radius of convergence), then the cancelation ought to happen at each order in α s as well.
To answer this question, we need multi-loop corrections to the PDF and quasi-PDF.However, complete results beyond one-loop does not exist yet because of their technical difficulty.Therefore, we resort to large n F expansion again and only include the bubble-chain diagrams.As we will see in Fig. 4(b), the R-scheme could indeed introduce large cancellation of the bubble-chain diagrams at each order in α s .This is consistent with what Ref. [147] advocated and demonstrated that R-scheme indeed improved the convergences of QCD perturbation.Now we show how to extract the fixed order result from the bubble summed result.The bubble-chain contribution to the (n + 1)-th loop quasi-PDF in the MS scheme can be extracted by expanding the Borel series of Eq.( 12) in powers of w: where The bubble-chain contribution to the (n + 1)-th loop PDF in the MS scheme is where The results for q(n) , q (n) , together with the MS to MS matching kernel for n = 1, 2, 3 are listed in the Appendix.These matching kernels are free from IR divergence from diagram-by-diagram cancellations.Note that we should have also included the convolution term Z (1) ⊗ q (1) in the Z (2) kernel and the Z (2) ⊗ q (1) + Z (1) ⊗ q (2) term in the Z (3) kernel were we to include all the n-loop diagrams rather than just the bubble-chain diagrams.But here we only investigate how the bubble-chain diagrams which could give rise to the renormalon ambiguities converge at higher loops.
To check the effect of the bubble-chain contribution at the n-loop, we convolute the CJ12 proton isovector PDF [146] with Z (n) of Eq.( 34) to obtain the corresponding isovector quasi-PDF of the proton at n-loop in the MS scheme.The result up to 3-loops is shown in Fig. 3.We see that the two-loop contribution is smaller than one-loop but about the same size as the three-loop.Therefore the convergence is already slow at three-loop for the MS quasi-PDF.However, as shown in Fig. 4, the convergence is much better when we use the R-scheme of Eq.( 29), especially when a larger P z is used.The need to use a large P z can be understood because for a fixed order expansion, the kernel only has FIG.3: Isovector proton quasi-PDF's the MS scheme derived from convoluting the CJ12 proton PDF [146] (shown as the tree level result) with the matching kernels computed with bubble-chain diagrams with α s = 0.283, P z = 1.5 GeV, µ = 3 GeV.The convergence of the series expansion is slow.powers of ln P z dependence but not 1/P 2 z dependence.Hence when we take P z → 0, R-scheme reduces to the usual MS scheme shown in Fig. 3 and has slow convergence.Therefore, larger P z tends to yield larger cancellation of the bubble-chain diagrams.
After establishing the power of R-scheme in removing the bubble-chain diagram contributions in 2-and 3-loop diagrams, we turn to the RI/MOM scheme which is typically used in lattice QCD and see how bubble-chain diagrams contribute in this scheme.In RI/MOM, all the loop corrections to the matrix element of a single quark state with momentum p µ are subtracted non-perturbatively at an off-shell kinematics −p 2 = µ 2 R and p z = p z R .In momentum space, this amounts to the subtraction [13] with η = p z /p z R .The RI/MOM renormalized quasi-PDF q(n) R is UV finite, so it does not matter what UV regulator is used to compute q(n) of Eq. (35).The UV regulator will be removed at the end to obtain q(n) R .We will just replace q(n) in Eq.( 34) by q(n) R to get the MS to RI/MOM matching kernel.It is worth commenting that all the IR renormalon ambiguity comes from the first term in Eq. (35).The second term, the counterterm, takes the external quark off-shell, hence the Feynman diagrams do not experience IR divergence using the gluon propagator of Eq. (11).Therefore, the subtraction in the RI/MOM scheme does not add additional renormalon ambiguity.
The effect of applying the RI/MOM renormalization to the proton isovector quasi-PDF up to three-loop bubblechain diagrams is shown in Fig. 5. Convergence is seen for the whole range of x-the possible slow convergence due to the renormalon effect does not appear up to three-loops in the RI/MOM scheme.Of course, this result is not conclusive for QCD due to the non-convergence problem of the 1/n f expansion mentioned above.However, the convergence pattern in the large n f world is still interesting on its own.It gives some hope that QCD might have a similar convergence pattern.But one can only know after carrying out the complete higher loop calculations.
Finally, we comment on whether we can take advantage of choosing the BLM scales [148] to improve the convergence FIG.5: Isovector proton quasi-PDF's derived from convoluting the proton PDF of Ref. [146] with the matching kernels from bubble chain diagrams for (a) quasi-PDF's in the MS scheme (b) RI/MOM renormalized quasi-PDFs, with α s = 0.283, P z = 1.5 GeV, µ = 3 GeV, µ R = 2.4 GeV, and p z R = 1.201GeV.
of the series expansion.The BLM approach is based on the observation that Hence by choosing a different renormalization scale for each order in the expansion, one can completely remove all the α n+1 MS β n 0 dependence in the expansion to speed up the convergence.Interestingly, the α n+1 MS β n 0 dependence is exactly what the bubble-chain diagrams yield (see e.g.Eq.( 30)).Hence the BLM approach can cancel the renormalon ambiguity from the bubble-chain diagrams to improve the convergence.However, unlike the factor D of Eq.( 36) is a constant, the prefactors of our matching kernel depend on the momentum fraction.It is not clear how to absorb them into renormalization scales.Even if we proceed by demanding the bubble-chain diagrams be canceled only for a specific momentum fraction, we still need to benchmark the convergence of this approach with the complete multi-loop results which are not yet available.Therefore, we do not pursuit this program here in this work.

V. CONCLUSION
We have investigated the renormalon ambiguity in the flavor non-singlet quasi-PDF of a hadron.We follow the usual practice to study the diagrams in the large n f (the number of fermion flavors) limit with powers of (α s n f ) summed to all orders [143].Although QCD is not asymptotically free in this limit, the qualitative features of the renormalon ambiguity are expected to remain in QCD [140].Also, taking the β function of QCD as a reference, it might be possible that the higher order 1/n f corrections of our calculation are about the same magnitude as the leading order.In that case, our large n f analysis could still be useful although conclusive results can only be drawn with explicit full QCD calculations.
The bubble-chain diagrams were computed in momentum space and the result agreed with the coordinate space computation of Ref. [1] up to one possible typo in a prefactor.We confirmed the assertion of Ref. [1] that the leading IR renormalon ambiguity from bubble-chain diagrams was an O(Λ 2 QCD /x 2 P 2 z ) effect on the quasi-PDF.In addition, we have also found an O(δ(x)Λ 2 QCD /P 2 z ) term with a divergent prefactor such that quark number conservation is not broken.This ambiguity is supposed to be cancelled by power corrections associated with higher-twist contributions.Hence, if the power corrections were not included in the analysis, then the error was O(Λ 2 QCD /x 2 P 2 z , δ(x)Λ 2 QCD /P 2 z ), which is quite significant at small enough x, rather than O(Λ 2 QCD /P 2 z ) from dimensional analysis.To remove this leading IR renormalon ambiguity, we have investigated the proposed R-scheme [147], which is designed to cancel all the O(1/P 2 z ) contributions.This cancelation does not rely on the bubble-chain approximation.It is applicable in QCD.Furthermore, since the R-scheme cancel the renormalon ambiguity non-perturbatively for different values of α s , it is possible that the cancellation also happens at each order in α s to improve the convergence of QCD perturbation [147].Because the full multi-loop results in QCD are not available, we use the bubble-chain diagrams to demonstrate this up to three-loops in Fig. 4(b).
After establishing the power of R-scheme in removing the bubble-chain diagram contributions, we turn to the RI/MOM scheme which is typically used in lattice QCD and see how bubble-chain diagrams contribute to the proton isovector quasi-PDF in Fig. 5. Convergence is seen for the whole range of x-the possible slow convergence due to the renormalon effect does not appear up to three-loops in the RI/MOM scheme.Of course, this result is not conclusive for QCD due to the non-convergence problem of the 1/n f expansion mentioned above.But it gives some hope that QCD might have a similar convergence pattern.The final result can only be known after carrying out the complete higher loop calculations.

One-loop result
The 1-loop correction to quasi-PDF of a single quark state is where Taking the ρ → 0 limit and isolating the logarithmic IR divergence, we have The one-loop correction to the lightcone PDF of a single quark state is non-vanishing only when 0 < x < 1, and where The matching kernel for q(1) (x, ρ → 0) and q (1) (x, −µ 2 /p 2 ), both in the MS scheme, is The IR divergence is cancelled between the quasi-PDF and PDF in the matching kernel as we expect that the kernel only compensates the UV difference between the quasi-PDF and PDF.
The kernel has the asymptotic behavior ).This leading contribution is cancelled in the RI/MOM to MS scheme matching which yields better convergence numerically [13].

(A8)
The first term comes from the sub-divergence of the gluon vacuum polarization bubble.Taking the limit ρ → 0 and isolating the logarithmic IR divergence, we have , 0 < x < 1 .

FIG. 1 :
FIG. 1: Bubble-chain diagrams for PDF and quasi-PDF for single quark states.The single lines are quark propagators, curly lines are gluons propagators and double lines are Wilson lines.The gray bubbles are gluon vacuum polarization diagrams at one loop.