Evolution of parton showers and parton distribution functions

Initial state evolution in parton shower event generators involves parton distribution functions. We examine the probability for the system to evolve from a higher scale to a lower scale without an initial state splitting. A simple argument suggests that this probability, when multiplied by the ratio of the parton distributions at the two scales, should be independent of the parton distribution functions. We call this the PDF property. We examine whether the PDF property actually holds using Pythia and Deductor. We also test a related property for the Deductor shower and discuss the physics behind the results.


I. INTRODUCTION
Parton shower algorithms for simulation of hadronhadron collisions use "backward evolution" for the initial state part of the shower [1]. The parton distribution functions (PDFs) for the initial state hadrons appear in the initial state splitting probabilities in the shower. In the simplest approximation, there is a property that relates two functions: (1) the probability for the state to evolve from a hard scale µ 2 h to a softer scale µ 2 s without an initial state splitting and (2) the ratio of the PDFs at the two scales. We will call this property (defined below) the PDF property.
Both PDF evolution and initial state parton shower evolution can be viewed as evolution under scale changes and both are calculated from perturbative splitting of initial state partons. Thus these two sorts of evolution are connected. The PDF property represents a selfconsistency condition for the shower in the sense that (as we shall see) if the property does not hold then there is some physics missing from the shower evolution. In this paper, we investigate this hypothesized property using the parton shower event generator Pythia [2]. Then we investigate the same property using our own shower event generator Deductor [3].
We state what the PDF property is in Sec. II and provide some alternative PDF sets for the purpose of testing the PDF property in Sec. III. Then we carry out the test using Pythia in Sec. IV and using Deductor with k T ordering in Sec. V. We test another property that applies to cross sections including threshold effects in Deductor in Sec. VI. In Sec. VII, we discuss the physics behind the results that we have seen. In Sec. VIII, we use Deductor to examine the previous properties using the default, virtuality based, shower ordering variable of Deductor and using angular ordering. Finally, we present * Zoltan.Nagy@desy.de † soper@uoregon.edu a brief summary in Sec. IX.

II. THE PDF PROPERTY
The property that we discuss is stated and derived in the widely used text of Ellis, Stirling, and Webber [4]. To keep the notation uncomplicated, we consider a simplified case in which there is only one kind of parton and there is only one hadron.
The parton distribution functions obey the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equation Here P is the unregulated evolution kernel. We have applied a + prescription, which is reflected in the subtraction at z = 1. Let us consider a parton shower algorithm based on a measure µ 2 of the hardness of parton splittings. For instance, µ 2 is often chosen to be a transverse momentum variable k 2 T . The scale µ 2 is used to order splittings within the shower. Harder splittings come first, then softer splittings.
In the parton shower, we define a probability not to split between two scales by Π(µ 2 s , µ 2 h , x) Here x is the momentum fraction of the initial state parton at the starting scale µ 2 h . Note the appearance of a ratio of PDFs in the exponent [1]. There should be limits, z − (µ 2 ) < z < z + (µ 2 ) for the z integration, but we ignore that here. We simply treat z − (µ 2 ) as being so close to 0 and z + (µ 2 ) as being so close to 1 that the limits do not matter. We also define Π pert (µ 2 s , µ 2 h , x) Here we have the perturbative splitting function but no PDFs.
We can relate these functions by using the evolution equation for the PDFs. We have This tells us that if we define Π alt (µ 2 s , µ 2 h , x) by then we will have Then Π alt (µ 2 s , µ 2 h , x) will be independent of which PDF set is used in its calculation. This is the PDF property in the simplified case. Now suppose we have two hadrons in the initial state and we have more than one kind of parton. We initiate a parton shower with two partons that create a hard scattering at scale µ 2 h . The two partons have momentum fractions x a , x b and flavors a, b. We define Π(µ 2 s , µ 2 h ; x a , a, x b , b) to be the probability that, according to the parton shower algorithm used, this state evolves to a lower scale µ 2 s with no parton splittings.
We now define an alternative probability function by The PDF property that we seek to investigate states that Π alt (µ 2 s , µ 2 h ; x a , a, x b , b) is independent of the PDF set that we use in the calculation. 1 1 We can also demand that Π alt (µ 2 s , µ 2 h ; xa, a, x b , b) depend on the parton momenta pa = xaP A and p b = x b P B but not on the hadron momenta P A and P B . This amounts to demanding that Π alt be unchanged if we use new PDFs with rescaled momentum fractions:f a/A (xa, µ 2 ) = λ A f a/A (λ A xa, µ 2 ) and Thus this is a special case of the demand that Π alt be independent of the PDFs.
The text by Campbell, Huston, and Krauss [5] (Section 5.3.1.5) also relates the evolution of PDFs, Eq. (1), to the no-splitting function in shower evolution, Eq. (2), using essentially the argument given above, but does not relate these to Eqs. (5) and (6), so that we do not encounter the PDF property directly.

III. CHOICES FOR PARTON DISTRIBUTION FUNCTIONS
In order to test the PDF property, we need some choices for PDFs. The choices that we use do not need to fit data, but they do need to obey the standard DGLAP evolution equations.
As a standard choice, we use the CT14 PDFs [6]. These do fit data. We define alternative "hard" and "soft" PDF sets by setting and at an initial scale µ I = 1.295 GeV. We define f hard a/p (x, µ 2 I ) = f soft a/p (x, µ 2 I ) = 0 for other flavors of partons. These functions obey the standard momentum and flavor sum rules. The first set is designated "hard" because the functions approach zero rather slowly as x → 1, while the second set is designated "soft" because the functions approach zero rather quickly as x → 1. We define f hard a/p (x, µ 2 ) and f soft a/p (x, µ 2 ) for µ 2 > µ 2 I by solving the first order DGLAP equation. We insert standard flavor thresholds as in the CT14 set and use a rescaled argument of α s according to the prescription in Eq. (45) below. We use α s (M 2 Z ) = 0.118.

IV. TEST OF THE PDF PROPERTY USING PYTHIA
In this section, we test the PDF property using Pythia [2] (version 8.2.43). The hardness parameter for Pythia is µ 2 = k 2 T with a particular definition of the transverse momentum k 2 T in a splitting. In Pythia, we set α s (M 2 Z ) = 0.118 to match the coupling used in the parton evolution.
We create a hard scattering event for proton-proton collisions at √ s = 13 TeV corresponding to the Drell-Yan process with off-shell Z 0 and γ production with an e + e − final state. The starting scale µ 2 h for the shower is the square of the e + e − mass. We accept events with e + e − mass in the range 2.5 TeV < µ h < 2.7 TeV. The parton momentum fractions x a and x b are determined by We accept events with 2/3 < x a /x b < 3/2. The parton flavors can be (a, b) = (u,ū), (ū, u), (d,d), (d, d), or sometimes other choices. For each event, we record x a , a, x b , and b. Let P (x a , a, x b , b) denote the probability density that the parton variables take the indicated values.
We use the "user hooks" mechanism of Pythia to determine the scale µ 2 s of the first initial state splitting. It is convenient to define a corresponding logarithmic variable for the "shower time" of the first splitting. Thus t = 0 corresponds to a splitting at the scale of the hard interaction and larger t corresponds to a softer first splitting. Let ρ(t; x a , a, x b , b)dt denote the probability that, for the indicated choice of parton variables, the first splitting happens between t and t + dt. Then the averaged probability that the first splitting happens between t and t+ dt isρ(t)dt wherē (11) Then it is of interest to plotρ(t) versus t. The plots for each of our choices of PDF sets are shown in Fig. 1. We would expect that with harder PDFs, it is more likely to have a first splitting at a smaller value of t. That is what we see.
For a specified choice of parton variables, the probability for no splitting to occur before shower time t is The average of this no-splitting probability is which equals We plot Π(t) for our three choices of PDF sets in Fig. 2. We expect that at any fixed t, Π(t) will be smallest for the hard PDF set and largest for the soft PDF set. That is what we see. The differences between different PDF sets is substantial. Now we can test the PDF property of initial state splittings. We define Π alt (µ 2 s , µ 2 h ; x a , a, x b , b) according to Eq. (7). Then in Fig. 3 we plot the average of Π alt (µ 2 s , µ 2 h ; x a , a, x b , b), for our three choices for PDF sets. If the PDF property holds for a Pythia shower, the plots Π alt (t) will be identical for the CT14 set, the soft set, and the hard set. For this to happen, Π alt (t; soft) − Π alt (t; CT14) needs to decrease compared to Π(t; soft) − Π(t; CT14) and Π alt (t; hard) − Π alt (t; CT14) needs to increase. This is what happens. However, the change is too pronounced, so that now We conclude that the PDF property does not hold for a Pythia shower in that the differences among the three PDF curves in Fig. 3 are not much smaller than the differences in Fig. 2. Should we have expected that the PDF property might not hold for a Pythia or for other parton shower Monte Carlo event generators? To see, we should look more closely at the derivation in Sec. II. We note, first of all, that the splitting function that governs an initial state splitting must be equal to P (z) for an splitting that is close to the collinear limit, but that the parton splitting function will not generally equal to P (z) away from the collinear limit. Second, we note that the limits on the z-integrals in Eqs. (2), (3), and (4) are z − (µ 2 ) < z < z + (µ 2 ). The derivation in Eq. (4) treated z − (µ 2 ) as being very close to 0 and z + (µ 2 ) as being very close to 1. The upper limit, z + (µ 2 ) is especially important. In an initial state splitting, the emitted gluon carries momentum fraction 1 − z. When (1 − z) → 0, the emitted gluon is becoming very soft. As explained in Ref.
[4], emissions with (1 − z) < (1 − z + (µ 2 )) are omitted from the integration because they are "unresolvable." But "unresolvable" means "unresolvable at scale µ 2 ." When µ 2 is large, then (1 − z + (µ 2 )) is not small. Just what the function z + (µ 2 ) is depends on the choice of the hardness parameter µ 2 used for ordering emissions in the shower and it depends on the definition of the kinematics used in the shower algorithm. But whatever the precise definitions are, we ought not to expect that z + (µ 2 ) is generally small. Thus we ought not to expect the PDF property to hold for initial state splittings in the shower. Alternative no-splitting function, Π alt (t), versus shower time according to Pythia.

V. TEST OF THE PDF PROPERTY USING DEDUCTOR
We now examine the PDF property using our own parton shower event generator, Deductor [3]. This uses a dipole shower algorithm that is in many ways similar to that of Pythia [2]. The default hardness variable µ 2 in Deductor is a variable Λ 2 that is proportional to the virtuality of the splitting. However, Deductor has the option to use k T ordering. In this section, in order to stay reasonably close to the algorithm of Pythia, we use k T ordering. We note, however, the definition of k 2 T in Deductor is not the same as the definition in Pythia. Some of the other differences between Deductor and Pythia are discussed in Ref. [7].
Using Deductor, we start initial state probability preserving parton shower evolution using a state with two incoming partons corresponding to a proton-proton collision with √ s = 13 TeV. The partons have momentum fractions x a = x b = 0.2 = 2.6 TeV/ √ s. We choose parton flavors a = u, b =ū. In the notation of Sec. IV, the function P (x a , a, x b , b) that gives the distribution of parton momentum fractions and flavors is simply a product of delta functions representing these choices.
In Fig. 4, we use the Deductor no-splitting operator and plot the alternative no-splitting function Π alt (t) as defined in Eqs. (7) and (15) for our three choices of PDF sets. If the PDF property holds for the Deductor shower, the three curves should be identical. They are not. In fact, the three Deductor curves in Fig. 4 are quite similar to the three Pythia curves in Fig. 3. We should not be surprised. The argument that the PDF property should not hold does not depend on features of a specific parton shower event generator. Rather, this reasoning applies quite generally.

VI. RESTORING THE MISSING PHYSICS
The results presented in the previous sections suggest that some physics may be missing in the shower evolution in Pythia and Deductor. In this section, we explore the missing physics. We describe an operator, in addition to the shower evolution operator, that is present in Deductor. We then provide numerical evidence that this operator provides the missing physics.
We need a certain amount of preparation to introduce the additional operator. For this, we follow the general theory of parton showers from Ref. [8]. The general theory works, in principle, at arbitrary order of perturbation theory. It recognizes that the splitting functions in a parton shower must match the infrared singularities of QCD but are rather arbitrary away from the singular regions. Nevertheless, the cross section corresponding to an infrared safe measurement should match between a perturbative QCD calculation and a calculation that uses a parton shower, up to the perturbative order available in both calculations. This matching requires a certain structure for the operators used in the parton shower calculation. Even when we use a shower with just order α s splitting functions for the shower, the analysis suggests something about the structure needed in the parton shower calcu-lation if we want to include potential large logarithms.
The theory we use for Deductor [3,[7][8][9][10][11][12][13][14] uses a vector space, the "statistical space," that describes the momenta, flavors, colors, and spins for all of the partons created in a shower as the shower develops. 2 The shower is then described using linear operators on this space. Following Ref. [8], we begin with an operator U pert (µ 2 s , µ 2 h ) that describes evolution from a harder scale µ 2 h to a softer scale µ 2 s . The operator U pert (µ 2 s , µ 2 h ) is constructed from Feynman diagrams with approximations that capture their collinear and soft limits. PDFs do not appear in U pert (µ 2 s , µ 2 h ). The full shower evolution operator is constructed [8] from There are two kinds of operators here that modify U pert .
The first is F (µ 2 ). This operator multiplies by PDFs is the number of colors associated with a parton of flavor i (3 or 8) and n s (i) is the corresponding number of spin states (2). We apply the operator F (µ 2 s ) to the parton state after U pert (µ 2 s , µ 2 h ) acts. We apply the inverse operator . This operator leaves the number of partons, their flavors, and their momenta unchanged. It is a necessary part of the general formalism. We write it as One could expand this operator perturbatively and keep only the first terms, but the exponent contains large logarithms, so we use the exponential form. When we use a first order shower, we evaluate S V up to order α s . The scale µ 2 f is an infrared cutoff scale of order 1 GeV 2 . The integrand in the exponent is convergent in the infrared if α s is held constant, so the result is not sensitive to the choice of µ 2 f . An approximation to S V (µ 2 ) at first order is part of Deductor. It contains perturbative factors and ratios of PDFs.
The operator U V (µ 2 f , µ 2 ) makes the complete shower evolution operator U(µ 2 s , µ 2 h ) probability preserving. In the notation [9] that we use, the total probability associated with a statistical state ρ is obtained by summing over the number of partons, integrating over momenta, summing over flavors, and taking the trace in color and in spin. This total probability is denoted as 1 ρ . With the U V factors included, one has so that the shower does not change the total probability. The showers in Pythia and other parton shower event generators are constructed to be probability preserving. We will discuss the method for arranging this at the end of this section. The operator U V (µ 2 f , µ 2 ) is also numerically important. It describes threshold logarithms [15][16][17][18][19][20][21][22] in the context of a parton shower. 3 We have analyzed U V (µ 2 f , µ 2 ) in Refs. [7,12].
We are now prepared to examine the probability not to have a splitting between two scales in the Deductor shower. The operator U(µ 2 , µ 2 h ) obeys an evolution equation of the form Here S split (µ 2 ) increases the number of partons and S no-split (µ 2 ) leaves the number of partons unchanged. This evolution equation can be solved in the form Here T indicates ordering of the exponential in the scale µ 2 , with lower scales to the left. The operator N (µ 2 s , µ 2 h ) is the no-splitting operator. The quantity 1 N (µ 2 s , µ 2 h ) ρ gives the probability for the state ρ to evolve from scale µ 2 h to scale µ 2 s without splitting. Then Eq. (21) says that the shower can either evolve from scale µ 2 h to scale µ 2 s without splitting or it can evolve to an intermediate scale µ 2 without splitting, then undergo a splitting at scale µ 2 , then evolve from µ 2 to µ 2 s with zero or more splittings.
The operator N (µ 2 s , µ 2 h ) leaves the number of partons and their momenta and flavors unchanged. Thus we call it the no-splitting operator. However, it has a non-trivial action on the parton colors. In order to deal with the complications of quantum color, Deductor starts with an approximate treatment of color, the LC+ approximation [10]. Then color beyond this approximation can be included perturbatively [13]. In this paper, we start with a quark-antiquark pair in a color singlet state and consider the probability to have no emissions between two scales. For this problem, the needed color treatment is trivial. However, a note of caution is in order: one should not imagine that the color content of N (µ 2 s , µ 2 h ) is always trivial.
The operator U pert (µ 2 , µ 2 h ) obeys a similar equation, The operators S pert split (µ 2 ) and S pert no-split (µ 2 ) are calculated from the singular behavior of Feynman diagrams. They do not involve PDFs. At first order, S pert split (µ 2 ) comes from diagrams with a real emission, while S pert no-split (µ 2 ) corresponds to virtual graphs.
The first order version of S pert no-split (µ 2 ) used in Deductor is approximate in that it omits the contributions from the imaginary part of virtual graphs. These contributions can be included perturbatively [13] or even in exponentiated form [14], but we find that they are not numerically important [13,14].
This evolution equation can be solved in the same form as above, Here the perturbative no-splitting operator is Now we can use the relation (17) between U(µ 2 s , µ 2 h ) and U pert (µ 2 s , µ 2 h ). Using this relation in Eqs. (21) and (24), we see that the no-splitting operators are related by Then the corresponding generators of splittings are related by This relation is simpler at lowest perturbative order. The perturbative expansions of both S split (µ 2 ) and S pert split (µ 2 ) begin at order α s , while the perturbative expansion of U V (µ 2 f , µ 2 s ) begins with U V (µ 2 f , µ 2 s ) = 1 + O(α s ). Thus the relation that connects the splitting functions used in a first order parton shower is An initial state splitting comes with the ratio of the PDFs after the splitting to the PDFs before the splitting. This, of course, is the standard form for a parton shower with backward evolution for the initial state partons. The relation (26) is of special interest for our study of the PDF property. We have, after noting that F commutes with U V , Recall that the operator N pert (µ 2 s , µ 2 h ) is constructed from perturbative Feynman diagrams but does not involve PDFs. Therefore the operator on the left-hand side of Eq. (29) is unchanged if we change PDFs.
Using the operator notation, the alternative nosplitting function is where ρ is the initial partonic statistical state. This function can depend on the PDF set used in its calculation. As we saw in Fig. 4, Π alt (t) does indeed depend on the PDF set used. Because of Eq. (29) and because N pert (µ 2 s , µ 2 h ) does not involve PDFs, the function is independent of the PDF set used. 4 (Here "opt" is for "optimal.") Deductor contains an approximate version of the operator U V (µ 2 f , µ 2 ). Using this operator, we can directly check the dependence of Π opt (t) on the parton distribution set used. The result is shown in Fig. 5.
In Deductor, there are some approximations in both N (µ 2 s , µ 2 h ) and in U V (µ 2 f , µ 2 ), so we cannot expect Π opt (t) as calculated to be exactly independent of the PDFs. Indeed, the three curves for the CT14, hard, and soft PDF sets are not exactly the same. Nevertheless, the agreement is quite good.
If we assume that the dependence on the parton distributions seen in Π alt (t), Fig. 4, is a sign that some physics is missing from its definition, then Fig. 5 suggests that the missing physics is contained in U V (µ 2 f , µ 2 ).

VII. DISCUSSION
Within the framework of Ref. [8], the shower evolution operator U(µ 2 s , µ 2 h ) is probability preserving thanks to the inclusion of the operator U V (µ 2 f , µ 2 ) in its definition (17). In Pythia and other parton shower event generators, the parton shower is probability preserving without the need for U V (µ 2 f , µ 2 ). The method for guaranteeing this property is simple. Consider the case of just one hadron and just one kind of parton as in Sec. II. When the initial state parton has momentum fraction x, the differential probability for the initial state parton to split at scale µ 2 with momentum fraction z and azimuthal angle φ is where G(µ 2 , z, φ) is the splitting function, approximately equal to the DGLAP splitting kernel P (z). The splitting function vanishes for for z > z + (µ 2 ), where z + (µ 2 ) is determined by the definition of the ordering variable and the shower kinematics. The shower algorithm needs a function giving the probability for the initial state parton not to split between a scale µ 2 h and a lower scale µ 2 s . The standard method is to define this function to be as in Eq. (2). The exponent is the negative of the inclusive probability to have a splitting, so the exponential is the probability not to have a splitting. This makes the shower probability preserving. We should note, however, that if we base the parton shower algorithm purely on the properties of perturbative QCD, then for a lowest order shower we have available real emission diagrams and one loop virtual diagrams. We then start with a perturbative no-splitting operator N pert (µ 2 s , µ 2 h ) as given in Eq. (25). In the exponent of N pert (µ 2 s , µ 2 h ), the operator S pert no-split (µ 2 ) is derived from one loop virtual diagrams. Since we know that virtual diagrams cancel the completely inclusive integral of real emission diagrams, Eq. (33) is a plausible approximation for the effect of the virtual diagrams. However, Eq. (33) cannot be exact because the virtual diagrams do not contain the ratio of PDFs in Eq. (33). In order to connect the evolution of the parton shower to the evolution of the PDFs, we need something like the approximations used in Sec. I to give us the PDF property. Thus, a good way to test the approximations that give us Eq. (33) is to test the PDF property. That is what we have done in the previous sections. This test can be applied quite easily to any parton shower event generator.
What is the practical effect of U V (µ 2 f , µ 2 )? The result of Ref. [8] is that the cross section corresponding to an infrared safe observable J is given by This cross section is determined by standard QCD perturbation theory and is independent (up to the perturbative order calculated) of the many choices used in the definition of the shower. There is a perturbative error term O(α k+1 s ) coming from uncalculated higher order corrections. 5 There is also a power suppressed error term ) that is small if the infrared cutoff scale µ 2 f is much smaller than the scale Q[J] 2 corresponding to the observable that we measure. There is an operator O J that specifies the observable to be measured, followed by a statistical bra state 1 that instructs us to integrate over all parton variables. There is a statistical state ρ h corresponding to a hard scattering at scale µ 2 h . If we calculate beyond leading order, then ρ h includes appropriate infrared subtractions. There is an operator F (µ 2 h ) that inserts the PDFs for the hard scattering.
There is the operator U(µ 2 f , µ 2 h ) that generates a probability preserving shower from the hard scale µ 2 h to the shower cutoff scale µ 2 f . At lowest order, and ignoring for simplicity the imaginary parts of virtual diagrams, the no-spitting operator in the shower is related to the shower splitting functions by the analogue of the simple relation (33). So far, the lowest order version of this is very standard.
Finally there is the operator U V (µ 2 f , µ 2 h ). As we have noted, this operator is needed to make the shower operator U(µ 2 f , µ 2 h ) probability preserving. It also allows the 5 The operators U (µ 2 f , µ 2 h ) and U V (µ 2 f , µ 2 h ) include contributions at all orders of perturbation theory, but the strict perturbative accuracy of Eq. (34) is limited to the accuracy at which the hard scattering, ρh , is calculated. cross section σ[J] to be independent, up to the perturbative order calculated, of the choices made in defining the shower and the subtractions in ρ h . The operator U V (µ 2 f , µ 2 h ) appears immediately after the statistical state representing the hard scattering.
Examination of U V (µ 2 f , µ 2 h ) [7,12] shows that, with k T ordering, this operator generates a summation of threshold logarithms corresponding to the hard scattering.
Threshold logarithms have been understood from an analytical perspective for a long time [15][16][17][18][19][20][21][22] and are known to have a substantial effect on cross sections in many important cases. Thus it is not a surprise that we see substantial differences between Fig. 4 and Fig. 5.

VIII. DEDUCTOR WITH Λ OR ANGLE ORDERING
In Secs. V and VI, Deductor results were obtained with k T ordering because that is most similar to the ordering used in Pythia. However, the default ordering variable in Deductor is Λ 2 , which is proportional to the virtuality in a splitting. Deductor also has the option to use angular ordering. In this section, we explore the PDF property using these other ordering variables. We will find that the ordering matters.

A. Scale variables
Consider a splitting of an initial state parton with momentum p a into a new initial state parton with momentump a and a new final state parton with momentum p m+1 . Let Q 0 denote a fixed vector equal to the total momentum of the final state partons just after the hard scattering that initiates the shower and let Q denote the total momentum of the final state partons just before the splitting. We take all partons to be massless. Let µ 2 1 denote the virtuality in the splitting, We also define the momentum fraction in the splitting to be Here p b is the momentum of the other initial state parton. With this notation, the default ordering variable Λ 2 is defined for an initial state splitting by For this same splitting, we define a transverse momentum variable by and use this as the ordering variable for a k T ordered shower. It is notationally convenient to define a corresponding scale variable by µ 2 0 = k 2 ⊥ . Then We also define an angle variable by Deductor takes the ordering variable for an angle ordered shower to be µ 2 2 = θ 2 (p a · Q 0 ) 2 /Q 2 0 . Then We could also use θ 2 = µ 2 2 Q 2 0 /(p a · Q 0 ) 2 directly as the ordering variable. This change would not affect the numerical results later in this section.
We now have scale variables µ 2 0 , µ 2 1 and µ 2 2 for a k Tordered, Λ-ordered, and angle-ordered shower respectively. These are related by

B. Parton distribution functions
Now consider the evolution of the PDFs, following the argument that we have used for Λ ordering [12]. We write the first order DGLAP kernels that apply to MS PDFs in the form where P reg aâ (z) is nonsingular as z → 1. Here C g = C A and C q = C F for q ∈ {u,ū, d, . . . }. We also define standard constants γ a by Finally we include a standard factor [17] in the argument of α s in the evolution equation for the PDFs.
Then we can write the DGLAP evolution equation in a form that implements the + prescription as We have included a cut µ 2 0 > m 2 0 (a). For light flavors a, we take m 2 0 (a) to be the scale, of order 1 GeV, at which we specify initial conditions for the PDFs. Then the cut turns off evolution for µ 2 0 smaller than m 2 0 (a), so that we can impose the initial conditions at any smaller scale. For charm and bottom quarks, we define m 0 (a) to be the corresponding quark mass, so that evolution starts at this scale. Now, the MS definition of the PDFs is that these functions are renormalized by subtracting poles in ǫ, where loop integrals are performed with dimensional regularization: µ 2ǫ 0 d 2−2ǫ k ⊥ · · · . This means that µ 2 0 is the scale for a k T integration.
This argument tells us that in the splitting functions of a k T ordered parton shower, the parton distribution functions should be MS parton distribution functions evaluated at the splitting scale µ 2 0 (or a constant of order 1 times µ 2 0 ). Now consider a Λ ordered parton shower, with scale variable µ 2 1 , or an angle ordered parton shower, with scale variable µ 2 2 . Then the scale variable is related to µ 2 0 by Eq. (42). To use in this shower, we need modified PDFs f a/A (x, µ 2 λ ), which are defined by a new factorization scheme [8] that matches the scheme used to define the parton shower and, in particular, matches the choice of ordering variable. As argued in Ref. [8], matching these schemes in such a way that the integrals in U V (µ 2 f , µ 2 ) are infrared finite requires that the shower oriented PDFs should obey an evolution equation obtained by making the variable substitution 6 (42) in the MS evolution equa-tion (46): One could determine the functions f a/A (x, µ 2 λ ) from data using the new factorization scheme. That would be a major undertaking, so what we do for Deductor is to use the same functions at a starting scale around 1 GeV as used for the CT14 PDFs, then use Eq. (47) to determine the PDFs at higher scales.
Note that at very large scales, the evolution equation (47) is nearly the same as its MS version. However, at smaller values of the scale variable, the two evolution equations are quite different. Thus f a/A (x, µ 2 ) for µ 2 ≫ 1 GeV 2 will be substantially different from f MŜ a/A (x/z, µ 2 ).

C. No-splitting probabilities
Consider now the no-splitting probability in a Deductor shower using one of the three ordering choices. As in some of the previous sections, we simplify the notation by assuming that there is only one initial state hadron and one kind of parton. Then the no-splitting probability for ordering type λ is a slightly more precise version of Eq. (33), Here µ 2 h,λ is c λ times the value of the ordering variable for a previous splitting or the ordering variable assigned to the start of the shower, with c 0 = 1, c 1 = 2p a · Q 0 /Q 2 0 and c 2 = 1. Similarly, µ 2 s,λ is c λ times the value of the ordering variable for a next splitting. In the exponent, we are integrating over the probability for a potential splitting that did not happen. We divide the splitting function into a part, G sing λ that is singular for (1 − z) → 0 and a part G reg λ that is regular for (1 − z) → 0. Then we set the argument of α s for the potential splitting to λ R (1 − z) λ µ 2 λ for the singular part and to λ R µ 2 λ for the regular part. These choices for the argument of α s affect the splitting functions only at order α 2 s . The PDFs are the ones appropriate to ordering type λ.
The function G λ (µ 2 λ , z, φ) is the shower splitting function. We define it by The function G is quite complicated, but we need only two of its features. First, we define G to contain a θ function that requires 7 This θ function acts to turn off the parton shower at the scale m 2 0 . One can then complement the parton shower by a hadronization model to cover physics below this scale. There is also a kinematic limit: G is nonzero only for For smaller values of 1 − z, an initial state splitting is not possible [12]. This is directly applicable to a k T -ordered shower. This translates to which is directly applicable to a Λ-ordered shower. For an angle-ordered shower, we need This is no restriction at all as long as µ 2 2 < Q 2 . That is, the angle-ordered shower is not really a hardness-ordered shower because an infinitely soft gluon emission can still have a large angle and thus a large µ 2 2 . For an angleordered shower, the only restriction on the z integral in Eq. (48) comes from the infrared cutoff in Eq. (50),  Alternative no-splitting fraction, Π alt (t), versus shower time according to Deductor with Λ ordering. This figure is analogous to Fig. 4, which shows Π alt (t) with kT ordering. The range of t values shown here is different since now t has a different meaning.
Consider next Λ ordering. The comparison for Π alt (t) is shown in Fig. 6. There is a significant dependence on the PDF set, but it is smaller than with k T ordering (Fig. 4).
In fact, the operator U V (µ 2 f , µ 2 ) is closer to the unit operator for Λ ordering than for k T ordering. This is because the restriction (52) for Λ ordering is less restrictive than the restriction (51) for k T ordering. The fact that U V (µ 2 f , µ 2 ) is closer to the unit operator for Λ ordering than for k T ordering reflects the fact that some of the result of summation of threshold logarithms moves from U V (µ 2 f , µ 2 ) to the change in PDFs compared to the MS PDFs [12].
The comparison for Π opt (t) is shown in Fig. 7. The dependence on the PDF set is barely discernible, as it was for k T ordering (Fig. 5).
Finally, consider angle ordering. 8 The restriction (53) for angle ordering provides no restriction at all. Now the upper limit for the z integration in the shower matches the upper limit for the z integration in the evolution of the PDFs. Because of this, we can expect the PDF property for Π alt (t) to work to a good approximation. The comparison for Π alt (t) is shown in Fig. 8. There are three curves for the three PDF sets, but the differences are small.
With angle ordering, the operator U V (µ 2 f , µ 2 ) is close to the unit operator. For the splitting of an up quark in the x region examined, the approximation to U V (µ 2 f , µ 2 ) used in Deductor makes Π opt (t) = Π alt (t) exactly. Thus the graph for Π opt (t) is the same as Fig. 8 and we do not display it. Now all of result of summation of threshold logarithms is contained in the change in PDFs compared to the MS PDFs [12].

IX. CONCLUSIONS
Initial state splittings in backward evolution in a parton shower event generator are characterized by the probability Π(t) not to have a splitting between two scales µ 2 h and µ 2 s = e −t µ 2 h . This function is the exponential of an integral of the shower splitting functions times a ratio of PDFs. One can consider another function, Π alt (t) obtained by multiplying Π(t) by the ratio of PDFs with these two scales according to Eq. (7). If one were to follow an argument that ignores some complications, one would conclude that the function Π alt (t) is independent of the parton distribution functions used in the calcula- tion. We call this conclusion the PDF property of the shower evolution.
One can quite easily test the PDF property numerically by running the parton shower event generator using different PDF sets. We use three PDF sets: CT14, soft, and hard. Each obeys the same evolution equation but the three sets start from different initial conditions. We test Pythia and we test Deductor with k T ordering.
We find that the PDF property does not hold. Thus we conclude that the complications that were ignored in the argument for the PDF property are not just trivial complications but are rather quite significant. It appears that the main difficulty comes from the upper endpoint of the integral over momentum fraction z in the exponent of Π(t). This upper endpoint was regarded as being close to 1, but it is not.
Deductor has an approximation to another operator, U V (µ 2 f , µ 2 ), that adjusts cross sections and has the effect of summing threshold logarithms. Using this operator, one can define another function, Π opt (t) that should be independent of the PDF set to the accuracy of the approximations to U V (µ 2 f , µ 2 ) and N (µ 2 s , µ 2 h ) in Deductor. We find that Π opt (t) is independent of the PDF set to a good accuracy.
We also test Deductor using its default virtuality based ordering variable Λ 2 and we test Deductor using angular ordering. In either case, inside the shower one must use shower oriented PDFs that obey a modified evolution equation appropriate to the ordering variable.
With Λ ordering, as expected, the PDF property fails to hold for Π alt (t) but Π opt (t) is quite accurately independent of the PDF set.
With angular ordering, we expect Π alt (t), calculated with PDFs that obey the evolution equation appropriate for angular ordering, to be almost independent of the choice of PDF set. We find that this is the case. Also, for angular ordering Π opt (t) should be nearly equal to Π alt (t). This is indeed the case.
It would be of interest to see the extent to which other parton shower event generators exhibit or do not exhibit the PDF property.
For the future, it would certainly be desirable to have a parton shower with splitting kernels evaluated at order α 2 s . This should come along with the operator U V (µ 2 f , µ 2 ) defined with an exponent calculated to order α 2 s . It would then be a good check of the consistency of the initial state evolution in the higher order shower that Π opt (t) defined in Eq. (31) is still independent of the PDF set used in its calculation.