pQCD ingredients

Strong Coupling

Implementation: Couplings.

We use perturbative QCD with the running coupling \(a_s(\mu_R^2) = \alpha_s(\mu_R^2)/(4\pi)\) given at 5-loop by [BCK17, CFHV17, HRU+17, LMMS17, LMMS16]

\[\frac{da_s(\mu_R^2)}{d\ln\mu_R^2} = \beta(a_s(\mu_R^2)) \ = - \sum\limits_{n=0} \beta_n a_s^{n+2}(\mu_R^2)\]

It is useful to define in addition \(b_k = \beta_k/\beta_0, k>0\).

We implement two different strategies to solve the RGE:

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

\[\begin{split}a^{\text{LO}}_s(\mu_R^2) &= \frac{a_s(\mu_0^2)}{1 + a_s(\mu_0^2) \beta_0 L_{\mu}} \\ a^{\text{NLO}}_{s,\text{exp}}(\mu_R^2) &= a^{\text{LO}}_s(\mu_R^2)-b_1 \left[a^{\text{LO}}_s(\mu_R^2)\right]^2 \ln\left(1+a_s(\mu_0^2) \beta_0 L_{\mu}\right) \\ a^{\text{NNLO}}_{s,\text{exp}}(\mu_R^2) &= a^{\text{LO}}_s(\mu_R^2)\left[1 + a^{\text{LO}}_s(\mu_R^2)\left(a^{\text{LO}}_s(\mu_R^2) - a_s(\mu_0^2)\right)(b_2 - b_1^2) \right.\\ & \hspace{60pt} \left. + a^{\text{NLO}}_{s,\text{exp}}(\mu_R^2) b_1 \ln\left(a^{\text{NLO}}_{s,\text{exp}}(\mu_R^2)/a_s(\mu_0^2)\right)\right] \\ a^{\text{N3LO}}_{s,\text{exp}}(\mu_R^2) &= a^{\text{NNLO}}_s(\mu_R^2) + \frac{a^{\text{LO}}_s(\mu_R^2)^4}{2 b_0^3} \left\{ \right. \\ & -2 b_1^3 L_{0}^3 + 5 b_1^3 L_{\text{LO}}^2 + 2 b_1^3 L_{\text{LO}}^3 + b_1^3 L_{0}^2 \left(5 + 6 L_{\text{LO}} \right) \\ & + 2 b_0 b_1 L_{\text{LO}} \left[ b_2 + 2 \left(b_1^2 - b_0 b_2 \right) L_{\mu} a_s(\mu_0^2) \right] \\ & - b_0^2 L_{\mu} a_s(\mu_0^2) \left[ -2 b_1 b_2 + 2 b_0 b_3 + \left( b_1^3 - 2 b_0 b_1 b_2 + b_0^2 b_3 \right) L_{\mu} a_s(\mu_0^2) \right] \\ & - 2 b_1 L_{0} \left[ 5 b_1^2 L_{\text{LO}} + 3 b_1^2 L_{\text{LO}}^2 + b_0 \left[b_2 + 2 \left(b_1^2 - b_0 b_2\right) L_{\mu} a_s(\mu_0^2)\right] \right ] \\ & \left. \right\}\end{split}\]

being:

\[\begin{split}L_{\mu} &= \ln(\mu_R^2/\mu_0^2) \\ L_{0} &= \ln(a_s(\mu_0^2)) \\ L_{\text{LO}} &= \ln(a^{\text{LO}}_s(\mu_R^2)) \\\end{split}\]

When the renormalization scale crosses a flavor threshold matching conditions have to be applied [CKS06, SS06]. In particular, the matching involved in the change from \(n_f\) to \(n_f-1\) schemes is presented in equation 3.1 of [SS06] for \(\overline{MS}\) 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.

QCD Splitting Functions

In the case in which only the QCD corrections are considered, the Altarelli-Parisi splitting kernels can be expanded in powers of the strong coupling \(a_s(\mu^2)\) and are given by [MVV04, VMV04]

\[\begin{split}\mathbf{P}(x,a_s(\mu^2)) &= \sum\limits_{j=0} a_s^{j+1}(\mu^2) \mathbf{P}^{(j)}(x) \\ {\gamma}^{(j)}(N) &= -\mathcal{M}[\mathbf{P}^{(j)}(x)](N)\end{split}\]

Note the additional minus in the definition of \(\gamma\).

Polarized Splitting Functions

Polarized Altarelli-Parisi splitting kernels are implemented up to NNLO 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 \(\gamma\) and not \(\Delta\gamma\) just like in the unpolarized case.

The LO and NLO kernels are given in [GRSV96] and the NNLO in [MVV14] and [MVV15].

At LO, 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 \(\gamma^{(0)}_{qq} (N=1) = \gamma^{(0)}_{qg} (N=1) = 0\).

At NLO, 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:

\[\gamma^{(1)}_{qq} =\gamma^{(1)}_{ps} + \gamma^{(1)}_{ns,+}\]

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:

\[\begin{split}\gamma^{(1)}_{ns,+} (N=1) &= 0 \\ \gamma^{(1)}_{qq} (N=1) &= 24 C_F T_R \\ \gamma^{(1)}_{qg} (N=1) &= 0 \\\end{split}\]

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

\[\gamma^{(2)}_{ns,v} =\gamma^{(2)}_{ns,-} + \gamma^{(2)}_{ns,s}\]

where \(\gamma^{(2)}_{ns,-}\) is the minus flavour asymmetry non-singlet and \(\gamma^{(2)}_{ns,s}\) the sea-like polarized non-singlet. The singlet entry \(\gamma^{(2)}_{qq}\) is defined as above in the NLO case.

Unified Splitting Functions

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

\[\mathbf{P}=\mathbf{\tilde{P}}+\mathbf{\bar{P}}\]

where \(\mathbf{\tilde{P}}\) are the usual QCD splitting kernels defined in the previous section, while \(\mathbf{\bar{P}}\) are given by

\[\mathbf{\bar{P}} = a \mathbf{P}^{(0,1)} + a_s a \mathbf{P}^{(1,1)} + a^2 \mathbf{P}^{(0,2)} + \dots\]

where \(a = \alpha/(4\pi)\). The expression of the pure QED and of the mixed QED \(\otimes\) QCD splitting kernels are given in [dFSR16a, dFSR16b]

Order specification

In the code order=tuple(int,int) specifies the QCD and QED perturbative orders of the splitting functions in terms of \(a_s = \alpha_s/(4\pi)\) and \(a_{em} = \alpha_{em}/(4\pi)\). The available perturbative expansions are the following:

order=(n,0): with \(n=1,2,3,4\) correspond to the pure QCD evolution at LO, NLO, NNLO and N3LO in which the QCD splitting functions are expanded up to \(\mathcal{O}(a_s^n)\) and the strong coupling is evolved using the n-th coefficient of the beta function, i.e. \(\beta_{n-1}\).
order=(n,m); with \(n=1,2,3,4\) and \(m=1,2\) corresponds to the mixed QED \(\otimes\) QCD evolution in which the splitting functions are expanded up to \(\mathcal{O}(a_s^na_{em}^m)\), 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 QED evolution or (if \(m > 0\)) no evolution at all.

Sum Rules

The Altarelli-Parisi Splitting functions have to satisfy certain sum rules. In fact QED \(\otimes\) QCD interactions preserve fermion number, therefore

\[\int_0^1dx P_{ns,q}^-(x)=0\]

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

\[\int_0^1dx x (2n_dP_{dg}(x)+2n_uP_{ug}(x)+P_{\gamma g}(x)+P_{gg}(x))=0\]

\[\int_0^1dx x (2n_dP_{d\gamma}(x)+2n_uP_{u\gamma}(x)+P_{\gamma \gamma}(x)+P_{g\gamma}(x))=0\]

\[\int_0^1dx x \Bigl(\sum_{q_i=q,\bar{q}} P_{q_iq_j}(x)+P_{\gamma q_j}(x)+P_{gq_j}(x)\Bigr)=0\]

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:

\[\gamma_{ns}^-(N=1)=0\]

\[\bigl(2n_d\gamma_{dg}+2n_u\gamma_{ug}+\gamma_{\gamma g}+\gamma_{gg}\bigr)(N=2)=0\]

\[\bigl(2n_d \gamma_{d\gamma}+2n_u \gamma_{u\gamma}+ \gamma_{\gamma \gamma}+ \gamma_{g\gamma})(N=2)=0\]

\[\Bigl(\gamma_{ns,q}^+ +2n_u\gamma^S_{uq}+2n_d\gamma^S_{dq} + \gamma_{\gamma q}+\gamma_{gq}\Bigr)(N=2)=0\]

that must be satisfied order by order in perturbation theory.

Heavy Quark Masses

In QCD also the heavy quark masses (\(m_{c}, m_{b}, m_{t}\)) follow a RGE 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 VFNS when more flavour schemes need to be joined (see matching conditions).

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 \(\overline{MS}\) 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 ThresholdAtlas and set the evolution path.

For each heavy quark \(h\) we solve for \(m_h\):

\[m_{\overline{MS},h}(m_h^2) = m_h\]

where the evolved \(\overline{MS}\) mass is given by:

\[m_{\overline{MS},h}(\mu^2) = m_{h,0} \exp \left[ - \int_{a_s(\mu_{h,0}^2)}^{a_s(\mu^2)} \frac{\gamma_m(a_s)}{\beta(a_s)} d a_s \right ]\]

and \(m_{h,0}\) is the given initial condition at the scale \(\mu_{h,0}\). Here there is a subtle complication since the solution depends on the value \(a_s(\mu_{h,0}^2)\) which is unknown and depends again on the threshold path. To overcome this issue, EKO initialize a temporary instance of the class Couplings with a fixed flavor number scheme, with \(n_{f_{ref}}\) active flavors at the scale \(\mu_{ref}\).

Then we check that, heavy quarks involving a number of active flavors greater than \(n_{f_{ref}}\) are given with initial conditions:

\[m_h (\mu_h) \ge \mu_h\]

while the ones related to fewer active flavors follow:

\[m_h (\mu_h) \le \mu_h\]

So for the former initial condition we will find the intercept between RGE 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 RGE solution \(m_{\overline{MS},h}(\mu^2)\) with a vanishing limit for \(\mu^2 \rightarrow \infty\).

Now, being able to evaluate \(a_s(\mu_{h,0}^2)\), there are two ways of solving the previous integral and finally compute the evolved \(m_{\overline{MS},h}\). In fact, the function \(\gamma_m(a_s)\) is the anomalous QCD mass dimension and, as the \(\beta\) function, it can be evaluated perturbatively in \(a_s\) up to \(\mathcal{O}(a_s^4)\):

\[\begin{split}\gamma_m(a_s) &= \sum\limits_{n=0} \gamma_{m,n} a_s^{n+1} \\\end{split}\]

Even here it is useful to define \(c_k = \gamma_{m,k}/\beta_0, k \ge 0\).

Therefore the two solution strategies are:

method = "exact": the integral is solved exactly using the expression of \(\beta,\gamma_m\) up to the specified perturbative order
method = "expanded": the integral is approximate by the following expansion:

\[\begin{split}m_{\overline{MS},h}(\mu^2) & = m_{h,0} \left ( \frac{a_s(\mu^2)}{a_s(\mu_{h,0}^2)} \right )^{c_0} \frac{j_{exp}(a_s(\mu^2))}{j_{exp}(a_s(\mu_{h,0}^2))} \\ j_{exp}(a_s) &= 1 + a_s \left [ c_1 - b_1 c_0 \right ] \\ & + \frac{a_s^2}{2} \left [c_2 - c_1 b_1 - b_2 c_0 + b_1^2 c_0 + (c_1 - b_1 c_0)^2 \right] \\ & + \frac{a_s^3}{6} [ -2 b_3 c_0 - b_1^3 c_0 (1 + c_0) (2 + c_0) - 2 b_2 c_1 \\ & - 3 b_2 c_0 c_1 + b_1^2 (2 + 3 c_0 (2 + c_0)) c_1 + c_1^3 + 3 c_1 c_2 \\ & + b_1 (b_2 c_0 (4 + 3 c_0) - 3 (1 + c_0) c_1^2 - (2 + 3 c_0) c_2) + 2 c_3 ]\end{split}\]

The procedure is iterated on all the heavy quarks, updating the temporary instance of 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 \(\mu_{ref}\).

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 \(\overline{MS}\) mass running when the heavy matching scales are crossed we need to apply non trivial matching from order \(\mathcal{O}(a_s^2)\) as described here [LS15].

We provide the following as an illustrative example of how this procedure works: when the strong coupling is given with boundary condition \(\alpha_s(\mu_{ref}=91, n_{f_{ref}}=5)\) then the heavy quarks initial conditions must satisfy:

\[\begin{split}& \mu_{b} \le \mu_{ref} \le \mu_t \\ & m_c (\mu_c) \le \mu_c \\ & m_b (\mu_b) \le \mu_b \\ & m_t (\mu_t) \ge \mu_t\end{split}\]

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 \(m_c(\mu_{c})\) 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.