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Analytical Evaluation of Certain On-Shell Two-Loop Three-Point Diagrams
A. I. Davydychev
Moscow / Mainz / Sugar Land

Based on common work with V. A. Smirnov

Typeset by FoilTEX

A.I. Davydychev

One-lo op three-p oint function q M P 0 m p

All external momenta are ingoing, P + p + q = 0. On-shell conditions: P 2 = M 2, p 2 = m2 .

J

dn r r2 [(P + r)2 - M 2] [(p - r)2 - m2]

P 2 =M 2 2 2 p =m

,

where n = 4 - 2 is the space-time dimension.

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A.I. Davydychev

Reduction to a 2-p oint function
A.I. Davydychev and M.Yu. Kalmykov, Nucl. Phys. B605 (2001) 266

In this on-shell limit, the 3-point function reduces to a 2-point function with the external momentum q , masses m and M , unit powers of the propagators, and the space-time dimension 2 - 2,

J=

2

·

q

M m

q
n 2 - 2

To get result for J , we need to expand the 2-point function up to the next term of the expansion.

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A.I. Davydychev

Log-sine function and Nielsen p olylogarithm
Any term of the expansion can be calculated in terms of the log-sine functions Lsj () = -
0

d ln

j -1

2 sin

, 2

whose analytic continuation yields Nielsen polylogarithms S (-1) (z ) = (a - 1)! b!
a + b- 1 0 1

a,b

d

ln

a-1

lnb(1 - z ) .

In particular, S
a,1

(z ) = Li

a+1

(z ) ,

S

0,b

(-1)b b (z ) = ln (1 - z ). b!

Higher terms of the expansion, as well as their analytic continuation, are given in
A.I. Davydychev, Phys. Rev. D61 (2000) 087701; A.I. Davydychev and M.Yu. Kalmykov, Nucl. Phys. B (PS) 89 (2000) 283.
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A.I. Davydychev

One-lo op result
Expanding exact result in and m2, we obtain J|
m0

= i

2-

(1 + )(M 2)

- 1-

m2 1 в- ln 2 + 2 ln 2 M 1 2 m2 1 2 - ln + Li2 (1 - ) + ln M2 2 4 +O(, m2 ln2(m2)), where M2 . 2 2 M -q

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A.I. Davydychev

One-lo op result: the m 0 limit
If we are interested use this result, since To regulate them, we dimensionally-regulated J|
m=0

in the case m = 0, we cannot there are ln m singularities. need to return to the exact representation and put m = 0: F 22 1 -, 1 + 1- , 1-

i 2-(1 + ) 1 =- (M 2 - q 2)1+ 2

where 2F1 is the Gauss hypergeometric function. Expanding in we get J|
m=0

= i

2-

(1 + )(M 2)

- 1-

+ O ( ) .

1 1 12 в - 2 - ln - ln - Li2 (1 - ) 2 2

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A.I. Davydychev

One-lo op result: other contributions
M. Beneke and V.A. Smirnov, Nucl. Phys. B522 (1998) 321; V.A. Smirnov, Phys. Lett. B465 (1999) 226

Adding (2c) contribution, i 2-(1 + ) 1 M 2 ( M 2 - q 2 ) 2 m2 M2
-

2

i 2-(1 + ) = 2 2 M (M - q 2 )

1 1 m2 1 2 m - ln + ln 22 2 M 2 4 M

2 2

+ O ( ) ,

we obtain nothing but the expanded exact answer.

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A.I. Davydychev

Two-lo op diagram with two masses q M M P 0 0 m m p
p 2 = m2 .

On-shell conditions: P 2 = M 2,
Applications:

· QED correction to muon decay · QCD correction to t W + b · QCD correction to t H + b

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A.I. Davydychev

Expansion by regions
F ( q 2 , M 2 , m 2 ; ) = dn k dn l . 2 - 2P l )(l2 - 2pl)(k 2 - 2P k )(k 2 - 2pk )k 2 (k - l)2 (l {M 2, |q 2|}.

We are interested in the case when m2

Choose n1,2 = ( 1 , 0, 0, 1 ), (2n1,2k = k± k0 ± k3), 2 2 2 k = (k1, k2), P = (M , 0) and p = n1 + m n2. The relevant regions are: hard (h), 1-collinear (1c), 2-collinear (2c), ultrasoft (us), kM; k+ m2/M , k- M , k m ; k+ M , k- m2/M , k m ; k m2/M .

List of regions (k , l) generating non-zero contributions to the expansion of F in the leading order: (h-h), (1c-h), (1c-1c) and (us-1c).
V.A. Smirnov, Phys. Lett. B465 (1999) 226
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A.I. Davydychev

Two-lo op planar diagram with m = 0
The (h-h) region generates Taylor expansion of the integrand in m2. In the leading order, this is just the diagram with m = 0:

q M M P 0 0 0 0 p
p2 = 0.

On-shell conditions: P 2 = M 2, Integration by parts?
F.V. Tkachov, Phys. Lett. B100 (1981) 65

K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159

Other approaches?
J. Fleischer, A.V. Kotikov and O.L. Veretin, Nucl. Phys. B547 (1999) 343
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A.I. Davydychev

MellinBarnes representation (m = 0)
Contour integral representation for the vertex with m = 0 and unit powers of all propagators: - в
4- 2

(M ) (1 - 2)

2 - 2- 2

1 (2 i)

i 4 - i

dz dz dt du ~

2+2+z +z ~

(-t) (-u) ~ (-z ) (-z ) (1 - t) (1 - u) (1 + t + u) (1 - 2 + t + u)

~ в(1 + + t + u + z ) (1 + - t - u + z )

в(- - u + z ) (- + u + z ) (- - t - z ) (- + t - z ) . ~ ~ Contour integrals are chosen so as to separate the right and left series of poles of the functions in all four variables z , z , t and u. For small negative values ~ of , this condition can be satisfied even by straight contours (parallel to the imaginary axes), if we choose, say, Rez = Rez = 1 , Ret = and Reu = 1 . ~2 4 How to extract singularities, etc.?
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A.I. Davydychev

Result of the calculation (m = 0)

4 - 2 - 2 E e

(M 2 )

-2-2 2+4

1 1 2 1 2 1 в +2 - ln - Li2 (1 - ) 124 12 4 2 + 1 913 53 3 ln + ln Li2 (1 - ) + 36 12 2 1 +S 2
1 ,2

3 (1 - ) - Li3 (1 - ) 2

179 4 7 2 2 19 4 9 - ln - + ln - ln2 Li2 (1 - ) 1440 24 48 4 7 2 7 5 - Li2 (1 - ) - ln S1,2(1 - ) + ln Li3 (1 - ) 2 2 12 13 7 2 - Li2 (1 - ) - S1,3(1 - ) + S2,2(1 - ) 2 2 5 - Li4 (1 - ) + O() 2 where ,

M2 2 . 2 M -q
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A.I. Davydychev

Collecting all contributions

4- 2 - 2 E e

(M 2 )

- 2- 2 2

1 8

2

m2 ln 2 + 2 ln M

2

1 1 3 m2 1 2 m2 ln - + ln ln + 2 2 M M 6 2

m2 2 1 2 + ln ln 2 12 4 M
3

m2 1 13 2 ln - ln + ln 2 + 2 ln Li2 (1 - ) + + M 6 2 6 13 4 m2 5 3 m2 ln + ln + ln + 2 2 M M 96 12 3 m2 1 2 m2 Li (1 - + 3 ln 2 + ln 22 2 4 M M 1 m2 3 54 1 + ln 2 ln + ln + 4M 24 2

5 2 2 m2 3 2 m2 2 ln + ln ln 2 2 16 M M 8 11 2 m2 ) + ln 2 ln 12 M m2 ln 2 + 6 ln S1,2(1 - ) M

m2 3 1 m2 - ln 2 + 2 ln Li3 (1 - ) + ln 2 ln Li2 (1 - ) M 2 2M 12 7 2 2 + ln Li2 (1 - ) + ln - 3 ln + 4S 2 12 -2S
2 ,2 1 ,3

(1 - )

1 (1 - ) + Li2 (1 - ) 2

2

4 + O ( ) + 72
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A.I. Davydychev

Conclusion

New class of 3-point 2-loop diagrams that can be analytically calculated in terms of the standard polylogarithmic functions. This suggests that other similar contributions (crossed diagrams, etc.) can also be calculated.

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