pQCD.rst

pQCD ingredients

Strong Coupling

Implementation: ~eko.couplings.Couplings.

We use perturbative with the running coupling a_s(μ_R²) = α_s(μ_R²)/(4π) given at 5-loop by Herzog:2017ohr,Luthe:2016ima,Baikov:2016tgj,Chetyrkin:2017bjc,Luthe:2017ttg

It is useful to define in addition b_k = β_k/β₀, k > 0.

We implement two different strategies to solve the :

• method="exact": Solve using scipy.integrate.solve_ivp. In we fall back to the expanded solution as this is already the true solution.
• method="expanded": using approximate solutions:

being:

When the renormalization scale crosses a flavor threshold matching conditions have to be applied Schroder:2005hy,Chetyrkin:2005ia. In particular, the matching involved in the change from n_f to n_f − 1 schemes is presented in equation 3.1 of Schroder:2005hy for masses, while the same expression for POLE masses is reported in Appendix A. For this reason the boundary conditions of eko.couplings.Couplings can be specified at scale_ref along with nf_ref and, the computed result can depend on the number of flavors at the target scale, see eko.couplings.Couplings.a_s An example how the evolution path is determined is given here</code/Utilities/>.

QCD Splitting Functions

In the case in which only the corrections are considered, the Altarelli-Parisi splitting kernels can be expanded in powers of the strong coupling a_s(μ²) and are given by Moch:2004pa,Vogt:2004mw

Note the additional minus in the definition of γ.

Polarized Splitting Functions

Polarized Altarelli-Parisi splitting kernels are implemented up to and expanded in powers of the strong coupling as in the previous section. They are used to evolve longitudinally polarized parton distribution functions. Unlike in the unpolarized case, where the probability of the splitting describes the momentum of parent and daughter partons with averaged spins, the polarized splitting functions describe the parent and daughter momentums along with their spins and thus take into account positive or negative helicities. Throughout, the anomalous dimensions are defined as above and are represented with γ and not Δγ just like in the unpolarized case.

The and kernels are given in Gluck:1995yr and the in Moch:2014sna and Moch:2015usa.

At , the non-singlet is the same in both the polarized and unpolarized case. Due to helicity conservation, the first moment of the anomalous dimension is γ_qq⁽⁰⁾(N = 1) = γ_qg⁽⁰⁾(N = 1) = 0.

At , the singlet entry of the quark-quark anomalous dimension can be decomposed into the pure singlet (consisting of the flavour independent quark-quark and quark-antiquark anomalous dimensions) and the plus flavour asymmetry non-singlet:

The non-singlet sector in the polarized case swaps the plus and minus non-singlet relative to the unpolarized case. This is because the polarized non-singlet splitting functions are defined as the difference between the probability of the polarized parton splitting into daughter partons of the same flavour and daughters splitting into a different flavours and opposite helicity. The first moments of the anomalous dimensions are:

At the non-singlet is further decomposed into the helicity difference quark-antiquark anomalous dimension called the valence polarized non-singlet and defined as:

where γ_ns, −⁽²⁾ is the minus flavour asymmetry non-singlet and γ_ns, s⁽²⁾ the sea-like polarized non-singlet. The singlet entry γ_qq⁽²⁾ is defined as above in the case.

Unified Splitting Functions

When the corrections are taken into account, equation take the form

where P̃ are the usual splitting kernels defined in the previous section, while P̄ are given by

a^2 mathbf{P}^{(0,2)} + dots

where a = α/(4π). The expression of the pure and of the mixed ⊗ splitting kernels are given in deFlorian:2015ujt,deFlorian:2016gvk

Order specification

In the code order=tuple(int,int) specifies the and perturbative orders of the splitting functions in terms of a_s = α_s/(4π) and a_em = α_em/(4π). The available perturbative expansions are the following:

• order=(n,0): with n = 1, 2, 3, 4 correspond to the pure evolution at , , and in which the splitting functions are expanded up to 𝒪(a_sⁿ) and the strong coupling is evolved using the n-th coefficient of the beta function, i.e. β_n − 1.
• order=(n,m); with n = 1, 2, 3, 4 and m = 1, 2 corresponds to the mixed ⊗ evolution in which the splitting functions are expanded up to 𝒪(a_sⁿa_em^m), the stromg coupling is evolved using up to the n-th coefficient of the beta function and the electromagnetic coupling is kept fixed.

Observe that the case n = 0 is not allowed, since it would correspond to the pure evolution or (if m > 0) no evolution at all.

Sum Rules

The Altarelli-Parisi Splitting functions have to satisfy certain sum rules. In fact ⊗ interactions preserve fermion number, therefore

Moreover, the conservation of the proton's momentum implies that

The reason why multiple conservation equations follow from a single conserved quantity (i.e. proton's momentum) is that one is free to choose a border condition in which there is only one parton, e.g. the gluon, and the momentum should be preserved. This is just a simple way to consider that anomalous dimensions are actually operators, and the conservation thus apply element by element in the first dimension (summing over the second one only).

Using the definition of anomalous dimensions the sum rules are written as:

that must be satisfied order by order in perturbation theory.

Heavy Quark Masses

In also the heavy quark masses (m_c, m_b, m_t) follow a and their values depend on the energy scale at which the quark is probed. Masses do not play any role in a single flavour patch, but are important in when more flavour schemes need to be joined (see matching conditions <Matching>).

EKO implements two strategies for dealing with the heavy quark masses, managed by the theory card parameter HQ. The easiest and more common option for PDFs evolution is POLE mass, where the physical quark masses are specified as input.

On contrary selecting the option MSBAR the user can activate the mass running in the scheme, as described in the following paragraph.

If the initial condition for the mass is not given at a scale coinciding with the mass itself (i.e. in the input theory card Qmh≠mh), EKO needs to compute the scale at which the mass running function intersects the identity function, in order to properly initiate the ~eko.threshold.ThresholdAtlas and set the evolution path.

For each heavy quark h we solve for m_h:

where the evolved mass is given by:

and m_h, 0 is the given initial condition at the scale μ_h, 0. Here there is a subtle complication since the solution depends on the value a_s(μ_h, 0²) which is unknown and depends again on the threshold path. To overcome this issue, EKO initialize a temporary instance of the class ~eko.couplings.Couplings with a fixed flavor number scheme, with n_fref active flavors at the scale μ_ref.

Then we check that, heavy quarks involving a number of active flavors greater than n_fref are given with initial conditions:

while the ones related to fewer active flavors follow:

So for the former initial condition we will find the intercept between and the identity in the forward direction ($m_{\overline{MS},h} \ge \mu_h$) and vice versa for the latter.

In doing so EKO takes advantage of the monotony of the solution $m_{\overline{MS},h}(\mu^2)$ with a vanishing limit for μ² → ∞.

Now, being able to evaluate a_s(μ_h, 0²), there are two ways of solving the previous integral and finally compute the evolved $m_{\overline{MS},h}$. In fact, the function γ_m(a_s) is the anomalous mass dimension and, as the β function, it can be evaluated perturbatively in a_s up to 𝒪(a_s⁴):

Even here it is useful to define c_k = γ_m, k/β₀, k ≥ 0.

Therefore the two solution strategies are:

• method = "exact": the integral is solved exactly using the expression of β, γ_m up to the specified perturbative order
• method = "expanded": the integral is approximate by the following expansion:

The procedure is iterated on all the heavy quarks, updating the temporary instance of ~eko.couplings.Couplings with the computed masses.

To find coherent solutions and perform the mass running in the correct patches it is necessary to always start computing the mass scales closer to μ_ref.

Eventually, to ensure that the threshold values are properly set, we add a consistency check, asserting that the $m_{\overline{MS},h}$ are properly sorted.

Note that also for mass running when the heavy matching scales are crossed we need to apply non trivial matching from order 𝒪(a_s²) as described here Liu:2015fxa.

We provide the following as an illustrative example of how this procedure works: when the strong coupling is given with boundary condition α_s(μ_ref = 91, n_fref = 5) then the heavy quarks initial conditions must satisfy:

and EKO will start solving the equation $m_{\overline{MS},h}(m_h^2) = m_h$ in the order h = t, b, c.

Since the charm mass will be computed only when both the top and bottom matching scales are known, the boundary condition m_c(μ_c) can be evolved safely below the scale $m_{\overline{MS},b}$ where the solution of $m_{\overline{MS},c}(m_c^2) = m_c$ is sitting.