Mathematica and Fortran programs for various analytic QCD couplings
MMathematica and Fortran programs for variousanalytic QCD couplings C´esar Ayala and Gorazd Cvetiˇc
Department of Physics, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso,ChileE-mail: [email protected]
Abstract.
We outline here the motivation for the existence of analytic QCD models, i.e.,QCD frameworks in which the running coupling A ( Q ) has no Landau singularities. Theanalytic (holomorphic) coupling A ( Q ) is the analog of the underlying pQCD coupling a ( Q ) ≡ α s ( Q ) /π , and any such A ( Q ) defines an analytic QCD model. We present thegeneral construction procedure for the couplings A ν ( Q ) which are analytic analogs of thepowers a ( Q ) ν . Three analytic QCD models are presented. Applications of our program(in Mathematica) for calculation of A ν ( Q ) in such models are presented. Programs in bothMathematica and Fortran can be downloaded from the web page: gcvetic.usm.cl.
1. Why analytic QCD?
Perturbative QCD (pQCD) running coupling a ( Q ) [ ≡ α s ( Q ) /π , where Q ≡ − q ] hasunphysical (Landau) singularities at low spacelike momenta 0 < Q < ∼ .For example, the one-loop pQCD running coupling a ( Q ) (1 − (cid:96). ) = 1 β ln( Q / Λ . ) (1) has a Landau singularity (pole) at Q = Λ . ( ∼ . ). The 2-loop pQCD coupling a ( Q ) (2 − (cid:96). ) has a Landau pole at Q = Λ . and Landau cut at 0 < Q < Λ . .It is expected that the true QCD coupling A ( Q ) has no such singularities. Why?General principles of QFT dictate that any spacelike observable D ( Q ) (correlators ofcurrents, structure functions, etc.) is an analytic (holomorphic) function of Q in the entire Q complex plane with the exception of the timelike axis: Q ∈ C \ ( −∞ , − M . ], where M thr . ∼ . ∼ M π ). If D ( Q ) can be evaluated as a leading-twist term, then it is afunction of the running coupling a ( κQ ) where κ ∼ D ( Q ) = F ( a ( κQ )). Then the argument a ( κQ ) is expected to have the same analyticity properties as D , which is not the case with thepQCD coupling in the usual renormalization schemes (MS, ’t Hooft, etc.).A QCD coupling A ( Q ) with holomorphic behavior for Q ∈ C \ ( −∞ , − M . ], represents ananalytic QCD model (anQCD). Preprint USM-TH-330. Based on the presentation given by G.C. at the 16th International workshop onAdvanced Computing and Analysis Techniques in physics research (ACAT 2014), Prague, Czech Republic,September 1-5, 2014. To appear in the proceedings by the IOP Conference Series publishing. a r X i v : . [ h e p - ph ] N ov uch holomorphic behavior comes usually together with (IR-fixed-point) behavior [ A (0) < ∞ ]. The IR-fixed-point behavior of A ( Q ) is suggested by: • lattice calculations [1, 2, 3]; calculations based on Dyson-Schwinger equations (DSE) [4, 5];Gribov-Zwanziger approach [6, 7]; • The holomorphic A ( Q ) with IR-fixed-point behavior was proposed in various analytic QCDmodels, among them:(i) Analytic Perturbation Theory (APT) of Shirkov, Solovtsov et al. [8, 9, 10, 11, 12];(ii) its extension Fractional APT (FAPT) [13, 14, 15];(iii) analytic models with A ( Q ) very close to a ( Q ) at high | Q | > Λ . : A ( Q ) − a ( Q ) ∼ (Λ . /Q ) N with N = 3 , a ( Q ) in specific schemes with IRfixed point; the condition of reproduction of the correct value of the (strangeless and massless)semihadronic τ lepton V + A decay ratio r τ ≈ .
20 strongly restricts such schemes [24, 25, 26].If the analytic coupling A ( Q ) is not perturbative, A ( Q ) differs from the pQCD couplings a ( Q ) at | Q | > ∼ ∼ /Q N or 1 / [ Q N ln K ( Q / Λ . )].An analytic QCD model which gives A (0) = ∞ was constructed in [27, 28, 29].
2. The formalism of constructing A ν in general anQCD Having A ( Q ) [the analytic analog of a ( Q )] specified, we want to evaluate the physical QCDquantities D ( Q ) in terms of such A ( κQ ).Usually D ( Q ) is known as a (truncated) power series in terms of the pQCD coupling a ( κQ ): D ( Q ) [ N ]pt = a ( κQ ) ν + d ( κ ) a ( κQ ) ν +1 + . . . + d N − ( κ ) a ( κQ ) ν + N − . (2) In anQCD, the simple replacement a ( κQ ) ν + m (cid:55)→ A ( Q ) ν + m is not correct, it leads to astrongly diverging series when N increases, as argued in [30]; a different formalism was needed,and was developed for general anQCD, first for the case of integer ν [31, 32], and then for thecase of general ν [33]. It results in the replacements a ( κQ ) ν + m (cid:55)→ A ν + m ( Q ) (cid:2) (cid:54) = A ( Q ) ν + m (cid:3) , (3) where the construction of the analytic power analogs A ν + m ( Q ) from A ( Q ) was obtained.The construction starts with logarithmic derivatives of A ( Q ) [where β = (11 − N f / / ˜ A n +1 ( Q ) ≡ ( − n β n n ! (cid:18) ∂∂ ln Q (cid:19) n A ( Q ) , ( n = 0 , , , . . . ) , (4) and (cid:101) A ≡ A . Using the Cauchy theorem, these quantities can be expressed in terms of thediscontinuity function of anQCD coupling (cid:101) A along its cut, ρ ( σ ) ≡ Im A ( − σ − i(cid:15) ) (cid:101) A n +1 ( Q ) = 1 π ( − β n Γ( n + 1) (cid:90) ∞ dσσ ρ ( σ )Li − n ( − σ/Q ) . (5) This construction can be extended to a general noninteger n (cid:55)→ ν (cid:101) A ν +1 ( Q ) = 1 π ( − β ν Γ( ν + 1) (cid:90) ∞ dσσ ρ ( σ )Li − ν (cid:18) − σQ (cid:19) ( − < ν ) . (6) his can be recast into an alternative form, involving A ( ≡ (cid:101) A ) instead of ρ ˜ A δ + m ( Q ) = K δ,m (cid:18) dd ln Q (cid:19) m (cid:90) dξξ A ( Q /ξ ) ln − δ (cid:18) ξ (cid:19) , (7) where: 0 ≤ δ < m = 0 , , , . . . ; K δ,m = ( − m β − δ − m +10 / [Γ( δ + m )Γ(1 − δ )]. Thisexpression was obtained from Eq. (6) by the use of the following expression for the Li − ν ( z )function [34]: Li − n − δ ( z ) = (cid:18) dd ln z (cid:19) n +1 (cid:20) z Γ(1 − δ ) (cid:90) dξ − zξ ln − δ (cid:18) ξ (cid:19)(cid:21) ( n = − , , , . . . ; 0 < δ < . (8) The analytic analogs A ν of powers a ν are then obtained by combining various generalizedlogarithmic derivatives (with the coefficients (cid:101) k m ( ν ) obtained in [33]) A ν = (cid:101) A ν + (cid:88) m ≥ (cid:101) k m ( ν ) (cid:101) A ν + m . (9)
3. The considered anQCD models
We constructed Mathematica and Fortran programs for three anQCD models: 1.) FractionalAnalytic Perturbation Theory (FAPT) [13, 14, 15]; 2.) 2 δ analytic QCD (2 δ anQCD) [19]; 3.)Massive Perturbation Theory (MPT) [20, 21, 22, 23]. These three models are described below. Application of the Cauchy theorem to the function a ( Q (cid:48) ) ν / ( Q (cid:48) − Q ) gives a ( Q ) ν = 1 π (cid:90) ∞ σ = − Λ . − η dσ Im( a ( − σ − i(cid:15) ) ν )( σ + Q ) , ( η → +0) . (10) In FAPT, the integration over the Landau part of the cut in the above integral is eliminated;since σ ≡ − Q , the Landau cut is − Λ . < σ <
0. This leads to the FAPT coupling A (FAPT) ν ( Q ) = 1 π (cid:90) ∞ σ =0 dσ Im( a ( − σ − i(cid:15) ) ν )( σ + Q ) . (11) δ QCD
Here, ρ ( σ ) ≡ Im A ( − σ − i(cid:15) ) is approximated at high momenta σ ≥ M by ρ (pt) ( σ ) [ ≡ Im a ( − σ − i(cid:15) )], and in the unknown low-momentum regime by two deltas: ρ (2 δ ) ( σ ) = πF δ ( σ − M ) + πF δ ( σ − M ) + Θ( σ − M ) ρ (pt) ( σ ) ⇒ (12) (cid:101) A (2 δ ) ν ( Q ) = ( − β ν Γ( ν +1) (cid:26) (cid:88) j =1 F j M j Li − ν (cid:32) − M j Q (cid:33) + 1 π (cid:90) ∞ M dσσ Im a ( − σ − i(cid:15) )Li − ν (cid:18) − σQ (cid:19) (cid:27) . (13) The parameters F j and M j ( j = 1 ,
2) are fixed in such a way that the resulting deviation fromthe underlying pQCD at high | Q | > Λ is: A (2 δ ) ν ( Q ) − a ( Q ) ν ∼ (Λ /Q ) . The pQCD-onsetscale M is determined so that the model reproduces the measured (strangeless and massless) V + A tau lepton semihadronic decay ratio r τ ≈ . a is chosen in 2 δ anQCD, for calculational convenience, inthe Lambert-scheme form a ( Q ) = − c − c /c + W ∓ ( z ± ) , (14) where: c = β /β ; Q = | Q | e iφ , the upper (lower) sign when φ ≥ φ < z ± = ( c e) − ( | Q | / Λ ) − β /c exp [ i ( ± π − β φ/c )] . (15) Nonperturbative physics suggests that the gluon acquires at low momenta an effective(dynamical) mass m gl ∼ A (MPT) ( Q ) = a ( Q + m gl2 ) . (16) Since m gl > Λ Lan . , the new coupling has no Landau singularities.The (generalized) logarithmic derivatives (cid:101) A (MPT) δ + m ( Q ) are then uniquely determined (cid:101) A δ + m ( Q ) = K δ,m (cid:18) dd ln Q (cid:19) m (cid:90) dξξ A (MPT) ( Q /ξ ) ln − δ (cid:18) ξ (cid:19) . (17)
4. Numerical implementation and results
Programs of numerical implementation in anQCD models: • for integer power analogs A n ( Q ) in APT and in “massive QCD” [35, 36]: Nesterenko andSimolo, 2010 (in Maple) [37], and 2011 (in Fortran) [38]; • for general power analogs A ν ( Q ) in FAPT: Bakulev and Khandramai, 2013 (inMathematica) [39]; • for general power analogs A ν ( Q ) in 2 δ anQCD, MPT and FAPT: the presented work inMathematica [40] and Fortran (programs in both languages can be downloaded from theweb page: gcvetic.usm.cl ).The basic relations for the numerical implementation of A ν are: in FAPT Eq. (11); in2 δ anQCD Eqs. (13) and (9); in MPT Eqs. (17) and (9).In Mathematica, Li − ν ( z ) is implemented as PolyLog[ − ν, z ]. In Mathematica 9.0.1 it isunstable for | z | (cid:29)
1. Therefore, we provide a subroutine Li nu.m (which is called by the mainMathematica program anQCD.m) and gives a stable version under the name polylog[ − ν, z ].This problem does not exist in Mathematica 10.0.1.In Fortran, program Vegas [41] is used for integrations. However, in Fortran, Li − ν ( z ) functionis not implemented for general (complex) z , and is evaluated as an integral Eq. (8). Therefore,the evaluation of (cid:101) A ν ’s in 2 δ anQCD is somewhat more time consuming in Fortran than inMathematica. Further, more care has to be taken in Fortran to deal correctly with singularitiesof the integrands. .001 0.01 0.1 1 10 1000.00.20.40.60.81.0 Q (cid:64) GeV (cid:68) A (cid:73) Q (cid:77) c (cid:61)(cid:45) N f (cid:61) (cid:72) a (cid:76) aA (cid:72) ∆ (cid:76) Q (cid:64) GeV (cid:68) A (cid:73) Q (cid:77) c (cid:61) c ; N f (cid:61) (cid:72) b (cid:76) A (cid:72) FAPT (cid:76) A (cid:72) MPT (cid:76) a Figure 1. A ≡ A in three anQCD models with ν = 1 and N f = 3, as a function of Q for Q >
0; theunderlying pQCD coupling a is included for comparison: (a) 2 δ anQCD coupling and pQCD coupling, in theLambert scheme with c = − . c j = c j − /c j − for j ≥ = 0 . ; MPT is with m = 0 . . Q (cid:64) GeV (cid:68) A . (cid:73) Q (cid:77) c (cid:61)(cid:45) N f (cid:61) (cid:72) a (cid:76) a A (cid:72) ∆ (cid:76) Q (cid:64) GeV (cid:68) A . (cid:73) Q (cid:77) c (cid:61) c ; N f (cid:61) (cid:72) b (cid:76) A (cid:72) FAPT (cid:76) A (cid:72) MPT (cid:76) a Figure 2.
The same as in Fig. 1, but with ν = 0 . A ν =0 . ). A . is calculated from (cid:101) A . m using the relation(9) with ν = 0 .
3, and truncation at (cid:101) A . in 2 δ anQCD, and at (cid:101) A . in MPT; and in FAPT using Eq. (11).Figs. 1 and 2 are taken from [40].
5. Main procedures in Mathematica for three analytic QCD models N l[ N f , ν, , | Q | , Λ , φ ] gives N -loop ( N = 1 , , ,
4) analytic FAPT coupling A (FAPT ,N ) ν ( Q , N f ) with real power index ν , with fixed number of active quark flavors N f ,in the Euclidean domain [ Q = | Q | exp( iφ ) ∈ C and Q (cid:54) < AFAPT N l[ N f, ν, , Q , L , φ ] = A (FAPT ,N ) ν [ Q | Q | , φ = arg( Q ); N f = N f ; L N f ]( N = 1 , , , N f = 3 , , , . N l[ N f , M, ν, | Q | , φ ] gives “ N -loop” 2 δ anQCD coupling A (2 δ ) ν + M ( Q , N f ), with powerindex ν + M ( ν > − M = 0 , , . . . , N − N f ,in the Euclidean domain. It is used in the N N − LO truncation approach [where in (9): ν (cid:55)→ ν and n (cid:55)→ M , and we truncate at (cid:101) A ν + N − ] A2d N l[ N f, M, ν, Q , φ ] = A (2 δ ) ν + M [ Q | Q | , φ = arg( Q ); N f = N f ] , ( N = 1 , , , , N f = 3 , , , M = 0 , , . . . , N − . .) AMPT N l[ N f , ν, Q , m gl2 , Λ N f ] gives N -loop ( N = 1 , , ,
4) analytic MPT coupling A (MPT ,N ) ν ( Q , m gl2 , N f ), with real power index ν (0 < ν <
5) and with number of active quarkflavors N f , in the Euclidean domain ( Q ∈ C and Q (cid:54) < AMPT N l[ N f, ν, Q , M , L
2] = A (MPT , N) ν [ Q Q ∈ C ; N f = N f ; M m gl2 ; L N f ]( N = 1 , , , N f = 3 , , ,
6) ; 0 < ν < . (18) Examples:Input scale of the underlying MS pQCD for FAPT and MPT is Λ = 0 . . The timesare for a typical laptop, using Mathematica 9.0.1; the first entry in the results is the time ofcalculation, in s .In[1]:= << anQCD.m;In[2]:= AFAPT3l[5, 1, 0, 10 , 0.1, 0] // TimingOut[2]= { . , . } In[3]:= AMPT3l[5, 1, 10 , 0.7, 0.1] // TimingOut[3]= { . , . } In[4]:= A2d3l[5, 0, 1, 10 , 0] // TimingOut[4]= { . , . } In[5]:= AFAPT3l[3, 1, 0, 0.5, 0.1, 0] // TimingOut[5]= { . , . } In[6]:= AMPT3l[3, 1, 0.5, 0.7, 0.1] // TimingOut[6]= { . , . } In[7]:= A2d3l[3, 0, 1, 0.5, 0] // TimingOut[7]= { . , . } In[8]:= AFAPT3l[3, 0.3, 0, 0.5, 0.1, 0] // TimingOut[8]= { . , . } In[9]:= AMPT3l[3, 0.3, 0.5, 0.7, 0.1] // TimingOut[9]= { . , . } In[10]:= A2d3l[3, 0, 0.3, 0.5, 0] // TimingOut[10]= { . , . }
6. Conclusions
We constructed programs, in Mathematica and Fortran, which evaluate couplings A ν ( Q ) inthree models of analytic QCD (FAPT, 2 δ anQCD, and MPT). These couplings are holomorphicfunctions (free of Landau singularities) in the complex Q plane with the exception ofthe negative semiaxis, and are analogs of powers a ( Q ) ν ≡ ( α s ( Q ) /π ) ν of the underlyingperturbative QCD. We checked that our results in FAPT model agree with those of Mathematicaprogram [39]. Acknowledgments
This work was supported by FONDECYT (Chile) Grant No. 1130599 and DGIP (UTFSM)internal project USM No. 11.13.12 (C.A and G.C).
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