A proof algorithm associated with the dipole splitting algorithm

We present a proof algorithm associated with the dipole splitting algorithm (DSA). The proof algorithm (PRA) is a straightforward algorithm to prove that the summation of all the subtraction terms created by the DSA vanishes. The execution of the PRA provides a strong consistency check including all the subtraction terms --the dipole, I, P, and K terms-- in an analytical way. Thus we can obtain more reliable QCD NLO corrections. We clearly define the PRA with all the necessary formulae and demonstrate it in the hadron collider processes $pp \to \mu^{+}\mu^{-},\,2\,jets$, and $n\,jets$.


Introduction
The present article is the accompany one of the article [1]. Then we would like to start with the summary of the article [1]. At the CERN Large Hadron Collider (LHC) the standard model higgs boson was discovered in the year 2012 during the run 1 with the colliding energy 7 and 8 TeV. The run 2 with the energy 13 TeV plans to start in the year 2015. In order to identify the discovery signals we need both of the precise experimental results and the precise theoretical predictions. For the precise theoretical predictions, at least the inclusion of the quantum chromodynamics (QCD) next-to-leading order (NLO) corrections is required. One of the most successful procedures to obtain the QCD NLO corrections for multi-parton leg processes is the Catani-Seymour dipole subtraction procedure [2,3]. The procedure has been already applied to the huge number of the processes happening at the LHC. The partial list of the achievements is collected in the reference of the article [1]. Now that the dipole subtraction has been applied to so many processes, we can have some drawbacks about the use of the procedure. Among the drawbacks we would like to point out the three difficulties. The first difficulty is to confirm whether the created subtraction terms are the necessary and sufficient ones, and whether the expression of each term includes no mistake. The second difficulty is for one person to reproduce the results in an article presented by the other person, especially the difficulty to specify all used subtraction terms. The difficulty is equally converted to the inverse case, namely, the difficulty for one person to tell the other person all the used subtraction terms in a reasonable short form without confusion. The third difficulty is something about the use of the computer packages in which the dipole subtraction procedure is automated. The publicly available packages are presented in the articles [4][5][6][7][8][9][10]. The users sometimes have the difficulties to understand the algorithms implemented in the packages and the outputs of the run.
In order to solve some parts of the three difficulties, it is required that a practical algorithm to use the dipole subtraction is clearly defined, and that the documentation of the algorithm includes the clear presentation of all the subjects in the wishlist, 1. Input, output, and creation order, 2. All necessary formulae on document, 3. Necessary information to specify each subtraction term, 4. Summary table of all created subtraction terms, 5. Associated proof algorithm.
We succeeded to construct an algorithm which allows the clear presentation of all the entries in the wishlist. It is named the dipole splitting algorithm (DSA). The DSA has been already presented with the focus on the entries 1∼ 4 at the wishlist in the article [1]. Then the purpose of this article is to present the last entry at the wishlist, ' 5. The proof algorithm associated with the DSA'.
The proof algorithm associated with the DSA means a straightforward algorithm to prove that the summation of all the subtraction terms created by the DSA vanishes. The proof algorithm is abbreviated as the PRA hereafter. What the PRA proves is expressed in a formula as follow. The QCD NLO corrections for arbitrary process at hadron collider are written as where the symbols, σ R , σ V and σ C , represent the real correction, the virtual correction, and the collinear subtraction term, respectively. In the framework of the dipole subtraction the NLO corrections are reconstructed as where the symbols, σ D , σ I , σ P , and σ K , represent the dipole, I, P, and K terms, respectively. The quantities, (σ R − σ D ), (σ V + σ I ), σ P , and σ K , are separately finite. When the NLO cross section in Eq. (1.1) is equated with the cross section in Eq. (1.2), we obtain the identity for arbitrary process as We call the relation the consistency relation of the subtraction terms. The PRA is a straightforward algorithm to prove the consistency relation in Eq. (1.3) for any given process, if all the subtraction terms, the dipole, I, P, and K terms are created by the DSA.
We would like to clarify the main advantages of the PRA. Since the relation in Eq. (1.3) includes all the created subtraction terms, the dipole, I, P, and K terms, the proof of the consistency relation gives the confirmation of all the terms. As mentioned above, the subtracted cross sections, (σ R − σ D ) and (σ V + σ I ), are separately finite. When we used any wrong collection or wrong expression for the dipole or the I term, the subtracted cross sections would diverge. In this way at least the divergent parts of the dipole and I terms are confirmed by the successful cancellation against the real and virtual corrections. Compared to the dipole and I terms, the P and K terms, σ P , and σ K , are separately finite themselves, and the confirmation by the cancellation is impossible. The PRA can provide the precious confirmation including the P and K terms. This is the first advantage of the PRA. The PRA is executed in analytical way and does not rely on any numerical evaluation. All the dipole terms are integrated over the soft and collinear regions of the phase space in analytical way in d-dimension. The expressions for the integrated dipole terms are available also in the original articles [2,3]. The PRA utilizes the integrated dipole terms and all the steps of the PRA are executed in analytical way which does not require any numerical evaluation. This is the second advantage of the PRA. In consequence we can have a strong consistency check of all the subtraction terms by the execution of the PRA, and we can obtain more reliable QCD NLO corrections as the results.
In the dipole subtraction procedure some algorithms to create the subtraction terms can be constructed. For all of them the construction of a straightforward algorithm to prove the consistency relation in Eq. (1.3) is not always possible. The cancellations in the consistency relation are realized between the cross sections which have the same initial states and the same reduced Born processes. At the DSA the created subtraction terms are classified by the real processes and the kinds of the parton splittings. The classification of the subtraction terms is converted to the classification by the initial states and the reduced Born processes.
Then we can easily identify the cross sections which cancel each other in Eq. (1.3). The systematical identification of the cancellations for arbitrary process makes possible for us to construct a straightforward proof algorithm of the consistency relation. In the original article of the Catani-Seymour dipole subtraction [2], the dipole terms are constructed to subtract the soft and collinear divergences from the real corrections, and then the dipole terms are analytically integrated over the soft and collinear regions in the phase space in d-dimension. The integrated dipole terms are transformed to the I, P, and K terms. The way 2 Proof algorithm 2

.1 Definition
The prediction of the cross section including the QCD NLO corrections is generally written where the symbol, σ LO , represents the leading order (LO) cross section or a distribution, and the symbol, σ NLO , represents the QCD NLO corrections to the LO cross section. The LO cross section does not appear in the present paper hereafter. For a given collider process the real emission processes which contribute to the collider process are written as R i . The set which consists of all the real emission processes is denoted as where the n real is the number of all the real processes. When the NLO corrections are treated within the framework of the DSA [1], they are expressed as In the DSA all the corrections are classified by the real processes R i , and each contribution is denoted as σ(R i ). The cross section σ(R i ) is defined as where the cross sections, σ R (R i ) and σ V (B1(R i )), represent the real and virtual corrections respectively. The process B1(R i ) is the Born process reduced from the R i by the rule, B1(R i ) = R i -(a gluon in final state), as defined in [1]. The cross sections, σ D (R i ), σ I (R i ), σ P (R i ), and σ K (R i ), represent the contributions of the dipole, I, P, and K terms respectively. The cross sections are factorized into the parton distribution function (PDF) and the subpartonic cross sections as where the f F(x a/b ) (x 1/2 ) represents the PDF and the subscript, F(x a/b ), denotes the field species of the initial state parton at the leg a/b as defined in [1]. The symbols,σ R (R i ) and σ V (B1(R i )), represent the partonic real and virtual corrections respectively. The quantities, σ(R i ), with the subscripts, D, I, P, and K, represent the contributions of the dipole, I, P, and K terms to the partonic cross sections respectively. The definitions of all the partonic cross sections are collected in Appendix A. 1. In order to specify the jet observables the corresponding jet functions, F (n/n+1) J , must be multiplied to all the cross sections. The use of the jet functions in the dipole subtraction is explained in [2]. For compact expression we do not show the jet functions explicitly in the present article.
In the original way of the calculation of the QCD NLO corrections, the corrections can be constructed as (2.6) where the symbol, σ orig (R i ), denotes each correction which belongs to the real process R i . The n real is the same number as in Eq. (2.3). The cross section, σ orig (R i ), consists of the three terms as where the symbols, σ R (R i ) and σ V (B1(R i )), are the same real and virtual corrections as appearing in Eq. (2.4). The σ C (R i ) represents the collinear subtraction term. When the NLO cross section in Eq. (2.6) is equated with the cross section in Eq. (2.3), we obtain the identity for arbitrary process as where the cross section σ subt (R i ) is defined as The cross section σ subt (R i ) includes all the subtraction terms. We call the relation in Eq. (2.8), the consistency relation of the subtraction terms. The aim of the PRA is to prove the consistency relation for arbitrary collider process. In order to construct the proof algorithm in the separated steps, we reconstruct the dipole term, σ D (R i ), into the four terms as The definitions of the four terms will be given in the following Sec. 2.3 ∼ 2.6, respectively. Using the four terms we can rewrite the σ subt (R i ) in Eq. (2.9) as The execution of the proof algorithm proceeds in the steps in such a way that the first three terms with the square bracket are calculated in turn. At this stage we can define all the six steps of the PRA as Step 1. Convert the dipole terms σ D (R i ) to the integrated form All the six steps will be separately explained in Sec. 2.2 ∼ 2.7, respectively. The premise for the execution of the PRA for a given process is that all the dipole, I, P, and K terms are created by the DSA in [1].

The
Step 1 of the PRA is to convert all the dipole terms, which are created by the DSA, into the integrated form. The contribution of each dipole term to the partonic cross section is generally written in d-dimension, where the R i is a real correction process and the dipj is the category to which the dipole term belongs. The S R i is the symmetric factor of the process R i . The spin degree of freedom, n s (a/b), is determined as n s (quark) = 2 and n s (gluon) = d − 2 = 2(1 − ǫ). Each dipole term is specified with the three legs, I, J, and K, of the real process R i . The dipole term is generally written as where the details of the notation to express the dipole terms are explained in the article of the DSA [1]. The category Dipole j (in short, dipj), and the sub-category of the splittings are shown at Fig. 1 in Appendix A.2. It is noted that in the DSA we introduce the field mapping, y = f (x), and in Eq. (2.14) the legs of the reduced Born process Bj, on which the color and helicity operators act, are specified with the elements (y emi , y spe ) of the set {y}.
The partonic cross section of the dipole term in Eq. (2.13) is converted to the integrated form aŝ where the overall factor A d is defined as The concrete expressions of the universal singular functions, where the Factor 1 and 2 are denoted as Once the concrete expressions of the Factor 1 and 2 are determined, the integrated dipole term is also uniquely determined. The Factor 1 is universal in the same category of the splitting with the spectator in final/initial state, for instance, the splitting, (1)-1, (3)-2, and so on. In order to summarize all the integrated dipole terms in a short form, it is sufficient that the information of the reduced Born process, the kind of the splitting, the Factor 1   and 2, is supplied at a table format. The summary tables for the dijet process are shown at  Tables 3 ∼ 13 in Appendix B. Those tables can be a template of the format for the summary   tables for arbitrary process. We here would like to see two examples at the real process, R 1 = uū → uūg, which contributes to the dijet process, pp → 2 jets + X. The creation of the dipole terms by the DSA is explained in the article [1].
The reduced Born process is taken as and the field mapping for this dipole term is made as (y a , y b ; y 1 , y 2 ) = (a, b ; 13, 2) . (2.24) The contribution to the cross section is written aŝ and the dipole term is written as The dipole term belongs to the type of the Final-Final dipole term and is converted to the integrated form in Eq. (A.20) aŝ where the color correlated Born squared amplitude is denoted as with the Lorentz scalar, s y 1 ,y 2 = 2 P(y 1 ) · P(y 2 ). The expression of the dipole term is further abbreviated in the fixed form in Eq. (2.20) with the Factor 1 and 2 as At the summary table in addition to the Factor 1 and 2, we can explicitly specify the reduced Born process and the kind of the splitting, which are the B1 and (1)-1 in the present case. Actually the integrated dipole term is shown with the above information at the first one, 1. (a, b ; 12, 3), in Table 3 in Appendix B.1.
The reduced Born process and the field mapping are determined as B3u = g(y a )ū(y b ) →ū(y 1 )g(y 2 ) and (y a , y b ; y 1 , y 2 ) = ( a1, b ; 2, 3) . (2.31) The cross section and the dipole term are written aŝ The dipole term belongs to the Initial-Initial dipole term and is converted to the integrated form in Eq. (A.56) aŝ with the Lorentz scalar, s xa, y b = 2 p a · p b . The Factor 1 and 2 in Eq. (2.20) are determined as The integrated dipole term is shown at 18. ( a1, b ; 2, 3) in Table 3.
In this way all the dipole terms can be converted into the integrated form. Then the summary tables of all the dipole terms created by the DSA, are converted to the summary tables of the integrated dipole terms with the necessary information. One original dipole term is converted to one integrated dipole term, and the total number of the dipole terms is conserved through the conversion. As the example, for the dijet process, all the summary tables of the dipole terms created by the DSA are shown at Tables 4∼14 in Appendix B.1 in [1]. All the summary tables are converted into the summary tables of the integrated dipole terms as Tables 3 ∼ 13 in Appendix B in the present article.

2.3
Step 2 : σ D (I) − σ I The Step 2 of the PRA is to prove the relation, The relation in Eq. (2.38) stands for arbitrary process and is regarded as an identity. The left-hand side in Eq. (2.38) is the first term with the square bracket in Eq. (2.11). We define the three cross sections, σ D (R i , I), σ I (R i ), and σ I (R i , (2) -1/2, N f V ff ), in Eq. (2.38) as follows.
First we define the cross section σ D (R i , I). At the Step 1 all the dipole terms are converted into the integrated form. Among them we take only the dipole terms in the category, Dipole 1. The Dipole 1 includes the splittings (1)∼(4) with the cases of the spectator in the final/initial state denoted as the sub-category, -1/2, which are shown at Fig. 2 in Appendix A.3. We extract the following parts from the whole expressions, depending on the splittings in Eq. (A.59). Then for both splittings (3)-1/2, the partonic cross sectionsσ D (R i , I, (3)-1/2) are extracted as the identical expression in Eq. (2.39). For the splitting (4)-1, we extract the part δ(1 − x) V g (ǫ) from V g,g (x ; ǫ) in Eq. (A.42), and for the splitting (4)-2, we extract the same quantity δ(1 − x) V g (ǫ) from V g,g (x ; ǫ) in V g,g (x; ǫ) in Eq. (A.60). Then for both splittings (4)-1/2, the cross sectionsσ D (R i , I, (4)-1/2) are defined aŝ with the Lorentz scalar, s y emi ,yspe = 2 P(y emi ) · P(y spe ). The partonic cross sectionσ D (R i , I) is the summation of all the existing partonic cross sectionsσ D (R i , I, (1) ∼ (4)-1/2) defined above. The formulae for theσ D (R i , I) are collected in Appendix A.3.
Next we define the cross section σ I (R i ). The partonic cross sectionσ I (R i ) is the contributions of the I terms which are created by the DSA [1]. The contribution of each I term is written asσ (2.43) where again the notation is defined at [1]. The S B1 is the symmetric factor of the reduced Born process B1(R i ). In the DSA the creation of the I terms is ordered by the species of the first leg I, (1)∼(4) which are shown at Fig. 7 in Appendix A.8. There are the four cases for the choices of the first leg I, and each case has the further choices of the second leg K in the final/initial state denoted as -1/2. The factor V F(I ) /T 2 F(I ) is determined in each case as with the Lorentz scalar, s IK = 2 p I · p K . The partonic cross sectionσ I (R i ) is the summation of all the created I terms asσ Similar to the case σ D (R i , I), the cross section σ I (R i ) is obtained by multiplying the PDFs to the partonic cross sectionσ I (R i ). The formulae for the I term σ I (R i ) are collected in Appendix A.8. The third cross section σ I (R i , (2) -1/2, N f V ff ) is defined as follows. We take only the I terms which belong to the splitting (2)-1/2 at Fig. 7. Among them we extract the part, N f V ff (ǫ), in the function V g (ǫ) in Eq. (A.45). The extracted part is defined as the partonic cross section,σ where the leg I is in the final state and the field species is gluon. By the definition the term exists only if the reduced Born process, B1(R i ), includes any gluon in the final state.
The statement is equivalent to that the term exists only if the real process R i includes two or more gluons in the final state. The partonic cross sectionσ I (R i , (2) -1/2, N f V ff ) is the summation of all the existing cross sections in Eq. (2.47). The formula is also added in Appendix A. 8. Now that all the three terms in Eq. (2.38) are defined, we can interpret the equation in such a way that the extracted part from the integrated dipole term,σ D (R i , I), cancels the I term, σ I (R i ), except for the part, σ I (R i , (2) -1/2, N f V ff ). The remaining part is canceled by a term created at the different real process. The mechanism of the cancellation will be clarified at the Step 6 in Sec. 2.7. Finally we see one example. We take the same real emission process as used at Step 1, The reduced Born process, B1(R 1 ) = uū → uū, does not include any gluon in the final state, and the right-hand side in Eq. (2.38) does not exist. Then the relation to be proved is written as First we construct the partonic cross sectionσ D (R 1 , I). All the integrated dipole terms of theσ D (R 1 ) are summarized at Table 3 in Appendix B.1. Among all the twenty-one dipole terms we take only the first twelve terms in the category Dipole 1. Following the definition of the σ D (R i , I) given above, we extract the cross section aŝ  with the symmetric factor S B1 = 1. Using the relations, S R 1 = S B1 = 1 and V f (ǫ) = V f g (ǫ), we prove the relation in Eq. (2.48). The Step 2 for the process R 1 is completed. The result is shown in Eq. (B.1) in Appendix B.1. As mentioned above in this example the right-hand side in Eq. (2.38) does not exist. The cases where the right-hand side exists will be seen at Sec. 4 and Sec. 5.

2.4
Step 3 : Step 3 of the PRA is to prove the relation, The left-hand side in Eq. (2.51) is the second term with the square bracket in Eq. (2.11). We define the three cross sections, σ D (R i , P), σ C (R i ), and σ P (R i ), in Eq. (2.51) as follows.
First we define the cross section σ D (R i , P). The quantity is extracted from the integrated dipole terms converted at Step 1 in the following way. We choose only the dipole terms with the splittings, (3), (4), (6), and (7), as shown at Fig. 3 in Appendix A.4. For the Initial-Final dipole terms, namely the splittings, (3)-, (4)-, (6)-, and (7)-1, we extract the factors, (−1/ǫ + ln x) · P f f, gg, f g, gf (x), in the functions, V f,f (x ; ǫ), V g,g (x ; ǫ), V f,g (x ; ǫ), and V g,f (x ; ǫ), in Eqs. (A.41)∼(A.44), respectively. For the Initial-Initial dipole terms, namely the splittings, (3)-, (4)-, (6)-, and (7)-2, we extract the same factors in the functions, V f,f (x; ǫ), V g,g (x; ǫ), V f,g (x; ǫ), and V g,f (x; ǫ), in Eqs. (A.59)∼(A.62). For both of the dipole terms with the splittings (3)-, (4)-, (6)-, and (7)-1/2, the partonic cross sections,σ D (R i , P), are defined as the universal expression, where the factor, Second we define the cross section σ C (R i ). The quantity is the collinear subtraction term which is introduced in the QCD NLO corrections as in Eq. (2.7). Some algorithms to create the collinear subtraction terms for arbitrary process may be available. We here introduce an algorithm which is analogous with the algorithm to create the P term in the DSA [1]. The input is taken at a real process R i . Same with the DSA, the R i defines the set {x} = {x a , x b ; x 1 , · · · , x n+1 }. The field species and the momenta are denoted as F({x}) = {F(x a ), F(x b ) ; F(x 1 ), · · · , F(x n+1 )} and {p a , p b ; p 1 , · · · , p n+1 }. Then we check whether the process R i can have the splittings (3), (4), (6), and (7), shown at Fig. 6 in Appendix A.7. We start with the splitting (3) including the leg-a (x a ). When the process R i can have the splitting (3), a pair (x a , x i ) is chosen and the new element, x ai , is created with the field species, F(x ai ), which is the species of the root of the splitting, in the present splitting (3), a quark. The reduced Born process B1 is taken at the same one as determined for the dipole terms in the DSA. The B1 associates the set {y} = {y a , y b ; y 1 , · · · , y n }, the field species F({y}) = {F(y a ), F(y b ) ; F(y 1 ), · · · , F(y n )}, and the momenta, P({y}) = {P(y a ), P(y b ) ; P(y 1 ), · · · , P(y n )}. There are two possible cases, , which are denoted as y emi = y a or y emi = y b , respectively. For both of the cases the collinear subtraction terms with the leg-a (x a ) are created aŝ where the S B1 is the symmetric factor of the Born process B1 and the Altarelli-Parisi splitting . The symbol B1 represents the square of the matrix elements of the process B1 after the spin-color summed and averaged in d-dimension, which is written with the input momenta as B1 = | M B1 ( P(y a ), P(y b ) → P(y 1 ), · · · , P(y n ) ) | 2 . (2.54) In the case, y emi = y a or y emi = y b , the input momenta in the initial state are determined as (P(y a ), P(y b )) = (xp a , p b ) or (p b , xp a ), respectively. It is noted that when the final state of the R i includes the identical fields, one kind of splitting has some possible pairs (x a , x i ) as many as the number of the identical fields. In the case only one pair is taken and the other pairs must be discarded. For instance, the process has three pairs (x a , x 1 ), (x a , x 2 ), and (x a , x 3 ), for the splitting (3). Among them only one pair, for instance, (x a , x 1 ), is taken and the others (x a , x 2 ), and (x a , x 3 ), must be discarded.
The discard rule is same as the creation algorithm of the P term in the DSA. The creation algorithm shown above is similarly applied for the leg-b (x b ) and for the other splittings (4), (6), and (7). We summarize the general formulae for the collinear subtraction term for arbitrary process as follows. The collinear subtraction terms consists of the terms with the different splittings, equivalently, the different reduced Born processes aŝ The collinear subtraction term with the reduced Born process Bj and with the leg-a/b is universally written aŝ where the splitting function P F(x a/b )F(y emi ) (x) is determined as shown in Eq. (A.105). The formulae for the collinear subtraction term are collected in Appendix A.7. The third cross section σ P (R i ) is the contribution of the P terms created by the DSA [1].
The input for the creation is taken at a real process R i and the creation of the P terms are ordered by the splittings, (3), (4), (6), and (7), and also by the spectators in final/initial states denoted as -1/2, which are shown at Fig. 8 in Appendix A.9. The partonic cross sectionσ P (R i ) is written in 4-dimension as the universal form, where the factor A 4 is denoted as A 4 = α s /2π and the S B j is the symmetric factor of the process B j . The definition of the factor, , is same as the factor for the cross sectionσ D (R i , P) in Eq. (A.77). The definition of the Lorentz scalar, s x a/b ,yspe , is also same as in Eq. (A.79). The symbol y emi , y spe represents the color correlated Born squared in 4-dimension as y emi , y spe = Bj |T y emi · T yspe | Bj 4 . The formulae for the P term σ P (R i ) are collected in Appendix A.9.
We see one example at the same process as used in Step 1 and 2, R 1 = uū → uūg. The relation to be proved for the process is written as (2.59) As shown at Table 3 in Appendix B.1 the dipole terms include the splittings, (3), (6)u, and (6)ū, which have the reduced Born processes, respectively as Then the relation in Eq. (2.59) is divided into the three independent relations, for the dipj = dip 1, 3u, and 3ū. First we construct the σ D (R 1 , P). We show the expressions only for the terms, σ D (R 1 , P, dip1, x a ) and σ D (R 1 , P, dip3u), for convenience as with the symmetric factor S R 1 = 1. In Eq. (2.64) the color correlated Born squared amplitudes are denoted as where the Lorentz factors are written as s xa, y 1/2 = 2 p a · P(y 1/2 ) and s xa, y b = 2 p a · p b . The corresponding quantities in Eq. (2.65) are written in such a way that the B1 is replaced with the B3u in Eq. (2.66). Next we construct the collinear subtraction term σ C (R 1 ). The algorithm to create the collinear subtraction term is executed as follows. The process R 1 associates the set {x} as . The process R 1 can have the splitting (3) with the leg-a and the pair (x a , x 3 ) is chosen. The reduced Born process B1 associates the set {y} as B1 = u(y a )ū(y b ) → u(y 1 )ū(y 2 ), and the relation, F(x a3 ) = F(y a ) = u, stands. The process R 1 can have also the splitting (6)u with the leg-a and the pair (x a , x 1 ) is chosen. The reduced Born process B3u associates the set {y} as B3u = g(y a )ū(y b ) →ū(y 1 )g(y 2 ) and the relation, F(x a1 ) = F(y a ) = g, stands. Then we write down the collinear subtraction terms aŝ with the symmetric factors S B1 = S B3 = 1. Then we construct the third term σ P (R 1 ). The P terms created by the DSA are summarized at Table 20 in Appendix B.3 in [1]. We write down the terms, σ P (R 1 , dip1, x a ) and σ P (R 1 , dip3u), aŝ Now that all the terms are explicitly written down, we start the calculation from the summation, where the relation, S R 1 = S B1 = 1 is used. We expand the factor with Lorentz scalar as s −ǫ = 1 − ǫ ln s, and expand also the squared amplitude B1 by using the color conservation The use of the color conservation low at the dipole subtraction is explained in the article [2].
Then we obtain the expression aŝ Since the expression is now finite in 4-dimension, it is safely reduced back to 4-dimension, which becomes nothing but the termσ P (R i , dip1, x a ) in Eq. (2.69). In this way the relation,

2.5
Step 4 : Step 4 of the PRA is to prove the relation, The left-hand side in Eq. (2.76) is the third term with the square bracket in Eq. (2.11). We define the three cross sections in Eq. (2.76) as follows.
We first define the cross section σ D (R i , K). The integrated dipole terms are separated into the four terms shown in Eq. (2.10). Among them we have already defined the terms, is defined as the dipole term with the splitting (5) in the category Dipole 2, which is concretely shown at Sec. 2.6. Then the term σ D (R i , K) is defined as the remaining term in Eq. (2.10) as (2.77) All the poles 1/ǫ 2 and 1/ǫ in the integrated dipole term σ D (R i ) are extracted by the three terms, σ D (R i , I/P/dip2). Then the cross section σ D (R i , K) is finite and defined at 4dimension. The term σ D (R i , K) is classified by the splittings shown at Fig. 4 in Appendix A.5.
We start with the category Dipole 1 which includes the splittings (1) For the splittings (3)/(4)-1, the remaining terms V    (6) and (7), respectively. For the splittings (6)/(7)-1, the terms V Next we define the cross section σ K (R i ). The quantity is the contribution of the K terms created by the DSA [1]. The K terms are classified into the category, Dipole 1 and 3/4 with the sub-categories, (3)/(4)-0/1/2 and (6)/(7)-0/2, respectively, as shown at Fig. 9 in Appendix A. 10. The contributions to the partonic cross sections are denoted aŝ This term has a special feature that the factor γ F(yspe) /T 2 F(yspe) is determined by the field species of the spectator, F(y spe ), unlike the other terms.
The third cross section, , is defined as follows. When the cross sectionσ K (R i , dip1, (3)/(4)-1) in Eq. (2.81), which is in the category Dipole 1 with the splitting (3)/(4)-1, has the gluon in the final state of the reduced Born process as the spectator, that is, F(y spe ) = gluon, the factor (2.82) Now all the three cross sections in Eq. (2.76) are defined.
The relation in Eq. (2.76) is divided into the three independent ones, where the argument R i at the cross sections is omitted asσ(R i , X) →σ(X) for the compact notation. The cross section σ D (R i , K, dip1) is reconstructed into the four terms as a, 1 + a, 2 + · · · + a, n + a, b . (2.90) The color conservation low can be applied to the summation and the color correlations are factorized as Step 4 is divided into the two sub-steps as where again the argument R i is abbreviated. The cross section σ D (R i , K, dip3/4) is reconstructed into the two terms as where the right-hand side in Eq. (2.76) does not exist, because the reduced Born process, B1(R 1 ) = uū → uū, does not include any gluon in the final state. The integrated dipole terms σ D (R 1 ) are summarized at Table 3 in Appendix B.1. The dipole terms include the three categories, the Dipole 1, 3u, and 3ū, and the relation in Eq. (2.95) is divided into the three as First we prove the relation for the Dipole 1 in Eq. (2.96). In order to prove it, we have the three steps as shown at Eqs. (2.86)∼(2.88), with leg-a is written as the summation of the possible splittings, We explicitly prove the relations in Eqs. (2.99)∼(2.101) with the leg-a as follows. The relations with the leg-b are also similarly proved. Theσ D (R 1 , K, dip1, x a ) is reconstructed into the four terms in Eq. (2.89) as where each term is written aŝ Here the argument x a is suppressed. Next we write down the cross sections of the K terms, Then we execute from the step 4-1. We calculate the summation, is calculated aŝ is written as the summation, which are reconstructed aŝ The two terms are written aŝ Referring to the K terms with splittings (6)u-0/2 at Table.20 in [1], the contributions of the K terms are written down aŝ (2.120) Then we start from the step 4-1. The left-hand in Eq. (2.113) is calculated aŝ Step 2 and 4 in Sec. 2.3 and 2.5 respectively. We here define the cross section σ D (R i , dip2) as follow.
The term σ D (R i , dip2) is nothing but the integrated dipole terms in the category Dipole 2 with the splitting (5)-1/2 at Fig. 1 in Appendix A.2. The term is written as the summation of the two terms with the splittings (5)-1 and -2, The expressions for the two terms are shown in Eqs. (A.96) and (A.97) respectively in Appendix A.6. For the use at the last step, Step 6 we reconstruct the two terms into the two different ones aŝ The first termσ D (R i , dip2, (5)-1/2, V ff ) is defined in d-dimension with the color correlated Born squared amplitude, [ y emi , y spe ] = (s y emi , yspe ) −ǫ · y emi , y spe d . The second term σ D (R i , dip2, (5)-2, h) is finite at 4-dimension, and so the squared amplitude is defined at 4-dimension as y emi , y spe 4 . The formulae for the term σ D (R i , dip2) are collected in Appendix A.6. As mentioned at the Step 2 and 4, the terms, σ I (R i , (2) -1/2, N f V ff ) and Finally we see one example at the process used in the previous sections, R 1 = uū → uūg. We have seen the results for the R 1 at the Step 2, 3, and 4, in Eqs. (2.48), (2.59), and (2.95) respectively. In Eq. (2.11) we substitute the three terms with the square bracket by the proved three relations, and obtain the expression for the σ subt (R 1 ) as (2.127) Since the process R 1 includes only one gluon in the final state, the first two terms at the right-hand side in Eq. (2.122) do not exist. The cross section σ D (R 1 , dip2) is written as the summation of the two terms as in Eq. (2.123). Referring to Table 3 in Appendix B.1, the two terms in the category Dipole 2 are written aŝ where the reduced Born process is determined as B2 = u(y a )ū(y b ) → g(y 1 )g(y 2 ). The dip2) is reconstructed into the two terms in Eq. (2.124) aŝ

2.7
Step 6 : For a given collider process the set of all the real emission processes is denoted as Step 1∼5 are repeated over all the real processes, R 1 , R 2 , · · · , R n real , and we obtain the corresponding cross sections σ subt (R i ) as Each cross section is obtained in the expression in Eq. (2.122). The Step 6 of the PRA is to prove that the summation of all the cross sections σ subt (R i ) vanishes as We can prove the relation in Eq. (2.133) in a systematic way as follows. For the process R i which has two or more gluons in the final state, we introduce a set, Con(R i ), defined as The element of the process R C,qq i is defined in such a way that two gluons in the final state of the R i are replaced with the qq-pair as We call the process, R C,qq i , the ggqq conjugation of the process R i . The process R C,qq i exists for all the massless quark flavors, for instance, q = u, d, s, c, and b, at the energy scale of the LHC. Then we introduce the cross section σ(Con(R i )) as which leads to the cancellation of the σ(Con(R i )) as σ(Con(R i )) = 0 . (2.138) We introduce the set, {Con(R i )}, which consists of all the possible sets, Con(R i ), for the processes {R i }. We further introduce the set, Self. The set Self contains the process R i , of which the final state includes one or no gluon, and the final state includes no qq-pair.
As mentioned at the previous sections, for any element R i of the set Self the cross section σ subt (R i ) itself vanishes as Using the sets, {Con(R i )} and Self, the left-hand side in Eq. (2.133) is always reconstructed where the symbol, {Con(R i )} , represents the summation over all the sets, Con(R i ), of the set {Con(R i )}, and the symbol, Self ⊃ R j , represents the summation over all the processes R j of the set Self. Due to the cancellations, σ(Con(R i )) = 0 and σ subt (R j ) = 0, the relation in Eq. (2.133) is proved. We briefly explain why the terms, σ I (R i , , dip2), is transfered into the K term σ K (R i ), and identified as the term σ K (R i , dip1, (3)/(4) -1, N f h) at the construction of the dipole subtraction by the authors in [2]. The relation is represented in Eq. (2.137). In this way the cancellation σ(Con(R i )) = 0 is understood. The above explanation simultaneously becomes also the explanation why the σ subt (R j ) for any R j ⊂ Self completely cancels inside itself as σ subt (R j ) = 0.
We see one example of the cancellation, σ(Con(R i )) = 0. We take the process, R 8u = uū → ggg, at Table 10 in Appendix B.8 at the dijet process. After the execution of the Step 1∼5, we obtain the cross section σ subt (R 8u ) in Eq. (B.58) aŝ where the two terms are written in Eqs. (B.54) and (B.57) aŝ (2.143) For the process R 8u , the set Con(R 8u ) is determined as Con The gg -uū conjugation of the process R 8u is written as R C,uū 8u = uū → uūg, which is nothing but the process R 1 used at the examples in the previous sections. Then the cross section σ(Con(R 8u )) is written down as , dip2) must be symmetrized about the color factor insertion operators on the identical fields. In the present case the summation in the termσ D (R 1 , dip2) must be symmetrized about the color factor operators, T y 1 and T y 2 , on the two gluon legs, y 1 and y 2 , as [1,2]  Then we prove the cancellation as σ(Con(R 8u )) = 0. We will see the full treatment of the Step 6. i σ subt (R i ) = 0 at the dijet process at Sec. 4. 3 Drell-Yan : pp → µ + µ − + X

Results of the DSA
The subtraction terms, the dipole, I, P and K terms, for the Drell-Yan process have been created by the DSA at Sec. 3 in [1]. Here we summarize the results. There are the three real emission processes as The dipole, I, and P/K terms which belong to the process R 1 , are summarized at Tables 1, 2, and 3, respectively in [1]. The dipole and P/K terms which belong to the process R 2 are summarized at Tables 1 and 3 respectively in [1]. The ones for the R 3 are summarized at Tables 1 and 3 in [1].

Execution of the PRA
For the subtraction terms created by the DSA we execute the PRA as follows. We start with the execution for the process R 1 . Since the final state of the R 1 does not include two or more gluons, nor any qq-pair, the relation to be proved is written as To prove the relation we start from the Step 1. The dipole terms created by the DSA at Table 1 in [1] are converted to the integrated dipole terms shown at Table 1 in the present article. Then we proceed to the Step 2 where we prove the relation, Referring to Table 1 and Appendix A.3, the integrated dipole termσ D (R i , I) is written aŝ Referring to Table 1 and Appendix A.4, the integrated dipole term σ D (R 1 , P) is written aŝ With Appendix A.7 the collinear subtraction term is created aŝ Then we calculate the summation aŝ The summation at 4-dimension is shown to be equal to the P term σ P (R 1 ) which is created at Table 3 in [1] and is written down with Appendix A.9. In this way the relation in Eq. (3.5) is proved. At the Step 4 we prove the relation (3.9) With Table 1 and Appendix A.5, the termσ D (R 1 , K) is written aŝ (3.10) With Table 3 in [1] and Appendix A.10, the termσ K (R 1 ) is written aŝ The present process, R 1 = uū → µ − µ + g, is so simple that we do not have to divide the Step 4 into the sub-steps in Eqs. Next we apply the PRA to the process R 2 = ug → µ − µ + u. The relation to be proved is written as Referring to Table 1 and Appendix A.4, the termσ D (R 2 , P) is written aŝ (3.14) With Appendix A.7 the collinear subtraction term σ C (R 2 ) is created, and with Table 3 in [1] and Appendix A.9, the P term σ P (R 2 ) is written down. It is shown that the three terms, σ D (R 2 , P),σ C (R 2 ), andσ P (R 2 ), satisfy the relation in Eq. (3.13). At the Step 4 we prove the With Table 1 and Appendix A.5, the termσ D (R 2 , K) is written aŝ With Table 3 in [1] and Appendix A.10, the termσ K (R 2 ) is written down, which is shown to be equal to theσ D (R 2 , K) in Eq. (3.16). At the Step 5 we obtain the relation in Eq. (3.12).
In the similar way we can prove the relation for the process R 3 as Finally we come to the last step, Step 6. All the real processes in the set {R i } belong to the set Self as Self = {R 1 , R 2 , R 3 }, and any set Con(R i ) does not exist. Then we calculate the summation of the cross sections σ subt (R i ) as 4 Dijet : pp → 2 jets + X

Results of the DSA
The DSA for the dijet process has been executed at Sec. 4 in the article [1]. We here summarize the results. The real processes are denoted as

Execution of the PRA
We start from the Step 1. The dipole terms σ D (R i ) for the processes, R 1 , · · · , R 11 , at Tables 4∼14 in Appendix B.1 in [1], are converted into the integrated ones at Tables 3∼13 in Appendices B.1∼B.11 in the present article, respectively. The results of the Step 2∼5 for the R 1 , · · · , R 11 , are shown at the next pages to Tables 3∼13 in Appendices B.1∼B.11, respectively. Here we write down the results of the Step 5 aŝ where the expressions for the cross sections at the right-hand sides are all written in Appendix B. Then we proceed to the Step 6. We construct the sets, Con(R i ) and Self, as We introduce also the set as {Con(R i )} = {Con(R 8u ), Con(R 9u ), Con(R 11 )}. Referring to the explicit expressions for theσ subt (R i ) in Appendix B, we can prove the cancellations of the σ(Con(R i )) in Eq. (2.135) for the R i = R 8u , R 9u , and R 11 as σ(Con(R i )) = 0 . (4.7) Among them the cancellation σ(Con(R 8u )) = 0 is concretely shown at Sec. 2.7. Then we obtain the cancellation of the summation σ subt (R i ) as Thus the execution of the PRA for the dijet process is completed. 5 n jets : pp → n jets + X In this section we treat with the three real processes among all ones, which contribute to the collider process pp → n jets + X, as R 1 = uū → (n + 1)-g , and prove the cancellation of the cross section as σ(Con(R 1 )) = 0 at the of the integrated dipole termσ D (R i ). We explain them at the following two paragraphs respectively.
The first notice is about the expressions of the integrated dipole terms converted at the Step 1. When the reduced Born process includes the identical fields, the field mapping, which is determined for each dipole term, has the freedom to choose the the pair of the emitter and the spectator (y emi , y spe ) among the identical fields in the set {y}. Using the freedom of the field mapping we can transform the integrated dipole terms into an identical expression. We call the operation the unification of the integrated dipole terms. We show one example at the process, R 1 = u(x a )ū(x b ) → g(x 1 )g(x 2 ) · · · g(x n+1 ). The creation of the dipole terms starts from the splitting (2)-1, where the reduced Born process is fixed with the set {y} as B1(R 1 ) = u(y a )ū(y b ) → g(y 1 )g(y 2 ) · · · g(y n ) .

(5.2)
Then we can choose the three legs (x i x j , x k ) in the final state in the set {x} as The total number of the pairs is n+1 C 2 · (n − 1). One field mapping is fixed for each pair (x i x j , x k ), and specifies the two legs of the emitter and the spectator in the set {y} as (y emi , y spe ). All the specified pairs (y emi , y spe ) for the pairs (x i x j , x k ) in Eq. (5.3) are written in the expression, (y α , y β ), where the indices α and β take any value among 1, · · · , n, with the condition α = β. Using the freedom of the field mapping we can reconstruct all the field mappings in such a way that they have the identical pair (y emi , y spe ), for instance (y 1 , y 2 ). We call the operation the unification. After the operation of the unification, the summation of the integrated dipole terms in the splitting (2)-1 is rewritten aŝ where the degeneracy factor, n deg , is determined as n deg = n+1 C 2 · (n − 1). We can apply the operation of the unification to all the other integrated dipole terms as well.  Table 14. Referring to Table 14, for instance, we can read out the integrated dipole termσ D ( R 1 , I ) used at the Step 2 aŝ The advantage of the table format in the unification expression is that the length of the table is shorter than the original format. The disadvantage is that the one-to-one correspondences between the original dipole terms and the integrated dipole terms are lost . The format of   Tables 14, 15, and 16, can be a template to show the results in the unification expression.
The second notice is about the symmetric expression of the integrated dipole terms used for the Step 2, 3, and 4. In order that the integrated dipole termsσ D (R i , I/P/K) cancel the I, P, and K termsσ I/P/K (R i ) at the level of the squared amplitude [y emi , y spe ] on each phase space point, the summation of the color correlated Born squared amplitudes [y emi , y spe ] in the integrated dipole termsσ D (R i , I/P/K) must be symmetrized over the legs of the identical fields. We call the operation the symmetrization of the integrated dipole terms. To demonstrate the symmetrization we take the same process R 1 . The reduced Born process B1(R 1 ) in Eq. (5.2) has the n identical fields at the legs, y 1 , · · · , y n . Then the color correlated Born squared amplitude [1,2], for instance, in Eq. (5.5) can be symmetrized over the legs y 1 , · · · , y n as with the condition i = k. We call the operation the symmetrization. The symmetrization is allowed by the freedom of the field mapping over the identical fields. The freedom is the same one by which the operation of the unification is allowed as explained above. When the symmetrization is applied to the whole terms in the integrated dipole termσ D (R 1 , I) in Eq. (5.5), the expression is transformed aŝ where we use the relation of the symmetric factors, S R 1 = (n+ 1) · S B 1 = (n+ 1) ! . The I term σ I (R 1 ) is created by the DSA in such a expression that the factor V gg (ǫ)/2 in Eq. (5.7), is replaced with the factor, V gg (ǫ)/2 + N f V ff (ǫ). Then the integrated dipole termσ D (R 1 , I) in Eq. (5.7) is able to cancel the I termσ I (R 1 ) at the integrand level, which means that on each phase space point of the Φ(B1) d , the color correlated Born squared amplitudes [y emi , y spe ] in the two terms,σ D (R 1 , I) andσ I (R 1 ), cancel each other. In this way we prove the relation at the Step 2 asσ where the termσ I (R 1 , (2)-1/2, N f V ff ) is written in Eq. (C.2). In the similar way at the Step 3 and 4 the integrated dipole termsσ D (R 1 , P/K) are symmetrized and can be canceled against the P/K termsσ P/K (R 1 ) as shown in Eqs. (C.3)/(C.4), respectively.

Summary
In the dipole subtraction procedure we create the subtraction terms and write down the expressions for the phase space integration as shown in Eq. (2.5). While creating the subtraction terms and writing down the expressions, we sometimes have the chance to make mistakes. The main reason is that many subtraction terms exist for the multi-parton processes and each term is not so simple. Among the subtraction terms the singular parts of the dipole and I terms are confirmed by the cancellation against the real and virtual corrections during the calculation of the NLO corrections in Eq. (2.5). The P and K terms are finite and the confirmation by the cancellation during the calculation in Eq. (2.5) is impossible. The summation of all the created subtraction terms must vanish as in Eq. (2.8). We call the relation the consistency relation of the subtraction terms. The proof of the consistency relation provides one confirmation of the P and K terms as well as all the other subtraction terms. The cancellations at the consistency relation are realized between the subtraction terms which have the same initial states and the same reduced Born processes. In the article [1] we presented the dipole splitting algorithm (DSA) which is an algorithm to create the subtraction terms. At the DSA the subtraction terms are classified by the real processes {R i } and the kinds of the splittings. The classification can be translated as the classification by the initial states and the reduced Born processes. Thanks to such a classification at the DSA, we can construct a straightforward algorithm to prove the consistency relation. In this article we presented the proof algorithm (PRA) with the necessary formulae and demonstrated the PRA at the example processes. The PRA is defined in Sec. 2 and all the formulae are collected in Appendix A. The PRA is demonstrated at the Drell-Yan, the dijet, and the n jets processes at Sec. 3, 4, and 5, respectively. The results of the PRA for the dijet and the n jets are summarized at Appendices B and C respectively.
The PRA is the algorithm to prove the consistency of the subtraction terms created by the DSA. We can reduce the number of the potential mistakes by the execution of the PRA. At the Step 1 of the PRA the dipole terms are converted to the integrated dipole terms.
We showed the two templates for the tables to represent the integrated dipole terms at Appendices B and C. At the tables in Appendix B one integrated dipole term corresponds to one original dipole term. At the tables in Appendix C the integrated dipole terms are transformed to the identical expressions as many as possible. The transformed expression is called the unification expression. The executions of the PRA at the Drell-Yan and the dijet processes can be easily completed by hand manipulation. For more complicated multi-parton processes the execution by hand manipulation will be too long and may not be realistic. Then the automation of the PRA as a computer code is wished and may be realized in future. The beginners who start to use the dipole subtraction may have some difficulties to create the subtraction terms and to confirm them. Then the DSA in [1] and the PRA in the present article may serve as the supplementary materials by which the beginners can learn how to create the subtraction terms and how to confirm them. The reason is that the DSA and the PRA are well defined with all the formulae on the documents and what the user does is just to follow the steps in the straightforward algorithm. We hope that the DSA and the PRA will help the users to obtain the reliable predictions at the QCD NLO accuracy.
A Formulae for PRA A.1 Cross sections:σ(R i ) The partonic cross sections in Eq. (2.5) are defined aŝ The phase spaces including the flux factors are defined as The exact definitions of the factors and the symbols are given at the DSA in the article [1].
The jet functions, F (n/n+1) J (p 1 , · · · , p n/n+1 ), must be multiplied to the partonic cross sections in Eqs. (A.1)∼(A.6). For the real correction in Eq. (A.1) the jet function with the (n + 1) fields is multiplied aŝ is multiplied and the n momenta of the arguments are identified with the n reduced momenta (P(y 1 ), · · · , P(y n )).
The use of the jet functions in the dipole subtraction is explained in the article [2]. For compact notation we do not show the jet functions explicitly in the present article.
We summarize the PRA and clarify the places in Appendix A, where the various cross sections appearing in the PRA are defined. The cross section σ subt (R i ) is defined as (A.11) The term σ D (R i ) is the integrated dipole term which is converted at Step 1, and the formulae are collected in Appendix A.2. The integrated dipole term σ D (R i ) is separated into the four terms as The formulae for the four terms, σ D (R i , I/P/K) and σ D (R i , dip2), are collected in Appendix A.3, A.4, A.5, and A.6, respectively. Then the σ subt (R i ) is reconstructed as The formulae for the terms σ C (R i ) and σ I/P/K (R i ), are collected in Appendix A.7, A.8, A.9, and A.10 respectively. At the Step 2, 3, and 4, the following relations are proved, where the terms, σ I (R i , , are defined in Appendix A.8 and A.10 respectively. We substitute the first three terms with square bracket in Eq. (A.13) by the three relations in Eq. (A.14), and obtain the σ subt (R i ) in the expression, At the last step we prove that the summation of all the terms σ subt (R i ) vanishes as Step 6.
The integrated dipole term is universally written aŝ where the overall factor A d is defined as The integrated dipole term is classified into the four types aŝ (A.20) Definition of the symbols Universal singular functions : Phase space : Color correlated Born squared amplitude : [ y emi , y spe ] = (s y emi , yspe ) −ǫ · Bj |T y emi · T yspe | Bj d . (A.26) Lorentz scalar : s y emi ,yspe = 2 P(y emi ) · P(y spe ) .
(A.56) Definition of the symbols Universal singular functions : Color correlated Born squared amplitude : Definition of the symbols Universal singular functions : Color correlated Born squared amplitude : Lorentz scalar : s y emi ,yspe = 2 P(y emi ) · P(y spe ) .
Splitting functions : Definition of the symbols Common factor : A d same as in Eq. (2.16).

Definition of the symbols
Color correlated Born squared amplitude : y emi , y spe same as in Eq. (A.112).
Relations of the functions, K f f /gg (x), V f,f /g,g other (x ; ǫ), and g(x) : The factor T 2 F(yspe) /γ F(yspe) : : F(y spe ) = quark , B Summary for dijet process Step2σ which is separated into the three relations for the Dipole 1, 3u, and 3ū aŝ Step4σ which is separated into the three relations for the Dipole 1, 3u, and 3ū aŝ Step5σ Step3σ which is separated into the two relations for the Dipole 1 and 3u aŝ Step4σ which is separated into the two relations for the Dipole 1 and 3u aŝ Step5σ Step2σ which is separated into the three relations for the Dipole 3u, 4u, and 4ū aŝ which is separated into the three relations for the Dipole 3u, 4u, and 4ū aŝ Step5σ (B.23) Step2σ which includes only the Dipole 1. Step4σ which includes only the Dipole 1.