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CHIRON v0.51 Manual and User Guide

1. Bbeps Bbnum Bo ET EJ Cbeps Cb 21 CirandomlargeNc2 CirandomlargeNc Cirandom ci 19 DINTEGRAL LiCiBE14 dat LirandomlargeNc2 LirandomlargeNc Lirandon I8 Li 18 SINTEGRAL 25 WINTEGRAL changescale dcomplex 6 feta4L feta4R feta4Vb 38 feta4Vt feta4 33 feta6C feta L feta6R feta6VLb feta6VLt feta6VRb feta6VRt feta6Vb feta6Vt 38 feta6 finitevolumeoneloopintegrals cc finitevolumeoneloopintegrals h fk4L fk4R fk4Vb fK4Vt fk4 fk6C fk6L fk6R 33 fk6VLb fk6VLt 37 fk6VRb fk6VRt fk6Vb fk6Vt fk6 fpi4L fpi4R fpi4Vb fpi4vt 87 fpi4 fpi6C 33 fpi6L 52 fpi6R hh27dVt fpi6VLb 37 hh31 25 fpi6VLt hhVb fpi6VRb hhVt fpi6VRt hhdVb fpi6Vb 3 hhdVt fpi6Vt hhd fpi6 hh getBOmhat getBOms jbdad2 jbdad3 get 0 16 17 jbdbesio 6 getfpimeta cc jbdbesit 6 getfpimeta h 34 jbdbesko 6 getfpimeta4 jbdbesk1 6 getfpimeta6 jbdbesk2 7 getfpimeta jbdbesk3 7 getfpi jbdbesk4 7 getmeta jbdcauch2 getmko jbdcauch getmk jbdgauss2 getmpo jbdgauss 9 getmpi T6 jbdli2 6 getmu jbdguad15 getprecisiononeloopintegrals 23 jbdguad21 10 getprecisionguenchedsunsetintegrals jbdsing15 getprecisionsunsetintegrals jbdsing21 hhiVb jbdtheta2d02 9 hhiVt jbdtheta2dom1 J hhidVb jbdtheta2d0 hhidVt jbdtheta30m1 hhid hhi jbdtheta30
2. hh21dVt misq m2sq m3sq psq L mu2 returns 0 0p HYE mi m2 m2 p L p hh22dVt misq m2sq m3sq psq L mu2 returns 0 0p HP m m m2 p L u hh27dVt misq m2sq m3sq psq L mu2 returns 0 0p HY m m m2 p L u Evaluation using Bessel functions double double double double double double double double double double hhVb misq m2sq m3sq psq L mu2 returns HV m1 m2 m2 p L u hhiVb misq m2sq m3sq psq L mu2 returns H1 m2 m2 m2 p L p hh21Vb misq m2sq m3sq psq L mu2 returns HYF mi m3 m2 p L u hh22Vb m1sq m2sq m3sq psq L mu2 returns HYF mi m3 m2 p L u hh27Vb m1sq m2sq m3sq psq L mu2 returns HY mi m m2 p L u hhdVb n1sq m2sq m3sq psq L mu2 returns 0 0p HV m m2 m2 p L u hhidVb misq m2sq m3sq psq L mu2 returns 9 0p HYE mi m2 m2 p L u hh21dVb misq m2sq m3sq psq L mu2 returns 0 0p HF mi m m2 p L u hh22dVb misq m2sq m3sq psq L mu2 returns 0 0p HE mi m m2 p L u hh27dVb m1sq m2sq m3sq psq L mu2 returns 0 0p HY m m2 m2 p L u For all cases discussed both methods via Bessel or generalized Jacobi theta functions give the same results The derivatives w r t p for all the integrals were compared with taking a numerical derivative Note that the sunset functions at finite volume call the tadpole integrals evaluated with the same method Do not forget to set precision for th
3. p H 2 2 2 2 I _ H 2 2 i Sid 2 pap Mi Ma Ma P H ruryr5 PuPvPp ai mi m5 M3 p A GuvPp GupDv 9pvPy H m m3 m3 p p 20 The needed integrals with s replacing some of the r in the definitions can be related to those without s as descibed in 19 The evaluation of these sunsetintegrals has been done in 19 Further references can be found there We extract the parts the divergent parts and the parts containing C via 2 2 2 pedi 2 1 2 2 2 2 2 2 H mi mb mas p 16m2 02 2 mi Ma mi 1 2 mid log mi u m3 1 log m2 u2 m3 1 log m n p 2 H mi m3 ma p 4 Ole 21 2 2 2 42 eN 1 m m m H m m3 m3 p u 1672 09 4 2 m3 4 8 2m7 m 1 4log m p2 m3 1 4log m3 12 2p 3 E Hj mi m3 m p W ka O e gt 22 2 2 2 2 25 3 1 2 2 2 Hy mj ma m3 p i 1652 2 6 m3 F m3 41 36 3m m 2 12log m n m 2 121og m3 u2 3p 2 HE mi m3 ml p i 0 0 23 2 2 REN EN 1 2 2 2 Hay mj m5 m3 p i 1622 02 8 m3 m3 41 96 4m m 3 24log m n m3 3 241og m3 2 12p 5 HE mi m3 m3 p i 0 9 24 The routines for the sunset integrals calculate the value at p 0 and the derivative there analytically The remainder is then calculated with a rather smoot integral valid below threshold for the doub
4. 7 jbdtheta32 hh21Vb 30 jbdtheta34 8 hh21Vt jbnum1ib hh21dVb 30 jbwgauss2 T3 hh21dVt hh21d hh21 hh22Vb hh22Vt massdecayvevV h hh22dVb massesdecayvev cc hh22dVt massesdecayvev h hh27Vb metal hh27Vt 30 hh27dVb jbwgauss jbwguad15 jbwquad21 lomass meta4R 32 meta4Vb 93 meta4Vt meta4 32 meta6C meta6L meta6R meta6VLb meta6VLt meta6VRb meta6VRt meta6Vb meta6Vt 37 meta6 mkaL 32 mk4R mk4Vb mk4Vt mk4 mk6C mk6L mk6R mk6VLb mk6VLt 36 mk6VRb mk6VRt 36 mk6Vb mk6Vt mk6 mpi4L mpi4R mpi4Vb mpi4Vt mpi4 mpi6C 32 mpi6L mpi R 32 mpi6VLb mpi6VLt mpi6VRb mpi6VRt mpi6Vb mpi6Vt mpi6 oneloopintegrals cc oneloopintegrals h 21 23 out EEEE O physmass qqstrange4L 35 qqstrange4R qqstrange4 qqstrange6C ggstrangeGL qqstrange R qqstrange6 qqup4L qqup4R aqupa 35 qqup6c qqup6L 35 qqup6R qqup6 quarkmass quenchedsunsetintegrals cc quenchedsunsetintegrals h setBOmhat setBOms setci set o I6 7 setfpi setli T8 setmeta setmko setmk setmpo setmpi setmu setname setprecisionfinitevolumeoneloopb setprecisionfinitevolumeoneloopt 28 setprecisionfinitevolumesunsetb setprecisionfinitevolumesunsett 31 setprecisiononeloopintegrals setprecisionguenchedsunsetintegrals setprecisionsunsetintegrals sunsetintegrals cc
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6. S double qqstrange6C physmass Li returns 0 qq 0 S double qqstrange6R physmass returns 0 qq 0 S The functions are defined in massesdecayvev h implemented in massesdecayvev cc and examples of use are in testmassesdecayvev cc 8 2 Masses and decay constants at finite volume The expressions treated in this section have been derived in 26 A general remark is that care should be taken to set the precision in the loop integrals sufficiently high For the one loop integrals setting it very high is usually no problem For the sunset integrals the evaluation can become very slow It is strongly recommended to play around with the settings and compare the outputs for the two ways to evaluate the integral The theta and Bessel function evaluation approach the correct answer differently For most cases it is possible to have rsacc set smaller than racc For many applications it is useful to calculate the very time consuming parts those labeled 6RV once and store them They only depend nontrivially on the masses and size of the finite volume The decay constant dependence is very simple an overall factor at each order and there is dependence on the NLO LECs L7 35 The results presented in this section are with periodic boundary conditions and an infinite extension in the time direction They are also restricted to the case where the particle is at rest i e p 0 8 2 1 Masses at finite volume The finite volume corrections to the ma
7. AAN 8 4 2 3 4 jbdtheta hj 8 1235 jbdtheta2dO 8 4 2 3 0 jbdthetazdUml 4 xke oO ow SSG a Ba 9 4 2 3 jedtheta2d02 9 T 9 4 3 1 One dimensi n tell 9 ewe a x yeu x 9 1d od bo dob SQA WIR duh CLARE mik RG 9 A usuta sos XO gg 304 wx UR EGRE ETC XC 10 43 L3 A ARI 10 AAA AI 10 4 3 2 One dimension real with singularity 11 4 3 2 AA E 11 4 3 2 jbdcauch2 ARRE 11 4 3 2 9 O A Hund El AE EE AN z a a ee eS OR ras es ee E 4 3 3 2 jJbwgauss2 AE 43 3 4 jJbwquadi5 on ex x 30x99 4 x OE ee bi VB 4 3 3 4 jbwquad2l 4 xo Rho oko a 4 3 4 Two dimensions real ia 4 amp amp wwe BRA ay OM Ba ewe Wess 13 41 jbdad2 e 4 3 5 Three dimensions Teal uud sek Se hee A eae E xg Soha a debe BAe k hs d Rear A 5 Chiral Perturbation Theory 6 Data structures 61 Three flavour Oh e aes Baers Se 4e 39 ae d c mob eed Wc Ol EA AI deer ge ne fee EE See eee E ee RA AS 614 NLO LECs Classkil 6 1 5 NLO LECs ClassClil 7 1 Tadpole or one propagator integralj 7 2 Bubbles or two propagator integralsj 7 2 1 MDefinitlong 7 2 2 Analytical implementationj 7 2 3 Numerical implementationj 7 3 Sunset integralj 7 3 1 Definition 2 23223 KA or A s 7 3 2 Functions 7 4 Sunsetintegrals with di
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10. be exactly in the format provided by the output stream checks for equality within relative precision of 1077 An error will occur if any of the data members is zero Defined in inputs h implemented in inputs cc examples of use in testinputs cc 6 1 2 Class lomass lomass mp0in 0 135 mk0in 0 495 f0in 0 090 muin 0 77 mpOin mkOin fOin muin const double lomass const quarkmass mass Private data double mp0 mk0 f0 mu Physical quantities lowest order pion mass lowest order kaon mass lowest order pion decay constant and subtraction scale p Relevant physical case three flavour ChPT isospin limit The constructor from a quarkmass is provided such that conversions can be used Input member functions void setmpO const double mp0in 0 135 void setmkO const double mk0in 0 495 void setfO const double f0in 0 09 void setmu const double muin 0 77 Output member functions exist in two varieties Those that return all or a subset of values using references or those that return one value as the function value void out double amp mpOout double amp mkOout double amp fOout double amp muout double getmpO void double getmk0 void double getf0 void double getmu void 16 Operators defined lt lt gt gt and lt lt and gt gt are defined such that output and input streams work as expected The input stream should be exactly in the format provided by the output stream checks for equality within relat
11. double L returns Bo double feta6VLt const physmass massin const Li Liin const double L returns He double feta6VRt const physmass massin const double L returns RE Eta decay constant Bessel function method double feta4Vb const physmass massin const double L returns Bo double feta6Vb const physmass massin const Li Liin const double L returns BO double feta6VLb const physmass massin const Li Liin const double L returns n double feta6VRb const physmass massin const double L returns pes All these are defined in massdecayvevV h and implemented in massdecayvevV h Exam ples of use are in testmassdecayvevV cc Acknowledgements This work is supported in part by the Swedish Research Council grants 621 2011 5080 and 621 2013 4287 I thank all my collaborators in the various applications for which my version of the program made it into this collection and especially Ilaria Jemos who has tested many of the earlier versions in the course of 15 A GNU GENERAL PUBLIC LICENSE Version 2 June 1991 Copyright 1989 1991 Free Software Foundation Inc 51 Franklin Street Fifth Floor Boston MA 02110 1301 USA Everyone is permitted to copy and distribute verbatim copies of this license document but changing it is not allowed 38 Preamble The licenses for most software are designed to take away your freedom to share and change it By contrast the GNU General Public License is intended to guarantee your freedom to share
12. double fpi4Vt const physmass massin const double L returns FY double fpi6Vt const physmass massin const Li Liin const double L returns h double fpi6VLt const physmass massin const Li Liin const double L returns EVO double fpi6VRt const physmass massin const double L returns EVO Pion decay constant Bessel function method double fpi4Vb const physmass massin const double L returns pie double fpi6Vb const physmass massin const Li Liin const double L returns pV double fpi6VLb const physmass massin const Li Liin const double L returns p double fpi6VRb const physmass massin const double L returns FA Kaon decay constant theta function method double fk4Vt const physmass massin const double L returns pre double fk6Vt const physmass massin const Li Liin const double L returns FY double fk6VLt const physmass massin const Li Liin const double L returns FYO 37 double fk6VRt const physmass massin const double L returns FYO Kaon decay constant Bessel function method double fk4Vb const physmass massin const double L returns F double fk6Vb const physmass massin const Li Liin const double L returns poe double fk6VLb const physmass massin const Li Liin const double L returns FY double fk6VRb const physmass massin const double L returns FS Eta decay constant theta function method double feta4Vt const physmass massin const double L returns EVO double feta6Vt const physmass massin const Li Liin const
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14. input streams work as expected The input stream should be exactly in the format provided by the output stream checks for equality within relative precision of 1077 An error will occur if any of the data members is zero Defined in inputs h implemented in inputs cc examples of use in testinputs cc 17 6 1 4 NLO LECs Class Li Li 11r 0 12r 0 13r 0 14r 0 15r 0 16r 0 17r 0 18r 0 19r 0 110r 0 hir 0 h2r 0 mu 0 77 Name nameless Li const double lir 110r hir h2r mu const string Name Private data double L1r L2r L3r L4r L5r L6r L7r L8r L9r L10r H1r H2r mu and string name Physical quantities the 12 LECs L7 H of which two are so called contact terms of three flavour ChPT as introduced in and the subtraction scale u Relevant physical case three flavour ChPT Input member functions void setli const int n const double lin void setli const double lin const int n Set the value of the LECs with index n n 11 12 correspond to Hr Hor setmu const double muin Sets the scale u to the value muin This does not change the LECs for that use changescale setname const string namein Sets the name of the set of LECs Output member functions double out const int n returns the value of the n th LEC void out exists in many varieties 13 double references and a string returning all private data 13 double references returning all LECS and the subtraction scale 12 double refer ences returning all LEC
15. sunsetintegrals h testfinitevolumeoneloopintegrals cc testgetfpimeta cc testintegralscomplex cc testintegralsreal cc S testintegralsrealsingular cc 54 testmassdecayvev cc testmassdecayvewV cc 37 38 testmassesdecayvev cc testoneloopintegrals cc testquenchedsunsetintegrals cc testsunsetintegrals cc zhhid zhh1 zhh21d zhh21 zhh31 25 zhhd zhh 25 Bubbles Files Guidelines Installation Introduction Loop integrals Ouenched sunsets Sunsets Tadpoles Testroutines 35
16. the argument Uses the recursion relations for Bessel functions and jbdbesk0 and jbdbesk1 Defined in jbnumlib h implemented in jbdbesik cc 4 2 2 6 jbdbesk3 double jbdbesk3 const double x Returns the modified Bessel function K3 for real values of the argument Uses the recursion relations for Bessel functions and jbdbesk0 and jbdbesk1 Defined in jbnumlib h implemented in jbdbesik cc 4 2 2 7 jbdbesk4 double jbdbesk4 const double x Returns the modified Bessel function K for real values of the argument Uses the recursion relations for Bessel functions and jbdbesk0 and jbdbesk1 Defined in jbnumlib h implemented in jbdbesik cc 4 2 3 Theta and related functions 4 2 3 1 jbdtheta30 double jbdtheta30 const double g Returns the value of the function n n so q 1 2 m gas x gel 2 n 1 00 T 00 00 This function is related to the third Jacobi theta function For small g the summation in is used directly For larger q the identity 030 q EY e 3 with A z log q is used instead This is related to the modular invariance for the higher dimensional case Precision can be judged by comparing the two series to each other Same idea as used in the CERNLIB routine DTHETA Defined in jbnumlib h implemented in jbdtheta30 cc 4 2 3 2 jbdtheta30m1 double jbdtheta30m1 const double q Returns the value of the function Oz0 4 1 2 Y q gt g 4 n 1 00 n Z nz0 Implementation as for j
17. without permission under this Public License b To the extent possible if any provision of this Public License is deemed unenforceable it shall be automatically reformed to the minimum extent necessary to make it enforceable If the provision cannot be reformed it shall be severed from this Public License without affecting the enforceability of the remaining terms and conditions c No term or condition of this Public License will be waived and no failure to comply consented to unless expressly agreed to by the Licensor d Nothing in this Public License constitutes or may be interpreted as a limitation upon or waiver of any privileges and immunities that apply to the Licensor or You including from the legal processes of any jurisdiction or authority 49 References CU ma C2 NH N 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 J Bijnens arXiv 1412 0887 hep ph XXX add published XXX http creativecommons org licenses by 4 0 http www gnu org licenses gpl 2 0 htmlhttp www gnu org licenses gpl 2 0 html G t Hooft M J G Veltman Nucl Phys B153 1979 365 401 J Bijnens E Bostr m and T A Lahde JHEP 1401 2014 019 arXiv 1311 3531 hep lat A van Doren and L de Ridder J Comput Appl Math 2 1976 207 217 S Weinberg Physica A 96 1979 327 J Gasser and H Leutwyler Annals Phys 158 1984 142 J Gasser and H Leutwyler Nucl Phys B 250 1985 465 J Bij
18. 12 The definition at two loop order can be found in 18 We define for subtraction purposes 1 d 4 2e Celn 4r tl A c6 M 2A0 C Ag AM C 10 The d dimensional Feynman integrals do not depend directly on the subtraction scale However renormalization will always introduce the correct dependence We define the one loop integrals multiplied by an extra factor of 2 and the two loop integrals with an extra factor of j This introduces the u dependence in the expressions given below References are to places where the integrals are defined and or the method used elaborated 7 1 Tadpole or one propagator integrals These are defined by 4 d d H d q 1 A n m P 27 q EE m2 gt A m y A 1 m B m y A Q m gt C m y A 3 m 11 20 The expansions in e are given by see e g 19 E a a B m i B m i eB m P O C m ji C m2 p e m 2 OE 12 The O e terms are further expanded as m 1 m E Am 9 s 507 ole 7 A m 1 1 m E BS C Clog O Bm m 3 53 5 0873 m u 1 C ae CE s 2 C 2 2 13 ny ges Cmm 13 The analytical expressions are m m Ge _m 1 z 1 m A m u log A log 1 m Se 1 uw 1 m m Bim p 1 log gt B m u log 1 1 1 e 1 1 1 m C m u C 1 14 double Ab const double
19. CHIRON v0 51 Manual and User Guide Johan Bijnens Department of Astronomy and Theoretical Physics Lund University Solvegatan 14A SE 22362 Lund Sweden Manual version of January 2015 Abstract This manual and user guide describes the classes and functions contained in the ChPT program collection CHIRONv0 51which includes the numerical library jbnumlib and the ChPT routine library chiron This text is licensed under the creative commons license CC BY 4 0 http creativecommons org licenses by 4 0 except some parts which are under their own license Contents 3 2 Guidelines 3 21 Main comments stats cx O GR x Re we aOR Wc dE etek AA me 3 2 2 Some caution for use ee sa ae xo das xo ORR X b 4 E EH aS 4 4 A AAA 4 SU see ee Ok A Rae AS dee oe ae Q KU a 5 Dag LE ROE uos x mop w e u Bos ee Ree ee Se DEY Ce S eR eee E 5 4 jbnumlib 5 E gub AAA h eee aa ee eee 5 T oa alee oh SD SG NS ah a Z SAS GP ES ae Sa oe 6 AA SEPTEM 6 4 2 2 Besselfunctions ld 4 4 X9 he eS be BRS ES OR 6 422 L JAP SO lt s Ls es 3 gos o A BAe a we Bea E 6 4 2 2 sise ARETES 6 4 2 2 JDABESKOJ ica lt Ca dre A S E AA re as ee 8 6 42 24 jbdbeski eo as GR erm nO ge ORE 8 3 6 1 225 BHBBSEP v Qux oe Se ees Be dee CE CE E 7 NA ee a ee a ro Qe dock SRR ede a Rec T ar po as IA JA 7 4 2 3 Theta and related functions T 4 2 3 1 A a 7 4 2 3 2 jbdthetadOml spa Book aa wad RE A 8 4 23 5 A
20. Gauss rule for the error estimate Adaptive with a subdivision strategy appropriate for high precision Defined in jbnumlib h implemented in jbdquad15 cc 4 3 1 4 jbdguad21 double jbdguad21 f a b eps f double 4f const double x The double precision function to be integrated over a b eps const double a Lower limit of integration b Upper limit of integration eps Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Uses 21 point Gauss Kronrod rule for the estimate and the difference with the embedded 10 point Gauss rule for the error estimate Adaptive with a subdivision strategy appropriate 10 for high precision Defined in jbnumlib h implemented in jbdguad21 cc 4 3 2 One dimension real with singularity The interface of these routines is identical so they can be simply interchanged For most problems the speed decrases as jbdguad15 or jbdguad21 jbdgauss2 jbdgauss but this is somewhat dependent on the function integrated and the precision reguested An example program that shows the relative speeds is in testintegralsrealsingular cc 4 3 2 1 jbdcauch double jbdcauch f a b s eps f double 4f const double x The double precision function to be integrated over a b s eps const double a Lower limit of integration b Upper limit of integration s Place of the singularity eps Precision attempted to be reached relative preci
21. L At finite volume there are more Lorentz structures possible The tensor t the spatial part of the Minkowski metric g is needed for these The functions for A are Al m L 12 gu Ay m L u tu Ag m L po 28 v In infinite volume Ag is related to A and A54 vanishes The relation in finite volume is given by _ _ s dA a m L y 3A5 m2 L y m AY m L p 29 The full integrals are now split in the infinite volume part which was defined earlier in Sect 7 1 and the finite volume remainder as Aem A m L p r A m2 u A m2 L e As m2 2 AV m L u OE _ BY m L p 5 B m i B m L e B m2 p BV m L 2 OE AV 2 2 Aom A 2 2 AY m2 2 2 Ve 2 2 As m L u 11672 A m u Ago m L As m u Az Mm L p 27 0 ii V AY m L u A 5 m 2 eASs m D O 30 7 5 2 Functions The integration routines needed can be set using the macro DINTEGRAL for the real inte gration default is jbdgauss Any of the similar routines in jbnumlib can be used instead const double msq L msq is m and L is the size L of the spatial dimension Evaluated with theta functions double AbVt msq L returns A m L double BbVt msq L returns B me L double A22bVt msq L returns A m L double A23bVt msq L returns A m2 L Evaluated with Bessel functi
22. PROGRAM TO THE EXTENT PERMITTED BY APPLICABLE LAW EXCEPT WHEN OTHERWISE STATED IN WRITING THE COPYRIGHT HOLDERS AND OR OTHER PARTIES PROVIDE THE PROGRAM AS IS WITHOUT WARRANTY OF ANY KIND EITHER EXPRESSED OR IMPLIED INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE THE ENTIRE RISK AS TO THE QUALITY AND PERFORMANCE OF THE PROGRAM IS WITH YOU SHOULD THE PROGRAM PROVE DEFECTIVE YOU ASSUME THE COST OF ALL NECESSARY SERVICING REPAIR OR CORRECTION IN NO EVENT UNLESS REQUIRED BY APPLICABLE LAW OR AGREED TO IN WRIT ING WILL ANY COPYRIGHT HOLDER OR ANY OTHER PARTY WHO MAY MODIFY AND OR REDISTRIBUTE THE PROGRAM AS PERMITTED ABOVE BE LIABLE TO YOU FOR DAMAGES INCLUDING ANY GENERAL SPECIAL INCIDENTAL OR CONSEQUEN TIAL DAMAGES ARISING OUT OF THE USE OR INABILITY TO USE THE PROGRAM INCLUDING BUT NOT LIMITED TO LOSS OF DATA OR DATA BEING RENDERED INAC CURATE OR LOSSES SUSTAINED BY YOU OR THIRD PARTIES OR A FAILURE OF THE PROGRAM TO OPERATE WITH ANY OTHER PROGRAMS EVEN IF SUCH HOLDER OR OTHER PARTY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGES END OF TERMS AND CONDITIONS 43 Appendix How to Apply These Terms to Your New Programs If you develop a new program and you want it to be of the greatest possible use to the public the best way to achieve this is to make it free software which everyone can redistribute and change under these terms To do so attach the follo
23. allows to rewrite all in terms of B m m3 p u These relations are used for the analytical evaluations given below The final evalaution is done by using a Feynman parameter z to combine the propagators and use the results for the tadpoles The z integral needed can be done analytically or numerically The functions are then all expanded in terms of e The arguments of the various Bubble integrals are not written out A _ a B FcB O e Ao 1 Bu 55 Bi eBi Ole Se Ao 1 Ba 16 23 By cB5 O A m2 me Ao 1 By 16724 By eB3 Ole pi A m m Ba 167 z 9 7 pens sa 7 2 2 Analytical implementation The functions in this section are all implemented fully analytically 22 const double msq misg m2sq psg mu2 these are m m2 m3 p u dcomplex Bb misq m2sg psg mu2 returns B mi m3 p u dcomplex Bb msg psg mu2 returns B m m p u using the simpler equal mass formula dcomplex Bib misq m2sq psg mu2 returns B m m3 p 117 dcomplex B21b misq m2sq psg mu2 returns B m m3 p u dcomplex B22b misg m2sg psq mu2 returns B mj m3 p pu dcomplex B22b msq psg mu2 returns Ba2 m m p u using the simpler equal mass formula Defined in oneloopintegrals h implemented in oneloopintegrals cc examples of use in testoneloopintegrals cc 7 2 3 Numerical implementation The functions in this section are all imlemented
24. ally eguivalent rights anywhere in the world j You means the individual or entity exercising the Licensed Rights under this Public License Your has a corresponding meaning Section 2 Scope a License grant 1 Subject to the terms and conditions of this Public License the Licensor hereby grants You a worldwide royalty free non sublicensable non exclusive irrevocable license to exercise the Licensed Rights in the Licensed Material to A reproduce and Share the Licensed Material in whole or in part and B produce reproduce and Share Adapted Material 2 Exceptions and Limitations For the avoidance of doubt where Exceptions and Limitations apply to Your use this Public License does not apply and You do not need to comply with its terms and conditions 3 Term The term of this Public License is specified in Section 6 a 4 Media and formats technical modifications allowed The Licensor authorizes You to exer cise the Licensed Rights in all media and formats whether now known or hereafter created and to make technical modifications necessary to do so The Licensor waives and or agrees not to assert any right or authority to forbid You from making technical modifications necessary to exercise the Licensed Rights including technical modifications necessary to circumvent Effective Technological Measures For purposes of this Public License simply making modifications authorized by this Section 2 a 4 never produces
25. ass Ci returns El 6 double fpi6R physmass returns FO Kaon decay constant double fk4 physmass Li returns FS double fk4L physmass Li returns FO double fk4R physmass Li returns FS double fk6 physmass Li Ci returns FO double fk6L physmass Li returns FO double fk6C physmass Ci returns FO double fk6R physmass returns FO Eta decay constant double feta4 physmass Li returns ES double feta4L physmass Li returns ES L double feta4R physmass Li returns Ea double feta6 physmass Li Ci returns ES L C double feta6L physmass Li returns P double feta6C physmass Ci returns PY 6 double feta6R physmass returns F g The functions are defined in massesdecayvev h implemented in massesdecayvev cc and examples of use are in testmassdecayvev cc 33 8 1 3 getfpimeta A problem that occurs in trying to compare to lattice QCD is that many of the routines are written in terms of the physical pion decay constant and physical masses In particular the eta mass is treated as physical One thus needs a consistent eta mass and pion decay constant when varying the input pion and kaon mass This assumes we have fitted the LECs L and C7 with a known set of Mr Mx My Fr With that input we can obtain an eta mass and pion decay constant with as input values the original Liin Ciin and the massin The formulas used are and up to order p and p The solution is obtained by iteration and stops when six digits of precisio
26. atisfy the conditions by providing a URI or hyperlink to a resource that includes the reguired information 3 If reguested by the Licensor You must remove any of the information reguired by Section 3 a 1 A to the extent reasonably practicable 4 If You Share Adapted Material You produce the Adapter s License You apply must not prevent recipients of the Adapted Material from complying with this Public License Section 4 Sui Generis Database Rights Where the Licensed Rights include Sui Generis Database Rights that apply to Your use of the Licensed Material a for the avoidance of doubt Section 2 a 1 grants You the right to extract reuse reproduce and Share all or a substantial portion of the contents of the database b if You include all or a substantial portion of the database contents in a database in which You have Sui Generis Database Rights then the database in which You have Sui Generis Database Rights but not its individual contents is Adapted Material and c You must comply with the conditions in Section 3 a if You Share all or a substantial portion of the contents of the database For the avoidance of doubt this Section 4 supplements and does not replace Your obligations under this Public License where the Licensed Rights include other Copyright and Similar Rights Section 5 Disclaimer of Warranties and Limitation of Liability a Unless otherwise separately undertaken by the Licensor to the extent p
27. bdtheta30 but without the 1 Especially for small q often needed to keep accuracy in the finite volume applications in ChPT Defined in jbnumlib h implemented in jbdtheta30 cc 4 2 3 3 jbdtheta32 double jbdtheta32 const double q Returns the value of the function d 04 2 gt gr Y nag Nor so q 5 n 1 00 n 00 00 For small q the summation in is used directly For larger q the derivative of the right hand side of the identity is used Defined in jbnumlib h implemented in jbdtheta32 cc 4 2 3 4 jbdtheta34 double jbdtheta34 const double g Returns the value of the function 034 4 2 gt n ge gt nig az Oxo q 6 n 1 00 N 00 00 For small q the summation in 6 is used directly For larger q the appropriate derivative of the right hand side of the identity is used Defined in jbnumlib h implemented in jbdtheta34 cc 4 2 3 5 jbdtheta2d0 double jbdtheta2d0 const double a const double b const double c Returns the value of the function tante Yi og mane 7 n n25 00 00 There are many higher dimensional generalizations of the Jacobi theta functions The modular invariance properties of these are discussed in App B of and are used in the evaluation to speed up the calculation It should be noted that 00 a b c is fully symmetric in a b c Defined in jbnumlib h and implemented in jbdtheta2d0 cc 8 4 2 3 6 jbdtheta2d0m1 double jbdtheta2d0 const double a const double b co
28. distribution of the Program by all those who receive copies directly or indirectly through you then the only way you could satisfy both it and this License would be to refrain entirely from distribution of the Program If any portion of this section is held invalid or unenforceable under any particular circumstance the balance of the section is intended to apply and the section as a whole is intended to apply in other circumstances It is not the purpose of this section to induce you to infringe any patents or other property right claims or to contest validity of any such claims this section has the sole purpose of protecting the integrity of the free software distribution system which is implemented by public license practices Many people have made generous contri butions to the wide range of software distributed through that system in reliance on consistent application of that system it is up to the author donor to decide if he or she is willing to distribute software through any other system and a licensee cannot impose that choice This section is intended to make thoroughly clear what is believed to be a consequence of the rest of this License 8 If the distribution and or use of the Program is restricted in certain countries either by patents or by copyrighted interfaces the original copyright holder who places the Program under this License may add an explicit geographical distribution limitation excluding those countries so that di
29. e is a file LiCiBE14 dat that contains the last fit of the LECs I6 6 1 5 NLO LECSs Class Ci Ci Cr mu 0 77 Name nameless Ci Ci mu 0 77 Name nameless Ci const double mu const string Name Private data double Cr 95 mu and string name Physical quantities the 94 LECs C7 of which four are so called contact terms of three flavour ChPT as introduced in and the subtraction scale u The C7 are the dimensionless version Scale to the dimensionfull version with appropriate powers of Fo but in practice normally with F Relevant physical case three flavour ChPT Input member functions void setci const int n const double lin void setci const double lin const int n Set the value of the LECs with index n setmu const double muin Sets the scale u to the value muin This does not change the LECs for that use changescale setname const string namein Sets the name of the set of LECs Output member functions double out const int n returns the value of the n th LEC void out exists in many varieties with a double Cit 95 a double reference and a string returning all private data a double Cit 95 a double reference returning all LECS and the subtraction scale and a double Cit 95 returning the LECs only void changescale const double newmu Li amp Liin void changescale Li amp Liin const double newmu This changes the subtraction scale to the new value given by muin and changes the LECs according to th
30. e methods of 19 Further references can be found there The divergent parts and the parts containing C via taking derivatives w r t masses of 21 We thus define the functions HF n m2 m2 m2 p u for all cases above i O blank 1 21 The routines for the sunset integrals calculate the value at p 0 and the derivative there analytically The remainder is then calculated with a rather smoot integral valid below threshold for the double hh functions The functions returning the derivative w r t p calculate the value at p 0 analytically and the remainder via a numerical integration as above An added addition here is that case where the Kahlen function Mm m3 m2 y m m3 m3 4m3m2 vanishes is treated correctly 7 4 2 Functions The integration routines needed can be set using the macro DINTEGRAL for the real inte gration default is jbdgauss Any of the similar routines in jbnumlib can be used instead const int n the integer n labelling the powers of the propagators as defined in Tab const double misq m2sq m3sq psq mu2 these are nmi ma ms p p Valid below threshold double hh n misq m2sq m3sq psq mu2 returns HF n mime m3 p u double hhi n misq m2sq m3sq psq mu2 returns HT n mt ms ms p u double hh21 n misq m2sq m3sq psq mu2 returns Hin mima ms p2 u double hhd n misg m2sq m3sg psg mu2 returns 9 0p H n m2 m3 m2 p y double hhid n misq m2s
31. e running derived in 18 Note that it changes the scale of the values of the NLO LECs L7 in Liin as well Operators defined lt lt gt gt and 19 lt lt and gt gt are defined such that output and input streams work as expected The input stream should be exactly in the format provided by the output stream x allows to multiply a Ci by a double in either order The resulting value has all LECs multiplied by the value of the double and allow to add or subtract set of LECs The resulting value of all LECs is the sum respectively the difference A warning is printed of the scales are different Extra functions Ci Cirandom void Ci CirandomlargeNc void Ci CirandomlargeNc2 void These return a set of random NLO LECs The values are uniformly distributed between 1 167 for Cirandom CirandomlargeNc does the same except that it leaves all LECs that are not single trace terms zero CirandomlargeNc2 does the same but the non single trace terms get a LEC with a random value between 1 3 167 The random numbers are generated using the system generator rand so initializing using something like srand time 0 These latter functions were used in the random walks in the C7 in 15 Defined in Ci h implemented in Ci cc examples of use in testCi cc 7 Loop integrals Loop integrals are done with dimensional regulariztion and we use the standard ChPT variant of MS At one loop order it was defined in I1
32. fferent powers of propagators TAL Definition 7 4 2 Functions 7 5 Finite volume tadpole interalj 7 5 1 Mefinitiongg 45 2 Functions 7 6 Finite volume sunsetintegrals 7 6 1 Definitions 7 6 2 Functions 8 Three flavour isospin conserving results 8 1 Masses decay constants and vacuum expectation values 15 15 15 15 16 17 18 19 20 20 21 21 22 23 23 23 25 25 25 26 27 27 28 29 29 30 31 311 Masses 2 k v vas s 31 8 1 2 Decay constants Sex s ate a s y s w ew oe a RO s ws S sa 32 B getfpimeta 4o ced ex je oes X ERA C YO s A wie d 34 8 1 4 Vacuum expectation values 34 8 2 Masses and decay constants at finite volume j 35 8 2 1 Masses at finite volume 4 24 av ees eee ed awe es 36 8 2 2 Decay constants at finite volume 37 A GNU GENERAL PUBLIC LICENSE 38 B Creative Commons Attribution 4 0 International Public License 45 50 52 1 Introduction This is the manual and user guide for the Chiral Perturbation Theory package CHIRON v0 51 It also defines the functions included in a more extended fashion as compared to the published short description 1 There is obviously a large overlap with that publication The numerical routines are described in Sect 4 The remaining sections a
33. gations under Article 11 of the WIPO Copyright Treaty adopted on December 20 1996 and or similar international agreements 45 e Exceptions and Limitations means fair use fair dealing and or any other exception or limitation to Copyright and Similar Rights that applies to Your use of the Licensed Material f Licensed Material means the artistic or literary work database or other material to which the Licensor applied this Public License g Licensed Rights means the rights granted to You subject to the terms and conditions of this Public License which are limited to all Copyright and Similar Rights that apply to Your use of the Licensed Material and that the Licensor has authority to license h Licensor means the individual s or entity ies granting rights under this Public License i Share means to provide material to the public by any means or process that reguires permis sion under the Licensed Rights such as reproduction public display public performance distribution dissemination communication or importation and to make material available to the public including in ways that members of the public may access the material from a place and at a time individually chosen by them j Sui Generis Database Rights means rights other than copyright resulting from Directive 96 9 EC of the European Parliament and of the Council of 11 March 1996 on the legal pro tection of databases as amended and or succeeded as well as other essenti
34. ghts or contest your rights to work written entirely by you rather the intent is to exercise the right to control the distribution of derivative or collective works based on the Program In addition mere aggregation of another work not based on the Program with the Program or with a work based on the Program on a volume of a storage or distri bution medium does not bring the other work under the scope of this License You may copy and distribute the Program or a work based on it under Section 2 in object code or executable form under the terms of Sections 1 and 2 above provided that you also do one of the following a Accompany it with the complete corresponding machine readable source code which must be distributed under the terms of Sections 1 and 2 above on a medium customarily used for software interchange or b Accompany it with a written offer valid for at least three years to give any third party for a charge no more than your cost of physically performing source dis tribution a complete machine readable copy of the corresponding source code to be distributed under the terms of Sections 1 and 2 above on a medium cus tomarily used for software interchange or c Accompany it with the information you received as to the offer to distribute corresponding source code This alternative is allowed only for noncommercial distribution and only if you received the program in object code or executable form with such an offe
35. i returns me 2 6 double meta6R physmass returns mon The functions are defined in massesdecayvev h implemented in massesdecayvev cc and examples of use are in testmassdecayvev cc 8 1 2 Decay constants The decay constants of the pion kaon and eta at two loops in three flavour ChPT were obtained in 19 The pion and eta decay constants were done earlier with a different subtraction scheme and a different way to perform the sunset integrals in 28 The expressions for the decay constants for a z K 7 are given by Fapnys Fo 1 FO FO 38 32 The superscripts indicate the order of the diagrams in p that each contribution comes from Fo denotes the decay constant in the three flavour chiral limit The expressions were originally derived in 19 but note the description in the erratum of 29 The expressions corrected for the error can be found in the website 13 The normalization is such that F 92 MeV The contributions themselves are divided into the parts depending on the NLO LECs L7 on the NNLO LECs C7 and the remainder as lep er Barer L FO 39 a a a For the 7 the decay constant has been defined with the octet axial vector current Pion decay constant double fpi4 physmass Li returns po double fpi4L physmass Li returns FG T y y En double fpi4R physmass Li returns T 6 T double fpi6 physmass Li Ci returns double fpi6L physmass Li returns El also double fpi6C physm
36. ils The hypothetical commands show w and show c should show the appropriate parts of the General Public License Of course the commands you use may be called something other than show w and show c they could even be mouse clicks or menu items whatever suits your program You should also get your employer if you work as a programmer or your school if any to sign a copyright disclaimer for the program if necessary Here is a sample alter the names 44 Yoyodyne Inc hereby disclaims all copyright interest in the program Gnomovision which makes passes at compilers written by James Hacker signature of Ty Coon 1 April 1989 Ty Coon President of Vice This General Public License does not permit incorporating your program into proprietary programs If your program is a subroutine library you may consider it more useful to permit linking proprietary applications with the library If this is what you want to do use the GNU Library General Public License instead of this License B Creative Commons Attribution 4 0 International Public License By exercising the Licensed Rights defined below You accept and agree to be bound by the terms and conditions of this Creative Commons Attribution 4 0 International Public License Public License To the extent this Public License may be interpreted as a contract You are granted the Licensed Rights in consideration of Your acceptance of these terms and conditions and the Licen
37. ive precision of 1077 An error will occur if any of the data members is zero Defined in inputs h implemented in inputs cc examples of use in testinputs cc 6 1 3 Class quarkmass quarkmass BOmhatin 0 01 BOmsin 0 25 f0in 0 090 muin 0 77 BOmhatin BOmsin fOin muin const double quarkmass const lomass mass Private data double BOmhat BOms f0 mu Physical quantities Born Boms lowest order pion decay constant and subtraction scale ju The quantities Bgm and Bom are the LEC Bo 12 multiplied by the up down quark mass and strange quark mass respectively These are independent of the QCD scale The lowest order pion and kaon masses are given by rn ro V2Bom and mgro y Bol ms Relevant physical case three flavour ChPT isospin limit The constructor from a lomass is provided such that conversions can be used Input member functions void setBOmhat const double BOmhatin 0 01 void setBOms const double BOmsin 0 25 void setfO const double f0in 0 09 void setmu const double muin 0 77 Output member functions exist in two varieties Those that return all or a subset of values using references or those that return one value as the function value void out double amp BOmhatout double amp BOmsout double amp fOout double amp muout double getBomhat void double getBOms void double getf0 void double getmu void Operators defined lt lt gt gt and lt z lt lt and gt gt are defined such that output and
38. le hh functions and with a dispersive method for the dcomplex zhh functions The latter is valid above and below threshold The functions returning the derivative w r t p calculate the value at p 0 analytically and the remainder via a numerical integration as above 24 7 3 2 Functions The integration routines needed can be set using the macro DINTEGRAL for the real inte gration default is jbdgauss and SINTEGRAL for the real integration with a singularity default is jbdcauch Any of the similar routines in jbnumlib can be used instead const double misq m2sq m3sq psq mu2 these are ie m3 qut Valid below threshold double hh misq m2sq m3sq psq mu2 returns HE m m qu m u double hhi misq m2sq m3sq psq mu2 returns HE mia ms m p u double hh21 misq m2sq m3sq psq mu2 returns H mnt m3 m p2 n double hh31 misq m2sq m3sq psq mu2 returns AE ont ma m p u double hhd misq m2sq m3sq psq mu2 returns 0 0p H mi m2 m2 p u double hhid misq m2sg m3sq psq mu2 returns 9 0p HE m3 m2 m2 p u double hh21d misq m2sq m3sq psg mu2 returns 9 0p HZ m2 m2 m3 p u Valid above and below threshold dcomplex zhh misq m2sq m3sq psg mu2 returns HU ntm m3 p u dcomplex zhhi misq m2sq m3sq psq mu2 returns AP m m2 m2 p u dcomplex zhh21 misq m2sq m3sg psg mu2 returns HT my qme m p u dcomplex zhh31 misq m2sq m3sq psq mu2 re
39. le mk6VRb const physmass massin const double L returns me Eta mass theta function method double meta4Vt const physmass massin const double L returns mv 2Note that in other papers the corrections to the mass itself are sometimes quoted 36 double meta6Vt const physmass massin const Li Liin const double L returns mivo double meta6VLt const physmass massin const Li Liin const double L returns me double meta6VRt const physmass massin const double L returns Me Eta mass Bessel function method double meta4Vb const physmass massin const double L returns mO double meta6Vb const physmass massin const Li Liin const double L returns mo double meta6VLb const physmass massin const Li Liin const double L returns mi double meta6VRb const physmass massin const double L returns np All these are defined in massdecayvevV h and implemented in massdecayvevV h Exam ples of use are in testmassdecayvevV cc 8 2 2 Decay constants at finite volume The finite volume corrections to the decay constants are defined as the difference of the decay constant in finite volume and in infinite volume AVE Pook FVO qm V 6 V 6 EVO PY rO 43 These definitions are for a 7 K 7 Note that the correction is defined to the value of the decay constant not divided by the the lowest order decay constant as in 38 The eta decay constant is defined with the octet axial current Pion decay constant theta function method
40. lowest order LO p or next to leading order NLO and p or next to next to leading order NNLO 6 Data structures This section describes a number of classes to deal with input parameters and LECs The default value mechanism of C is used to give them initial values if not specified These are visible below as zvalue in the definitions 6 1 Three flavour ChPT 6 1 1 Class physmass physmass mpiin 0 135 mkin 0 495 metain 0 548 fpiin 0 0922 muin 0 77 mpiin mkin metain fpiin muin const double Private data double mpi mk meta fpi mu Physical quantities pion kaon and eta mass pion decay constant and subtraction scale u Relevant physical case three flavour ChPT isospin limit Input member functions void setmpi const double mpiin 0 135 void setmk const double mkin 0 495 void setmeta const double metain 0 548 void setfpi const double fpiin 0 0922 15 void setmu const double muiin 0 77 Output member functions exist in two varieties Those that return all or a subset of values using references or those that return one value as the function value void out double amp mpiout double amp mkout double amp metaout double amp fpiout double amp muout double getmpi void double getmk void double getmeta void double getfpi void double getmu void Operators defined lt lt gt gt and lt lt and gt gt are defined such that output and input streams work as expected The input stream should
41. msq const double mu2 returns A m gu double Bb const double msq const double mu2 returns B m ju double Cb const double msg const double mu2 returns C m 27 double Abeps const double msq const double mu2 returns md p double Bbeps const double msq const double mu2 returns B m u double Cbeps const double msq const double mu2 returns C m ji double Ab const int n const double msg const double mu2 ine ae A m u B m 2 C m p for n 1 2 3 P Defined in oneloopintegrals h implemented in oneloopintegrals cc examples of use in testoneloopintegrals cc 7 2 Bubbles or two propagator integrals 7 2 1 Definitions We first define the abbreviation _ jo dig X mo oem ee 18 21 The bubble integrals themselves are defined by see e g 20 B mi ma yw 1 B mi ma p i qu pa Bi mj m p i B mj m3 p 4 2 quq lt PuPvBal mj m3 p 4 Gu Bal mi m2 p 12 Buwplmi mz p u qu99p pap pa Ba mi ma p Gime 9upPv drobi Bsa m m2 p p B 16 The methods of can be used to deduce the relations 1 B m ma p u 2p A mi u a A m i Ap m m mi 22 p B mi m p u 1 1 Am 2 2m4B mi ma p H 2 Bx mj m3 p u 2 d 1 m3 m p Bi mi m3 p i 1 Ba mi m5 p p p A m3 u m iB mi m p H 2 dB ml m p 1 17 This
42. n are reached This method was used in to obtain the consistent set of masses and decay constants used there physmass getfpimeta6 const double mpiin const double mkin const physmass massin const Li Liin const Ci Ciin returns a physmass containing mpiin mkin and the calculated compatibe meta fpi with the formulas including order pf i e to NNLO physmass getfpimeta4 const double mpiin const double mkin const physmass massin const Li Liin returns a physmass containing mpiin mkin and the calculated compatibe meta fpi with the formulas including order p i e to NLO The functions are defined in getfpimeta h implemented in getfpimeta cc and examples of use are in testgetfpimeta cc 8 1 4 Vacuum expectation values The corrections to the vacuum expectation values vevs 0 gq 0 for up down and strange quarks in the isospin limit were calculated at two loops in three flavour ChPT in 29 The expression for the up and down quark vev are identical since we are in the isospin limit We write the expressions in a form analoguous to the decay constant treatment 0 Gq 0 apnys Fo Bo 1 0 qq 0 0 qq 0 9 40 The superscripts indicate the order of the diagrams in p that each contribution comes from The lowest order values are F Bo Note that the vevs are not directly measurable quantities They depend on exactly the way the scalar densities are defined in QCD ChPT can be used for them when a massin dependent chiral s
43. n the cms frame tuv guv Pupv p but the separation appears naturally in the calculation 8 In addition it avoids singularities in the limit p 0 29 The finite parts are defined differently from the infinite volume case in 19 The parts with AV are removed here as well The functions HY can be computed with the methods of 8 They are obtained by adding the parts labeled with G and H in Sect 4 3 and the part of Sect 4 4 in 8 The derivatives w r t p can be treated using a simple adaptation of that method The method for evaluation works only below threshold The numerical evaluation is rather slow Playing with the precision settings for the specific case you need is very strongly recommended 7 6 2 Functions const double misq m2sq m3sq psq L mu2 These correspond to m2 m3 m3 p L p Evaluation using theta functions double double double double double double double double double double hhVt misq m2sq m3sq psq L mu2 returns HY m m3 m2 p L u hhiVt misq m2sq m3sq psq L mu2 returns H1 m m2 m2 p L p hh21Vt misq m2sq m3sq psq L mu2 returns HYF m m2 m2 p L 42 hh22Vt misq m2sq m3sq psq L mu2 returns HYF mi m m2 p L u hh27Vt misq m2sq m3sq psq L mu2 returns HY mi m m2 p L u hhdVt misq m2sq m3sq psq L mu2 returns 0 0p HV m2 m2 m2 p L u hhidVt misq m2sq m3sq psq L mu2 returns 0 0p HYE mi m2 m3 p L u
44. nens Prog Part Nucl Phys 58 2007 521 hep ph 0604043 J Bijnens and I Jemos Nucl Phys B 854 2012 631 arXiv 1103 5945 hep ph J Bijnens and G Ecker arXiv 1405 6488 hep ph XXX add published XXX J Bijnens G Colangelo and G Ecker JHEP 9902 1999 020 hep ph 9902437 J Bijnens G Colangelo and G Ecker Annals Phys 280 2000 100 hep ph 9907333 G Amor s J Bijnens and P Talavera Nucl Phys B 568 2000 319 hep ph 9907264 J Bijnens and P Talavera JHEP 0203 2002 046 hep ph 0203049 G Passarino and M J G Veltman Nucl Phys B 160 1979 151 J Bijnens N Danielsson and T A Lahde Phys Rev D 73 2006 074509 hep lat 0602003 J Bijnens and T A Lahde Phys Rev D 72 2005 074502 hep lat 0506004 J Bijnens and T A Lahde Phys Rev D 71 2005 094502 hep lat 0501014 J Bijnens N Danielsson and T A Lahde Phys Rev D 70 2004 111503 hep lat 0406017 J Bijnens and T Rossler arXiv 1411 6384 hep lat XXX add published XXX 50 27 J Bijnens and K Ghorbani Phys Lett B 636 2006 51 hep lat 0602019 28 E Golowich and J Kambor Phys Rev D 58 1998 036004 hep ph 9710214 29 G Amor s J Bijnens and P Talavera Nucl Phys B 585 2000 293 Erratum ibid B 598 2001 665 hep ph 0003258 sl A22bVb A22bVt A23bVb A23bVt AbVb AbVt Abeps Ab Bibnum Bib 23 B21bnun B21b B22bnun B22b B31bnum B32bnum BbVb BbVt
45. ns 4 2 1 jbdli2 dcomplex jbdli2 const dcomplex x Returns the complex dilogarithm or Spence function defined by izje f go 1 t where it converges and analytic continuation Cut defined on the positive real axis from 1 to oo Uses the properties of the dilogarithm to transform the argument and then the Bernouilly series as described in 6 Defined in jbnumlib h and implemented in jbdli2 cc 4 2 2 Bessel functions 4 2 2 1 jbdbesiO double jbdbesiO const double x Returns the modified Bessel function Jp for real values of the argument A simple port to C of CERNLIB routine DBESIO Defined in jbnumlib h implemented in jbdbesik cc 4 2 2 2 jbdbesil double jbdbesil const double x Returns the modified Bessel function T for real values of the argument A simple port to C of CERNLIB routine DBESI1 Defined in jbnumlib h implemented in jbdbesik cc 4 2 2 3 jbdbesko double jbdbeskO const double x Returns the modified Bessel function K for real values of the argument A simple port to C of CERNLIB routine DBESKO Defined in jbnumlib h implemented in jbdbesik cc 4 2 2 4 jbdbeski double jbdbeski const double x Returns the modified Bessel function K for real values of the argument A simple port to C of CERNLIB routine DBESKI Defined in jbnumlib h implemented in jbdbesik cc 4 2 2 5 jbdbesk2 double jbdbesk2 const double x Returns the modified Bessel function K for real values of
46. nset integrals in 28 The expressions for the physical masses for a 7 K 7 are given by Ma phys Mag MAD mao 35 The superscripts indicate the order of the diagrams in p that each contribution comes from The lowest order masses are a 2 m 2Boh mo Do ms ije z M 2m3 36 The higher order contributions are split in the parts depending on the NLO LECs L7 on the NNLO LECs C7 and the remainder as MI mid 4 mO mO 4 2 4 mO 37 31 The expressions for these can be found in and on 13 Note that when combining these with results from other sources one should be sure to use a compatible LO and NLO Pion mass double mpi4 physmass Li returns m2 2 4 double mpi4L physmass Li returns m7 I 2 4 double mpi4R physmass Li returns m double mpi6 physmass Li Ci returns me double mpi6L physmass Li returns mio 2 6 double mpi6C physmass Ci returns m 2 6 double mpi6R physmass returns mM Kaon mass double mk4 physmass Li returns mal double mk4L physmass Li returns mi double mk4R physmass Li returns mid double mk6 physmass Li Ci returns mi double mk6L physmass Li returns m9 double mk6C physmass Ci returns m9 double mk6R physmass returns mi Eta mass double meta4 physmass Li returns m2 double meta4L physmass Li returns m double meta4R physmass Li returns m 2 double meta6 physmass Li Ci returns mo double meta6L physmass Li returns mo double meta6C physmass C
47. nst double c Returns the value of the function abc 1 Y emi poo YO eriam g n1 n2 00 00 ni n2EZ n1 n2 A 0 0 Method as in jbdtheta2d0 but the 1 removed more accurate for small a b c as often needed in finite volume ChPT Defined in jbnumlib h and implemented in jbdtheta2d0m cc 4 2 3 7 jbdtheta2d02 double jbdtheta2d02 const double a const double b const double c Returns the value of the function o T hes o ME 5 00 a b c 9 ni n25 00 00 It should be noted that 0 a b c is symmetric in b c Method similar to jbdtheta2d0 Defined in jbnumlib h and implemented in jbdtheta2d02 cc 4 3 Integration routines 4 3 1 One dimension real The interface of these routines is identical so they can be simply interchanged For most problems the speed decrases as jbdquad15 jbdquad21 jbdgauss2 jbdgauss but this is somewhat dependent on the function integrated and the precision requested The routines do not use the endpoints so an integrable singularity at the endpoint can be done but an integrand transformation that removes the singularity will lead to a much better performance An example program that shows the relative speeds is in testintegralsreal cc 4 3 1 1 jbdgauss double jbdgauss f a b eps f double f const double x The double precision function to be integrated over a b eps const double a Lower limit of integration b Upper limit of integration eps Precision attempted to be reached
48. ons double AbVb msq L returns A m L double BbVb msq L returns B m L double A22bVb msg L returns A m L double A23bVb msq L returns A m2 L The last letter indicates whether they are computed with the theta function or Bessel func tion method The results were checked by comparing against each other and by comparing with the independent Bessel function implementation done in 27 void setprecisionfinitevolumeoneloopt const double Abacc 1e 10 const double Bbacc 1e 9 const bool printout true sets the precision for the fi nite volume integrals evaluated with theta function to Abacc for the tadpole integrals Bbacc for the bubble integrals The last variable printout is a logical variable which can be set to true or false default is false Default values are those indicated void setprecisionfinitevolumeoneloopb const int maxsum 100 const double Bbacc 1e 5 const bool printout true sets the precision for the fi nite volume integrals evaluated with Bessel functions The first argument indicates how far the sum over Bessel functions is taken Maximum at present is 1000 The second argument gives the precision of the numerical integration for the bubble integrals Defined in finitevolumeoneloopintegrals h implemented in finitevolumeoneloop integrals cc examples of use in testfinitevolumeoneloopintegrals cc 28 7 6 Finite volume sunsetintegrals 7 6 1 Definitions The sunset integrals are defined with p
49. ose as well 30 void setprecisionfinitevolumesunsett const double racc 1e 5 const double rsacc 1e 4 const bool printout true The double values sunsetracc and sunsetrsacc set the accuracies of the numerical inte gration needed when one or two loop momenta feel the finite volume Default values are le 5 and 1e 4 respectively The bool variable printout defaults to true and sets whether the setting is printed void setprecisionfinitevolumesunsetb const int maxsum1 100 const int maxsum2 40 racc 1e 5 rsacc 1le 4 printout The integers maxsum1 and maxsum2 give how far the sum over Bessel functions is used for the case with one or two loop momenta feeling the finite volume The first is maximum 1000 the second maximum 40 in the present implementation In the latter case a triple sum is needed hence the much lower upper bound The double values sunsetracc and sunsetrsacc set the accuracies of the numerical integration which is still needed after the sum for both cases For most applications it makes sense to have a higher precision for the case with one loop momentum quantized i e racc smaller than rsacc 8 Three flavour isospin conserving results 8 1 Masses decay constants and vacuum expectation values 8 1 1 Masses The masses of the pion kaon and eta at two loops in three flavour ChPT were evaluated in 19 The pion and eta mass were done earlier with a different subtraction scheme and a different way to perform the su
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51. pt this License since you have not signed it However nothing else grants you permission to modify or distribute the Program or its deriva tive works These actions are prohibited by law if you do not accept this License Therefore by modifying or distributing the Program or any work based on the Pro gram you indicate your acceptance of this License to do so and all its terms and conditions for copying distributing or modifying the Program or works based on it 6 Each time you redistribute the Program or any work based on the Program the recipient automatically receives a license from the original licensor to copy distribute or modify the Program subject to these terms and conditions You may not impose any further restrictions on the recipients exercise of the rights granted herein You are not responsible for enforcing compliance by third parties to this License 7 If as a conseguence of a court judgment or allegation of patent infringement or for any other reason not limited to patent issues conditions are imposed on you whether by court order agreement or otherwise that contradict the conditions of this License they do not excuse you from the conditions of this License If you cannot distribute so as to satisfy simultaneously your obligations under this License and any other pertinent obligations then as a consequence you may not distribute the Program at all For example if a patent license would not permit royalty free re
52. q m3sq psq mu2 returns 9 0p HT n m2 m2 m2 p p 26 double hh21d n misq m2sg m3sg psg mu2 returns 9 0p HE n m2 m2 m p u void setprecisionguenchedsunsetintegrals const double eps sets the precison to eps double getprecisionquenchedsunsetintegrals void returns the present precision The default is 1e 10 Defined in guenchedsunsetintegrals h implemented in quenchedsunsetintegrals cc examples of use in testquenchedsunsetintegrals cc 7 5 Finite volume tadpole interals 7 5 1 Definitions The methods used for these are derived in detail in 8 references to earlier literature can be found there The integrals used here are given in the Minkowski conventions as defined in 26 All of the integrals are available with two different methods one using a summation over Bessel function and the other an integral over a Jacobi theta function The versions included at present are using periodic boundary conditions all three spatial sizes of the same length Z and the time direction of infinite extent The tadpole integrals A and A are defined as se ded dir 1 r r AY m3 L u2 AV Li f ji Palys o 97 m gt 4 H iF MU Lp i T 27 r m The B tadpole integrals are the same but with a doubled propagator The subscript V on the integral indicates that the integral is a discrete sum over the three spatial components and an integral over the remainder The size of the spatial directions is
53. r ifail f double f double x The double precision function to be integrated over x 0 x 1 and x 2 contain the values of the three variables to be integrated over a double al a 0 a 1 and a 2 are the lower limits of integration 14 b double b b 0 b 1 and b 2 are the upper limits of integration releps const double requested relative precision of the integral relerr double amp returns the obtained relative precision via a reference ifail int amp returns an integer Zero indicates success if not zero the routine did not obtain the requested precision The function does a three dimensional integration over a hypercube The underlying routine is jbdadmul which is a simple port to C of the CERNLIB routine DADMUL This in turn was based on 9 Defined in jbnumlib h implemented in jbdadmul cc 5 Chiral Perturbation Theory The classic papers introducing ChPT are 12 References to lectures and intro ductions can be found in 13 Areview at two loop order is 14 The notation used here correspond to the notation introduced by Gasser and Leutwyler B F l7 Bo and Foly 12 for the two and three flavour case In general the decay constants are defined with a normalization of F gt 92 MeV The coupling constants in the higher order Lagrangians are usually referred to as low energy constants LECs Power counting is the usual dimen sional counting with orders referred to as p with alternatively p or
54. r in accord with Subsection b above The source code for a work means the preferred form of the work for making mod ifications to it For an executable work complete source code means all the source code for all modules it contains plus any associated interface definition files plus the scripts used to control compilation and installation of the executable However as a special exception the source code distributed need not include anything that is normally distributed in either source or binary form with the major components compiler kernel and so on of the operating system on which the executable runs unless that component itself accompanies the executable If distribution of executable or object code is made by offering access to copy from a designated place then offering equivalent access to copy the source code from the same place counts as distribution of the source code even though third parties are not compelled to copy the source along with the object code You may not copy modify sublicense or distribute the Program except as expressly provided under this License Any attempt otherwise to copy modify sublicense or distribute the Program is void and will automatically terminate your rights under this License However parties who have received copies or rights from you under this License will not have their licenses terminated so long as such parties remain in full compliance 41 5 You are not required to acce
55. re devoted to the chiron library This manual is released under the creative commons license CC BY 4 0 2 as reproduced in App B except for the parts in App A and App B which have their own licenses The software itself is released under the GNU General Public License GPL version 2 or later which is reproduced in App A Kheiron Xespwv or Chiron was the wisest and eldest of the Centaurs half horse men of Greek mythology His name comes from the Greek word for hand Kheir which is also the origin of the word chiral which is his name was deemed appropriate for this package 3 2 Guidelines 2 1 Main comments Most of these routines were produced during and after scientific research They are licensed under the GPL v2 or later see App 4 or the file COPYING in the main directory so you have very strong rights in using and modifying them However please respect the guidelines as described in the file GUIDELINES in the main directory A summary of these is e Citations are important in the academic world so when using these please both cite the relevant CHIRON publication 1 and the papers where the work itself was done 3 as guoted in the different chapters e Suspected bugs proposed fixes and suggestions should preferably be communicated to the author s so they can be added in future releases e If you distribute modified versions please indicate clearly the modifications in the source and at the point of distrib
56. relative precision if absolute value of the integral is above 1 otherwise absolute precision Subroutine translated from the CERNLIB routine DGAUSS Uses 8 and 16 point Gaus sian rules with the 16 point for the estimate and the difference for the error estimate Adaptive with a subdivision strategy Defined in jbnumlib h implemented in jbdgauss cc 4 3 1 2 jbdgauss2 double jbdgauss2 f a b eps f double 4f const double x The double precision function to be integrated over a b eps const double a Lower limit of integration b Upper limit of integration eps Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Uses 8 and 16 point Gaussian rules with the 16 point for the estimate and the difference for the error estimate Adaptive with a subdivision strategy Very similar to jbdgauss but the subdivision strategy is more appropriate for high precision Defined in jbnumlib h implemented in jbdgauss2 cc 4 3 1 3 jbdguad15 double jbdquadi5 f a b eps f double f const double x The double precision function to be integrated over a b eps const double a Lower limit of integration b Upper limit of integration eps Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Uses 15 point Gauss Kronrod rule for the estimate and the difference with the embedded 7 point
57. rm modification Each licensee is addressed as you Activities other than copying distribution and modification are not covered by this License they are outside its scope The act of running the Program is not restricted and the output from the Program is covered only if its contents constitute a work based on the Program independent of having been made by running the Program Whether that is true depends on what the Program does You may copy and distribute verbatim copies of the Program s source code as you receive it in any medium provided that you conspicuously and appropriately publish on each copy an appropriate copyright notice and disclaimer of warranty keep intact all the notices that refer to this License and to the absence of any warranty and give any other recipients of the Program a copy of this License along with the Program You may charge a fee for the physical act of transferring a copy and you may at your option offer warranty protection in exchange for a fee You may modify your copy or copies of the Program or any portion of it thus forming a work based on the Program and copy and distribute such modifications or work under the terms of Section 1 above provided that you also meet all of these conditions a You must cause the modified files to carry prominent notices stating that you changed the files and the date of any change b You must cause any work that you distribute or publish tha
58. s 11 double references returning L Lio and the subtraction scale and 10 double references returning Z Li void changescale const double newmu This changes the subtraction scale to the new value given by muin and changes the LECs according to the running derived in 12 Operators defined lt lt gt gt and lt lt and gt gt are defined such that output and input streams work as expected The input stream should be exactly in the format provided by the output stream allows to multiply an Li by a double in either order The resulting value has all LECs multiplied by the value of the double and allow to add or subtract set of LECs The resulting value of all LECs is the sum respectively the difference A warning is printed of the scales are different Extra functions Li Lirandom void Li LirandomlargeNc void Li LirandomlargeNc2 void These return a set of random NLO LECs The values are uniformly distributed between 18 1 167 for Lirandom LirandomlargeNc does the same except that it leaves L LZ and 7 zero LirandomlargeNc2 does the same but L Lg and L7 get a random value between 1 3 167 The random numbers are generated using the system generator rand so initializing using something like srand time 0 These latter functions were used in the random walks in the L7 in 15 Defined in Li h implemented in Li cc examples of use in testLi cc In the subdirectory test ther
59. s2 dip d ERA X v 2 A 27 4 27 r2 m2 s m2 r s p m2 31 The subscript V indicates that the spatial dimensions are a discrete sum rather than an integral The conventions correspond to those in infinite volume of and of Sect Integrals with the other momentum s in the numerator are related using the relations shown in 19 which remain valid at finite volume in the cms frame 8 In the cms frame we define the functions H Xv 32 Hj ry v Pa Hy a rury v pip Hy gu Hy T ty Hs The arguments of all functions in the cms frame are mz m3 mi p L u These functions satisfy in finite volume 8 HY AY m3 m3 mt p L e HY mami sp Lr SEV p HY dH 43H mH A m2 A m2 33 The arguments of the sunset functions in the relations if not mentioned explicitly are m ma mj p L pij We split the functions in an infinite volume part H and a finite volume correction HV with HY H 4 HY The infinite volume part has been discussed above For the finite volume parts we define HY O AY mb AY m A m A mi AY m3 A m3 167 1672 SE Ao 1 1 HV 0 2 Ve Ve VF A 16722 4 m A m 2 T 16 332 A m3 A m3 Hy 6 Ao 1 Ve 1 1 HY 0 2 Ve Ve VF Hy mz A m aed gas A Gr AY m3 HHP O A V ly Hy 2 M SA ma 1 1 zs Al FAK Ona ZAG mi HYP o 34 I
60. sion if absolute value of the integral is above 1 otherwise absolute precision Subroutine translated from the CERNLIB 7 routine DCAUCH Uses 8 and 16 point Gaussian rules with the 16 point for the estimate and the difference for the error estimate Adaptive with a subdivision strategy Integrates symmetrically around the singularity so it returns the integral in the sense of the principal value prescription Uses jbdgauss Defined in jbnumlib h implemented in jbdcauch cc 4 3 2 2 jbdcauch2 double jbdcauch2 f a b s eps f double f const double x The double precision function to be integrated over a b s eps const double a Lower limit of integration b Upper limit of integration s Place of the singularity eps Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Subroutine translated from the CERNLIB 7 routine DCAUCH Uses 8 and 16 point Gaus sian rules with the 16 point for the estimate and the difference for the error estimate Adaptive with a subdivision strategy more suitable for high precision Integrates symmet rically around the singularity so it returns the integral in the sense of the principal value prescription Uses jbdgauss2 Defined in jbnumlib h implemented in jbdcauch2 cc 11 4 3 2 3 jbdsing15 double jbdsing15 f a b s eps f double f const double x The double precision function to be integrated over a b s eps cons
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62. sses squared are defined as the difference of the mass squared in finite volume and in infinite volume AYM mi mV mYO07 p10 m2V00 EE mr 4 mv 42 These definitions are for a z K n Pion mass theta function method double mpi4Vt const physmass massin const double L returns m2Y double mpi6Vt const physmass massin const Li Liin const double L returns m double mpi6VLt const physmass massin const Li Liin const double L returns mr double mpi6VRt const physmass massin const double L returns m2 9 Pion mass Bessel function method double mpi4Vb const physmass massin const double L returns mv double mpi6Vb const physmass massin const Li Liin const double L returns mo double mpi6VLb const physmass massin const Li Liin const double L returns mr double mpi6VRb const physmass massin const double L returns mivo Kaon mass theta function method double mk4Vt const physmass massin const double L returns m2 0 double mk6Vt const physmass massin const Li Liin const double L returns mo double mk6VLt const physmass massin const Li Liin const double L returns mo double mk6VRt const physmass massin const double L returns mO Kaon mass Bessel function method double mk4Vb const physmass massin const double L returns me double mk6Vb const physmass massin const Li Liin const double L returns me double mk6VLb const physmass massin const Li Liin const double L returns mY O doub
63. stribution is permitted only in or among countries not thus excluded In such case this License incorporates the limitation as if written in the body of this License 42 9 10 11 12 The Free Software Foundation may publish revised and or new versions of the Gen eral Public License from time to time Such new versions will be similar in spirit to the present version but may differ in detail to address new problems or concerns Each version is given a distinguishing version number If the Program specifies a version number of this License which applies to it and any later version you have the option of following the terms and conditions either of that version or of any later version published by the Free Software Foundation If the Program does not specify a version number of this License you may choose any version ever published by the Free Software Foundation If you wish to incorporate parts of the Program into other free programs whose distribution conditions are different write to the author to ask for permission For software which is copyrighted by the Free Software Foundation write to the Free Software Foundation we sometimes make exceptions for this Our decision will be guided by the two goals of preserving the free status of all derivatives of our free software and of promoting the sharing and reuse of software generally No WARRANTY BECAUSE THE PROGRAM IS LICENSED FREE OF CHARGE THERE IS NO WARRANTY FOR THE
64. t double a Lower limit of integration b Upper limit of integration s Place of the singularity eps Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Subroutine similar to jbdcauch2 but uses a Gauss Kronrod 15 point rule for the estimate and the difference withe embedded 7 point Gauss rule for the error estimate Adaptive with a subdivision strategy more suitable for high precision Integrates symmetrically around the singularity so it returns the integral in the sense of the principal value prescription Uses jbdguad15 Defined in jbnumlib h implemented in jbdsingi5 cc 4 3 2 4 jbdsing21 double jbdsing21 f a b s eps f double f const double x The double precision function to be integrated over a b s eps const double a Lower limit of integration b Upper limit of integration s Place of the singularity eps Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Subroutine similar to jbdcauch2 but uses a Gauss Kronrod 21 point rule for the estimate and the difference withe embedded 10 point Gauss rule for the error estimate Adaptive with a subdivision strategy more suitable for high precision Integrates symmetrically around the singularity so it returns the integral in the sense of the principal value prescrip tion Uses jbdguad21 Defined in jbnumlib h implemen
65. t in whole or in part contains or is derived from the Program or any part thereof to be licensed as a whole at no charge to all third parties under the terms of this License c If the modified program normally reads commands interactively when run you must cause it when started running for such interactive use in the most ordinary way to print or display an announcement including an appropriate copyright notice and a notice that there is no warranty or else saying that you provide a warranty and that users may redistribute the program under these conditions and telling the user how to view a copy of this License Exception if the Program itself is interactive but does not normally print such an announcement your work based on the Program is not required to print an announcement These requirements apply to the modified work as a whole If identifiable sections of that work are not derived from the Program and can be reasonably considered independent and separate works in themselves then this License and its terms do not apply to those sections when you distribute them as separate works But when you distribute the same sections as part of a whole which is a work based on the Program the distribution of the whole must be on the terms of this License whose permissions for other licensees extend to the entire whole and thus to each and every part regardless of who wrote it 40 Thus it is not the intent of this section to claim ri
66. te and or modify the software Also for each author s protection and ours we want to make certain that everyone un derstands that there is no warranty for this free software If the software is modified by someone else and passed on we want its recipients to know that what they have is not the original so that any problems introduced by others will not reflect on the original authors reputations Finally any free program is threatened constantly by software patents We wish to avoid the danger that redistributors of a free program will individually obtain patent licenses in effect making the program proprietary To prevent this we have made it clear that any patent must be licensed for everyone s free use or not licensed at all The precise terms and conditions for copying distribution and modification follow TERMS AND CONDITIONS FOR COPYING DISTRIBUTION AND MODIFICATION 0 This License applies to any program or other work which contains a notice placed by the copyright holder saying it may be distributed under the terms of this General Public License The Program below refers to any such program or work and a work based on the Program means either the Program or any derivative work under copyright law that is to say a work containing the Program or a portion of it either verbatim or with modifications and or translated into another language 39 Hereinafter translation is included without limitation in the te
67. ted in jbdsing21 cc 4 3 3 One dimension complex The interface of these routines is identical so they can be simply interchanged For most problems the speed decrases as jbwguad15 or jbwguad21 or jbwgauss2 jbwgauss but this is somewhat dependent on the function integrated and the precision reguested An example program that shows the relative speeds is in testintegralscomplex cc 4 3 3 1 jbwgauss dcomplex jbwgauss f a b eps 12 f dcomplex 4f const dcomplex x The complex double precision function to be in tegrated over a b const dcomplex a Lower endpoint of integration b Upper endpoint of integration eps const double Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Subroutine translated from the CERNLIB routine WGAUSS Uses 8 and 16 point Gaus sian rules with the 16 point for the estimate and the difference for the error estimate Adaptive with a subdivision strategy The integration is the lineintegral over the straight line between a and b Defined in jbnumlib h implemented in jbwgauss cc 4 3 3 2 jbwgauss2 dcomplex jbwgauss2 f a b eps f dcomplex 4f const dcomplex x The complex double precision function to be in tegrated over a b const dcomplex a Lower endpoint of integration b Upper endpoint of integration eps const double Precision attempted to be reached relative precision if absolute value of the in
68. ted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Subroutinre similar to jbwgauss2 but uses a 21 point Gauss Kronrod rule for the estimate and the difference with the embedded 10 point Gauss rule for the error estimate Adap tive with a subdivision strategy better suited for high precision The integration is the lineintegral over the straight line between a and b Defined in jbnumlib h implemented in jbwguad21 cc 4 3 4 Two dimensions real 4 3 4 1 jbdad2 double jbdad2 f a b releps relerr ifail f double 4f double x The double precision function to be integrated over x 0 and x 1 contain the values of the two variables to be integrated over a double a a 0 and a 1 are the lower limits of integration b double b b 0 and b 1 are the upper limits of integration releps const double reguested relative precision of the integral relerr double amp returns the obtained relative precision via a reference ifail int amp returns an integer Zero indicates success if not zero the routine did not obtain the reguested precision The function does a two dimensional integration over a hypercube The underlying routine is jbdadmul which is a simple port to C of the CERNLIB routine DADMUL This in turn was based on 9 Defined in jbnumlib h implemented in jbdadmul cc 4 3 5 Three dimensions real 4 3 5 1 jbdad3 double jbdad3 f a b releps reler
69. tegral is above 1 otherwise absolute precision Subroutine translated from the CERNLIB Z routine WGAUSS Uses 8 and 16 point Gaus sian rules with the 16 point for the estimate and the difference for the error estimate Adaptive with a subdivision strategy better suited for high precision The integration is the lineintegral over the straight line between a and b Defined in jbnumlib h implemented in jbwgauss2 cc 4 3 3 3 jbwguad15 dcomplex jbwquadi5 f a b eps f dcomplex 4f const dcomplex x The complex double precision function to be in tegrated over a b const dcomplex a Lower endpoint of integration b Upper endpoint of integration eps const double Precision attempted to be reached relative precision if absolute value of the integral is above 1 otherwise absolute precision Subroutinre similar to jbwgauss2 but uses a 15 point Gauss Kronrod rule for the estimate and the difference with the embedded 7 point Gauss rule for the error estimate Adap tive with a subdivision strategy better suited for high precision The integration is the lineintegral over the straight line between a and b 13 Defined in jbnumlib h implemented in jbwguad15 cc 4 3 3 4 jbwguad21 dcomplex jbwguad21 f a b eps f dcomplex 4f const dcomplex x The complex double precision function to be in tegrated over a b const dcomplex a Lower endpoint of integration b Upper endpoint of integration eps const double Precision attemp
70. turns Ae mn m3 m p2 u dcomplex zhhd misq m2sq m3sq psg mu2 returns 9 0p2 H mi m3 m2 p u dcomplex zhhid misq m2sq m3sg psq mu2 returns 9 0p HF m2 m2 m3 p u dcomplex zhh21d misq m2sq m3sg psq mu2 returns 9 0p H m2 m2 m2 p p void setprecisionsunsetintegrals const double eps sets the precison to eps double getprecisionsunsetintegrals void returns the present precision The default is 1e 10 Defined in sunsetintegrals h implemented in sunsetintegrals cc examples of use in testsunsetintegrals cc 7 4 Sunsetintegrals with different powers of propagators 7 4 1 Definition We first define the abbreviation _ ut dir dis X kse i 27 27 4 r may s m2 r s p m2 Vo The translation of n to values for z j k is given in Tab 1 The sunset integrals themselves are defined by H n m Ma m p u Doa 25 m P RR NIN RNA N hh Ha RN NI C Ne NJO NN A N N N OO Table 1 The relation between the value of n and the powers i j k of the three propagators Hiti mi m m p u Tulin p Hjin m jemo m p Ll gt H n mi m2 m3 p u rutv n PyP Ha n UNEN ga Ha2 n mi ma m3 p u 26 The needed integrals with s replacing some of the r in the definitions can be related to those without s as descibed in The evaluation of these sunsetintegrals is by the generalziation of th
71. umlib a and libchiron a and also copy them to the lib subdirectory You might have to change the variables CC CFLAGS and CFLAGTESTS CC should specify the C compiler and the options to be used for everything CFLAGS can be used to specify addi tional options in compiling the libraries and CFLAGTESTS to specify additional options for the testing programs make clean can be used to remove many of the files created during compiling The actual instalation is by putting the contents of the include directory somewhere in the include path of your compiler and the two files libjbnumlib a and libchiron a somewhere in the library path For many C compilers the paths are given in the environment variables CPLUS INCLUDE PATH and LIBRARY PATH respectively 3 3 testroutines For every file xxx h and xxx cc included for chiron there is a testing example code testxxx cc in the subdirectory test These can be compiled using make testxxx in the main directory Executing the resulting file a out should then produce output identical up to the precision specified and possible randomly generated cases to the file testxxx dat in the subdirectory testoutputs 4 jbnumlib 4 1 Complex numbers Complex numbers are defined via the standard C library and an abbreviation provided as typedef std complex lt double gt dcomplex All variables declared complex will be of the this type and referred to as dcomplex in the remainder 4 2 Special functio
72. using a numerical complex integration over x The integration routine used can be specified using the macro WINTEGRAL which defaults to jbwgauss Any of the complex integration routines of jbnumlib can be used instead const double misg m2sg psq mu2 these are m7 m2 p p dcomplex Bbnum misq m2sq psg mu2 returns B mi mig u dcomplex Bibnum misq m2sq psq mu2 returns B m m2 ps u dcomplex B21bnum misq m2sg psg mu2 returns B3i m2 m2 p H dcomplex B22bnum mlsg m2sg psg mu2 returns Bas mi ma p y dcomplex B31bnum misq m2sq psg mu2 returns B3i m m2 p u dcomplex B32bnum misq m2sq psg mu2 returns Bon nd p H 2 Sio ta 2 The precision of the numerical integration can be set and obtained void setprecisiononeloopintegrals const double eps sets the precison to eps double getprecisiononeloopintegrals void returns the present precision The de fault is 1e 10 Defined in oneloopintegrals h implemented in oneloopintegrals cc examples of use in testoneloopintegrals cc 7 3 Sunset integrals 7 3 1 Definition We first define the abbreviation x gt f DuC 5 d x 19 i 2m 4 27 r mi s m3 r s p m3 23 The sunset integrals themselves are defined by H mi m3 m p p 0 gt H mi ma ma p o 2 m P E mi mo ms p u Hol mo mp p m Auf pup Ham m m3 p u gu Ha2 mi mi m p
73. utions However the preferred way to introduce changes is via future releases e To make published results reproducible the exact versions of the code that were used should be kept This includes the values of all parameters used including the precisions 2 2 Some caution for use These routines have been used and tested in a ChPT environment using units in powers of GeV Typical accuracies are set by default to relevant and obtainable values for that case In addition there are often special cases where the routines might not work often due to 0 0 or large cancelations Similar comments apply to the special functions included They are sufficiently accurate for the purposes they were used for originally and usually return values with a precision close to double precision but this is not guaranteed In some cases the large formulas have inherently large cancelations This might lead to degrading of precision in unexpected places Use common scientific sense to judge the quality of the results Finally there are a number of internal functions and extensions already present in the source code but not yet documented in this manual These might change and have not been tested as well as the doumented ones In particular interfaces etc might change 3 Files installation and testroutines The package can be downloaded from 5 There are ready to install libaries there for some cases but in general it is better to compile it for your own s
74. wing notices to the program It is safest to attach them to the start of each source file to most effectively convey the exclusion of warranty and each file should have at least the copyright line and a pointer to where the full notice is found one line to give the program s name and a brief idea of what it does Copyright C yyyy name of author This program is free software you can redistribute it and or modify it under the terms of the GNU General Public License as published by the Free Software Foundation either version 2 of the License or at your option any later version This program is distributed in the hope that it will be useful but WITHOUT ANY WARRANTY without even the implied warranty of MERCHANTABIL ITY or FITNESS FOR A PARTICULAR PURPOSE See the GNU General Public License for more details You should have received a copy of the GNU General Public License along with this program if not write to the Free Software Foundation Inc 51 Franklin Street Fifth Floor Boston MA 02110 1301 USA Also add information on how to contact you by electronic and paper mail If the program is interactive make it output a short notice like this when it starts in an interactive mode Gnomovision version 69 Copyright C yyyy name of author Gnomovision comes with ABSOLUTELY NO WARRANTY for details type show w This is free software and you are welcome to redistribute it under certain conditions type show c for deta
75. ymmetry respecting subtraction scheme is used MS in QCD satisfies this but there are other possibilities Even within a scheme Do and the quark masses depend on the QCD subtraction scale ocn is such a way that Bom is independent of it The higher order corrections in this case also depend on the LECs for fully local counter terms H1 H3 at order p and C C at p When the scalar density is fully defined measuring these quantities in e g lattice QCD and comparing with the ChPT expressions is a well defined procedure 34 The contributions at the different orders themselves are split in the parts depending on the NLO LECs L7 on the NNLO LECs C7 and the remainder as 0jgg 0 9 Olgqloy 0 gg 0 gt nm une 6 EN 6 6 0 qq O olgaloy O qq 0 2 0 qq 0 41 These are defined for q u s 0 74 0 phys double qqup4 physmass Li returns 0 qq 0 2 double qqup4L physmass Li returns 0 qq 0 2 double qqup4R physmass returns 0 qq 0 2 double qqup6 physmass Li Ci returns 0 qq 0 9 6 uL double qqup6C physmass Li returns 0 qq 0 9 double qqup6L physmass Li returns O gq 0 double qqup6R physmass returns 0 qq 0 gt O gq 0 s phys double qqstrange4 physmass Li returns o gg 0 5 double qqstrange4L physmass Li returns 0 qq 0 2 double qqstrange4R physmass returns 0 qq 0 2 double qqstrange6 physmass Li Ci returns oaa double qqstrange6L physmass Li returns 0 gq 0
76. ystem C can have a large overhead in calling classes and functions compared to FORTRAN Therefore always compile the library with optimization The interfaces are as much as possible defined with the keyword const to allow the compiler to optimize more efficiently 3 1 Files The gzipped tarred file chiron vvvv tar gz will produce a directory chiron vvvv with a number of subdirectories vvvv is version information The created directory is called the main directory in the remainder The main directory contains the files COPYING INSTRUCTIONS GUIDELINES and a Makefile The subdirectory contains the documentation The latest published article about CHI RON this manual manual tex a list of files filelist txt and a summary of things added since earlier versions Changelog txt The subdirectory lib will after compiling contain the compiled libraries libjbnumlib a and libchiron a The subdirectory include contains all the needed header files The subdirectory src contains the source files test contains the testing and example programs testoutputs contains the output the testprograms should produce Typically for each subject xxx there are files xxx h xxx cc testxxx cc and testxxx dat in the respective directories There are a few extra files around as well These typically contain inputs needed or large sets of constants 3 2 Installation The main steps are to run make in the main directory This should produce the files libjbn

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