TF 2 -Contributions to 3-Loop Deep-Inelastic Wilson Coefficients and the Asymptotic Representation of Charged Current DIS

Transcription

1 1/30 TF 2 -Contributions to 3-Loop Deep-Inelastic Wilson Coefficients and the Asymptotic Representation of Charged Current DIS Alexander Hasselhuhn RISC (LHCPhenonet) in collaboration with: J. Ablinger, J. Blümlein, M. Round, C. Schneider, F. Wißbrock

2 2/30 Inhalt Introduction New massive quark contributions and methods 1. Computing Graphs with two massive quark lines of equal masses 2. Moments from graphs with two massive lines of unequal mass 3. Convergent massive 3-loop graphs 4. Massive quarks in charged current DIS at 2 loops Conclusions

3 3/30 Deep-Inelastic Scattering (DIS) DIS [inclusive, unpolarized, electromagnetic] gives a clean probe of the proton substructure. Lepton l Proton P q l } } PF } L µν W µν known (QED) QCD dynamics kinematic variables: Q 2 = q 2, x = Q2 2P.q, y = P.q P.l parametrization of the hadronic tensor with structure functions W µν (q, P, s) = 1 4π = 1 2x d 4 ξ exp(iqξ) P, s [J em µ (ξ), J em ν (0)] P, s ( g µν q ) µq ν q 2 F L (x, Q 2 ) + 2x ( Q 2 P µ P ν + q ) µp ν + q ν P µ Q2 2x 4x 2 g µν F 2 (x, Q 2 ) contributions of massive and massless quarks

4 4/30 Light cone expansion of the current commutator: [Wilson 1969 PR, Zimmermann 1970, Brandt, Preparata 1971 NPB, Frishman 1971 AP] [ ] lim J(ξ), J(0) ( ) Λ c N ξ 2 i,τ (ξ 2, µ 2 ) ξ µ1...ξ µn O µ1...µm i,τ (0, µ 2 2 ) +O 0 }{{}}{{} Q 2. i,n,τ Wilson coefficients local operators At leading twist τ the structure functions factorize F (2,L) (x, Q 2 ) = ( ) C j,(2,l) x, Q2 µ 2, m2 µ 2 j f j (x, µ 2 ) into perturbative Wilson coefficients and nonperturbative parton densities (PDFs). denotes the Mellin convolution f(x) g(x) 1 0 dydz δ(x yz)f(y)g(z), which simplifies to a product upon Mellin transformation ˆf(N) := 1 0 dx x N 1 f(x). following computations in Mellin space

5 5/30 Divide the Wilson coefficients into massless and massive parts: ( ) ) ) C j,(2,l) N, Q2 µ 2, m2 µ 2 = C j,(2,l) (N, Q2 µ 2 + H j,(2,l) (N, Q2 µ 2, m2 µ 2 For Q 2 m 2 (Q 2 10m 2 für F 2 ) the massive Wilson coefficients factorize ( ) H j,(2,l) N, Q2 µ 2, m2 µ 2 = ( ( ) C i,(2,l) N, Q2 m 2 )A µ 2 ij µ 2, N, i [Buza, Matiounine, Smith, van Neerven 1996 NPB] into massless Wilson coefficients C i,(2,l) and massive operator matrix elements (OMEs); OMEs are local operators O i sandwiched between partonic states j = q, g ( ) m 2 A ij µ 2, N = j O i j. The C i,(2,l) are known at 3 loops [Moch, Vermaseren, Vogt, 2005]. ( ) m compute A 2 ij µ, N at 3 loops! 2

6 6/30 Importance of massive quark contributions at 3 loops Massive quark contributions: amount to 20 30% for small x contribute scaling violations which differ in shape from those of massless quarks are sensitive to the gluon and sea quark PDFs for small x allow for the determination of the strange PDF via charged current DIS necessary for the precision determination of α s at 3 loops note: asymptotic representation holds at the 1%-level for F 2

7 7/30 Status of massive quark contributions Leading Order: [Witten 1976, Babcock, Sivers 1978, Shifman, Vainshtein, Zakharov 1978, Leveille, Weiler 1979, Glück, Reya 1979, Glück, Hoffmann, Reya 1982] NLO: [Laenen, van Neerven, Riemersma, Smith 1993] Q 2 m 2 : via IBP [Buza, Matiounine, Smith, Migneron, van Neerven 1996] via p F q s, more compact [Bierenbaum, Blümlein, Klein, 2007] O(α 2 sε)-contributions (für allg. N) [Bierenbaum, Blümlein, Klein, Schneider 2008] [Bierenbaum, Blümlein, Klein 2009] NNLO: Q 2 m 2 moments of F 2 : N = (14) [Bierenbaum, Blümlein, Klein 2009] contributions to transversity: N = [Blümlein, Klein, Tödtli 2009] n f -contributions to F 2 (all-n): [Ablinger, Blümlein, Klein, Schneider, Wißbrock 2011] Known at 3 Loop: A PS qq,q, A qg,q: complete A NS,(TR) qq,q, A gq,q: complete ( A. De Freitas talk) A Qg, A PS Qq, A gg,q: all O(n f TF 2 C A/F )-contributions known A gg,q : O(TF 2 C A/F )-contributions (m 1 = m 2 ) this talk A gg,q : Moments for (m 1 m 2 )-contributions this talk

8 Computing Graphs with two massive quark lines (T 2 F -graphs, m 1 = m 2 ) 8/30

9 9/30 Hypergeometric series for T 2 F -graphs Graphs with two massive lines of equal masses (m 2 1 = m 2 2): Feynman parameterization contains (z 1 x + z 2 y(1 x)) a+bε Mellin-Barnes-Representation B(c + N σ, d + σ) unbalanced factor Γ(c + N σ) sum of residues diverges cannot be cured by subtracting finitely many residues Solution: observe: 1 ( 1 x B(c + N σ, d + σ) = dx x c+n 1 (1 x) d 1 0 x direction for closing the contour depends on x split x-integral into two parts x < 1 2 and x 1 2 no beta function, integrate later close contour differently convergent sum of residues ε-expansion and summation: SumProduction, EvaluateMultiSums, Sigma [C. Schneider] ) σ

10 10/30 Solving the last Feynman parameter integral Idea: solve the last integral in the space of cyclotomic HPLs The occurring cyclotomic HPLs are iterated integrals built from the alphabet: e.g.: H 1,(4,1),0,0 (x) = f 0 (x) = 1 x, f 1(x) = 1 1 x, f 1(x) = x, f (4,0) (x) = x 2, f (4,1)(x) = x 1 + x 2, x 0 dx 1 f 1 (x 1 ) x1 0 dx 2 f (4,1) (x 2 ) 1 2 ln2 (x 2 ). Furthermore, generalized letters occur: f [i,κ] (x) = f i (κx) many properties of cycl. HPLs and cycl. S-Sums are known [Ablinger, Blümlein, Schneider 2011 JMP] and implemented in the package HarmonicSums [J. Ablinger]

11 11/30 Solving the last Feynman parameter integral summation yields cyclotomic S-Sums ˆ= cycl. HPLs S 1 ( x, ) = H 1 (x) S (1,0,1),(2,1,1) ( x, 1; ) = 4H ( ) (4,0) x + H ( ) (4,1) x x 2x ( ) 15 2H (4,0),(4,0) x + 4 use generating function: [f(x)] N integrate 1 1 κf(x), 3 2 H ( ) (4,1) x yield representation a c ah a (κ) such that each term has a Taylor expansion for κ 0 determine N-th Taylor coefficient ( HarmonicSums [J. Ablinger]: Cauchy products, difference equations EvaluateMultiSums, Sigma [C. Schneider])

12 12/30 Results: A scalar graph { Res = ( 1)N ε 2 (N + 1) 1 ε [ S 1 (N) 90(N + 1) + 47N N 2 67N (N 1)N(N + 1) 2 105N 3 175N 2 ( ) + 56N N N + 4 j S 1 (j) ( 13440(N + 1) 2 (2N 3)(2N 1)4 N N 2j ) N j=1 j j 2 j=1 ( 5264N N N ) S 1 (N) (N + 1) 2 (2N 3)(2N 1) + S 3(N) S 2,1 (N) + 7ζ 3 420(N + 1) + ( 2j j 4 j ] ) 7ζ 3 j 3 + S 1(N) 2 + S 2 (N) + 3ζ 2 360(N + 1) P (N 1) 2 N 2 (N + 1) 3 (2N 3)(2N 1) } new nested sums occur

13 13/30 Results: A QCD graph ( ) [ N ( ) ] 2P 4 1 2N 4 j S 1 j N 4 j I 560 = ( 3N(N + 1) 2 (N + 2)(2N 5)(2N 3)(2N 1) 4 N N 2j ) ( j=1 j j 2 2j ) j=1 j j 3 7ζ3 N 2 + N + 2 [ ] + 144S 2,1(N) 36ζ 2S 1(N) 4S 1(N) S 2(N)S 1(N) + 88S 3(N) + 312ζ 27(N 1)N 2 3 (N + 1) 4P (N 2) 2 (N 1) 4 N 5 (N + 1) 4 (N + 2)(2N 5)(2N 3)(2N 1) 64 ( N 2 + N + 2 ) 9ε 3 (N 1)N 2 (N + 1) + 1 { 32P 6 ε 2 27(N 1) 2 N 3 (N + 1) 2 (N + 2) 32 ( N 2 + N + 2 ) } 9(N 1)N 2 (N + 1) S1(N) + 1 { 8P 7 ε 405(N 2)(N 1) 3 N 4 (N + 1) 3 (N + 2) 16P 6 27(N 1) 2 N 3 (N + 1) 2 (N + 2) S1(N) 8( N 2 + N + 2 )( ) S 1(N) 2 3S 2(N) + 3ζ 2 8 ( 55N N 2 52N + 20 ) } 9(N 1)N 2 (N + 1) 15(N 2)(N 1)N(N + 1) 2 (N + 2) ( ) 4P 5S 1(N) 81(N 1) 3 N 4 (N + 1) 3 (N + 2)(2N 5)(2N 3)(2N 1) 4P6 S1(N) 2 3S 2(N) + 3ζ 2 27(N 1) 2 N 3 (N + 1) 2 (N + 2) 4P 1 225(N 2) 2 (N 1) 2 N 2 (N + 1) 3 (N + 2)

14 14/30 T 2 F -contributions to A ggq for m 1 = m 2 inverse binomial sums occur: N 4 j S 1 (j) ) N j 2 j=1 ( 2j j j=1 4 j ( 2j j ) j 3 = 1 removable poles at N = 1/2, N = 3/2, N = 5/2 0 1 dx xn 1 1 dy 1 x x y 1 y [ ] ln(1 y) ln(y) + 2 ln(2) used for the calculation of all scalar graphs consistent with A (3) gg,q ( proof of principle) applied to QCD graphs in the T 2 F -contribution to A(3) gg,q [Ablinger, Blümlein, Hasselhuhn, Round, Schneider 2013] checks with MATAD [M. Steinhauser 2000] graphs are similar to the case m 1 m 2 same ideas are applicable with generalization

15 Moments from graphs with two massive lines of unequal mass (charm and bottom) 15/30

16 16/30 Moments for graphs with two massive lines (m 1 m 2 ) There are 8 different OMES: A Qg, A NS(TR) qq,q, APS Qq, A gg,q, A gq,q }{{} Contain contributions with b- and c-quarks OME # diagrams A (3) Qg 272 A (3),PS Qq 16 A (3),NS qq,q 4 A (3),NS,TR qq,q 4 A (3) gq,q 4 A (3) gg 76 Σ 376 A qg,q, A PS qq,q }{{} Completely known renormalization of the 2-mass case has been performed m 2 c/m 2 b 1/10 expansion parameter But: Charm cannot be treated massless at the scale µ m b.

17 17/30 Moments for graphs with two massive lines (m 1 m 2 ) { a (3) Qg (N = 6) = T F C A ζ3 + ζ x x x )[ + ln µ ζ x x ] x3 + ln 2 2 )[ µ x x ] x3 + ln 3 2 ) µ 2 2 ) ( + ln 2 m 2 )[ ln µ 2 µ x x ] x3 2 )[ + ln µ ) ln2 µ 2 2 ) ( + ln 2 m ln 2 2 ) µ 2 µ ζ x x ] x3 2 ) ln µ ln 2 2 )[ µ x x ] x3 + ln 3 2 ) µ { + TF 2 C F ζ3 + ζ x x x3 2 )[ + ln µ ζ x x ] x3 + ln 2 2 )[ µ x x ] x3 ln 3 2 ) ) µ ln2 µ 2 2 ) ( + ln 2 m 2 )[ ln µ 2 µ x x ] 2 ) x3 + ln 2 µ 2 2 )[ + ln µ ζ x x ] x3 + ln 2 2 )[ µ x x ] x3 ln 3 2 ) µ q2e/exp [Harlander, Seidensticker, Steinhauser 1999] } ln 2 2 ) µ } + O ( x 4 ln 3 (x) ) 2 ) ln µ

18 Convergent massive 3-loop graphs 18/30

19 19/30 The α-representation and graph polynomials Î 4(N) = dα 1 dα 2 dα 3 dα 4 dα 5 dα 6 dα 7 dα 8 N j=0 T N j 4α T j 4b U 2 V 2 T 4α = α 5α 7α 4 + α 2α 3α 5 + α 2α 5α 4 + α 3α 5α 7 + α 2α 5α 8 + α 8α 5α 4 + α 5α 7α 8 + α 2α 3α 8 + α 7α 2α 8 + α 6α 2α 8 + α 3α 7α 2 + α 2α 3α 6 + α 4α 2α 8 + α 2α 6α 4 + α 4α 7α 2 T 4b = + α 2α 5α 4 + α 4α 2α 8 + α 4α 7α 2 + α 2α 5α 8 + α 2α 3α 5 + α 7α 2α 8 + α 3α 7α 2 + α 8α 5α 4 + α 5α 7α 4 + α 4α 1α 8 + α 1α 7α 4 + α 3α 5α 7 + α 5α 7α 8 + α 8α 1α 7 + α 1α 3α 7 U = α 2α 5α 4 + α 2α 3α 5 + α 1α 3α 5 + α 5α 7α 4 + α 1α 6α 4 + α 1α 3α 6 + α 2α 3α 6 + α 2α 6α 4 + α 5α 6α 4 + α 1α 5α 4 + α 3α 5α 7 + α 1α 3α 7 + α 1α 7α 4 + α 3α 7α 2 + α 4α 7α 2 + α 3α 5α 6 + α 2α 3α 8 + α 2α 5α 8 + α 5α 7α 8 + α 8α 5α 4 + α 8α 5α 6 + α 5α 3α 8 + α 1α 8α 5 + α 1α 8α 6 + α 6α 2α 8 + α 1α 8α 3 + α 4α 1α 8 + α 4α 2α 8 + α 7α 2α 8 + α 8α 1α 7 V = α 1 + α 2 + α 3 + α 4 + α 6 + α 7 The integral above is a projective integral, one α-parameter may be set 1 use generating function ( N-th Taylor coefficient from HarmonicSums [J. Ablinger]) For the above example : after applying symmetry transformations α 1 x 1 α 2, α 3 x 2 α 4, α 5 x 5 α 6 α 2, α 4, α 6 are only contained in the operator polynomials and may be integrated out at this stage. Generalize Brown s algorithm for the massive case and local operator insertions.

20 20/30 Six Massive Lines and Vertex Insertion Î 4 = Q 1(N) 2(1 + N) 5 (2 + N) 5 (3 + N) + Q 2(N) 5 (1 + N) 2 (2 + N) 2 (3 + N) 2 ζ3 + ( 1)N ( N + 56N N 3 + N 4 ) S 2(1 + N) 2 (2 + N) 2 (3 + N) ( 24 5N + 2N 2 ) (2 + N) 2 (3 + N) 2 S3 1 2(1 + N)(2 + N)(3 + N) S2 2 + (2 + N)(3 + N) S2 1S 2 Q 4(N) + 4(1 + N) 3 (2 + N) 2 (3 + N) 2 S S5 Q 5(N) 6(1 + N) 2 (2 + N) 2 S3 2S 2, 3 2ζ3S 2 S 2,1S 2 (3 + N) 2 + ( 1)N ( N + 56N N 3 + N 4 ) ( N + 6N 2 ) S (1 + N) 2 (2 + N) 2 (3 + N) 2 2,1 + 2(1 + N)(2 + N)(3 + N) S4 + (5 + N) ζ3s1 (2) (1 + N)(3 + N) Q 6(N) 4(1 + N) 3 (2 + N) 2 (3 + N) 2 S2 ζ3s2 3 2 S3S2 2S2,1S2 + ( N + 190N N 3 + 7N 4 ) S 2(1 + N) 2 (2 + N) 2 2,1 (3 + N) Q 3(N) + (1 + N) 4 (2 + N) 4 (3 + N) 4 S1 (11 + 5N) (1 + N)(2 + N)(3 + N) ζ3s1 Q 7(N) 4(1 + N) 2 (2 + N) 2 S2S1 S2,3 (3 + N) 2 ( N) + 2(1 + N)(2 + N)(3 + N) S3S1 3(3 + 2N) (1 + N)(2 + N)(3 + N) S1S2,1 + ( 79 40N + N 2 ) S3,1 3S4,1 2(1 + N)(2 + N)(3 + N) ( ) + S 2,1, 2 + 2N+1 ( 28 25N 4N 2 + N 3 ) 1 S (1 + N) 2 (2 + N)(3 + N) 2 1,2 2, 1 ( 7 + 2N 2 ) (1 + N)(2 + N)(3 + N) S2,1,1 ( ) +5S 2,2,1 + 6S 3,1,1 + 2N ( 28 25N 4N 2 + N 3 ) 1 S (1 + N) 2 (2 + N)(3 + N) 2 1,1,1 2, 1, 1 (5 + N) (2, 12 ) (1 + N)(3 + N) S1,1,2 (5 + N) (2, 12 ) 2(1 + N)(3 + N) S1,1,1,1

21 Limit N The 2 N factors cancel in the large N limit (asymptotic expansion with HarmonicSums): [ ] Î4 ζ N 10 4N 9 20N N N N N N N 2 [( ) +ζ N 10 2N 9 2N N N N N N 5 ln(n) 3 2 N 2 ] N N N N N N N N N 2 [( ) ζ N 10 N 9 N N N 65 6 N N 5 4 N + 1 ln 2 (N) 3 N 2 ( ) N N N N N N N N 1 ln(n) 3 N 2 ] N N 9 840N N N N N N N 2 ( ) N 10 12N N 8 12N N N N N + 1 ln 3 (N) 3 6 N 2 ( N N N N N N N N + 1 ) ln 2 (N) 3 2 N 2 ( N N N N N N N N N N N N N N N N N N + O(N 11 ) 2 ) ln(n) 21/30

22 Massive quarks in charged current DIS at 2 loops 22/30

23 23/30 Structure functions of charged current DIS The cross section is parameterized introducing 3 structure functions. For symmetry reasons consider the combinations dσ ν dσ ν dxdy ± dxdy = G2 F s 4π { (1 + (1 y) 2 )F W + ±W 2 y 2 F W + ±W +(1 (1 y) 2 )xf W + ±W 3 The structure functions factorize in PDFs and Wilson coefficients { F W + +W ( Vdu 2 = 2 2 (d + d) + V su 2 (s + s) + V u (u + ū) ) (C W + +W,NS 2,q + L W + +W,NS + ( V dc 2 (d + d) + V sc 2 (s + s) ) [ ] H W + +W,NS 2,q + 2V c H W,PS 2,q Σ + H2,gG W [ ]} + 2V u (C W,PS 2,q + L W,PS 2,q )Σ + (C2,g W + LW 2,g )G, F W + +W 3 = 2 L }. 2,q ) { ( Vdu 2 (d + d) + V su 2 (s + s) V u (u + ū) ) (C W + +W,NS 3,q + L W + +W,NS 3,q ) + ( V dc 2 (d + d) + V sc 2 (s + s) ) [ H W + +W,NS 3,q + 2V c H W,PS 3,q Σ + H3,gG W where V i := V id 2 + V is 2, i = u, c, and Σ = q (q + q). ]}. factors ( 1) due to charge antisymmetry of corresponding contributions.

24 24/30 2-Loop corrections in the region Q 2 m 2 At 2-Loop order we derived the asymptotic representation using the transition to the variable flavor number scheme: L W + ±W,NS,(2) 2,q = A NS,(2) H W + ±W,NS,(2) qq,q 2,q = A NS,(2) qq,q + CW + ±W,NS,(2) 2,q (n f + 1) C W + ±W,NS,(2) 2,q (n f ), + CW + ±W,NS,(2) 2,q (n f + 1), L W,PS,(2) 2,q = C W,PS,(2) 2,q (n f + 1) C W,PS,(2) 2,q (n f ) = 0, H W,PS,(2) 2,q = 1 2 APS,(2) Qq + C W,PS,(2) 2,q (n f + 1), L W,(2) 2,g = A (1) gg,q CW,(1) 2,g (n f + 1) + C W,(2) 2,g (n f + 1) C W,(2) 2,g (n f ), H W,(2) 2,g = A (1) gg,q CW,(1) 2,g (n f + 1) + C W,(2) 2,g (n f + 1) + 1 ( ) A (2) Qg A(1) Qg CW +W,NS,(1) 2,q (n f + 1), 3,q = A NS,(2) + + CW ±W,NS,(2) 3,q (n f + 1) C W + ±W,NS,(2) 3,q (n f ), L W + ±W,NS,(2) H W + ±W,NS,(2) qq,q 3,q = A NS,(2) qq,q H W,PS,(2) 3,q = 1 2 APS,(2) H W,(2) 3,g = CW + ±W,NS,(2) 3,q (n f + 1), Qq, ( A (2) Qg A(1) Qg CW + +W,NS,(1) 3,q (n f + 1) ).

25 25/30 2-Loop corrections in the region Q 2 m 2 A previous drivation [Buza, van Neerven 1997 NPB] was corrected and completed. Using OMEs and massless Wilson coefficients from the literature, the massive Wilson coefficients in Mellin space and x-space have been constructed. The Mellin transformation implemented in HarmonicSums [J. Ablinger] was used. A FORTRAN implementation for the application to experimental data will be published soon. [Blümlein, Hasselhuhn, Pfoh 2013] The asymptotic representation is accurate since experimentally Q GeV 2.

26 26/30 Results W F 2, c ABM11_3n_NNLO Q GeV 2 Q GeV 2 Q 2 10 GeV 2 LO NLO NNLO x

27 27/30 Results W x F 3, c ABM11_3n_NNLO Q GeV 2 Q GeV 2 Q 2 10 GeV 2 LO NLO NNLO x

28 28/30 Results x F 3 W x F3 W ABM11_3n_NNLO Q GeV 2 Q 2 10 GeV 2 LO NLO NNLO x

29 29/30 Conclusions Heavy flavor contributions are needed at 3-loop order for the precise determination of α s from DIS data. 5 out of 8 massive operator matrix elements have been completed recently. A method for the calculation of graphs with two massive lines of equal masses and operator insertions was presented, and applied to the corresponding contributions to A (3) gg,q. The calculations will be finished soon. [Ablinger, Blümlein, Hasselhuhn, Round 2013] The method can be generalized to the case of unequal masses. The moments for N = 2, 4, 6 for all graphs with two quark lines of unequal masses are now known, as well as the renormalization of these contributions. Convergent massive three loop graphs can be calcualted in terms of iterated integrals, allowing for a study of the functions governing the more complicated topologies like ladder graphs, benz graphs and V-graphs.

30 30/30 Conclusions On the phenomenological side, the complete 2-loop heavy flavor contributions to charged current DIS were derived from light flavor Wilson coefficients and massive OMEs. They are implemented into a Mellin space program in FORTRAN and will be published soon [Blümlein, Hasselhuhn, Pfoh 2013]. The calculations of new 3-loop corrections can take advantage of rigorous computer algebra packages of our friends at RISC; at the same time techniques known to physicists or newly developed on practical problems lead to improvements in general purpose programs.