Hadron-Collider Factorization and Parton Luminosities

A hadron-collider prediction combines short-distance partonic scattering with one collinear distribution for each beam, when the observable admits collinear factorization. Parton luminosities reorganize the double convolution by the partonic invariant $\hat s=x_1x_2s$; they are weights, not cross sections. The worked example below turns a normalized toy luminosity into a photon-mediated Drell–Yan rate, with explicit color average, flux, mass/rapidity Jacobian, two beam assignments and correlated input errors. It is a leading-power Born calculation, inclusive in hadronic radiation, not a precision Standard Model prediction.

Required background. Collinear factorization and operator-defined PDFs supplies the incoming-hadron matrix elements and their scheme cancellation.

Helpful background. Phase-space integration and Monte Carlo estimators supplies measurement functions, mappings, and numerical validation.

The cut-aware hadronic convolution

Let $\mathcal M(\Phi_n)$ implement the observable, cuts, bin, jet definition, and any decay selection on $n$-body phase space. Assume that this observable and its scale hierarchy admit a collinear factorization theorem: the required inclusive sums, cancellation or factorization of soft interactions, and treatment of Glauber exchange must be justified. Under that hypothesis, the leading-power organization is

$$\begin{aligned} \sigma[\mathcal M] ={}& \sum_{i,j}\int_0^1dx_1\int_0^1dx_2\, f_{i/A}(x_1,\mu_F;\mathcal S) f_{j/B}(x_2,\mu_F;\mathcal S)\\ &\quad\times \int d\Phi_n\, \frac{d\widehat\sigma_{ij}} {d\Phi_n} (x_1P_A,x_2P_B;\mu_R,\mu_F;\mathcal S) \mathcal M(\Phi_n) +R_{\mathrm{pow}}. \end{aligned}$$

$i,j$ run over the active partons in the declared flavor scheme. $\widehat\sigma_{ij}$ includes flux, spin/color averages, virtual and real corrections, subtraction terms, and any perturbative decays assigned to the hard calculation. At higher orders the phase-space expression stands for the required sum over final-state multiplicities. $R_{\mathrm{pow}}$ denotes controlled power-suppressed terms in the stated hierarchy. A possible leading-power failure of factorization lies outside this formula; it cannot be hidden in $R_{\mathrm{pow}}$.

The measurement belongs inside the phase-space integral. Multiplying an inclusive cross section by an acceptance factor imported from a different theory setup can break spin correlations, recoil, channel mixtures, and uncertainty correlations.

Inclusive Drell–Yan sums over the unobserved hadronic final state. This is essential to the cancellation reasoning for soft spectator interactions; a pointwise soft approximation can encounter a pinched Glauber region. The argument and its limitations are explained in Collins, Soper, and Sterman 2004, §9.3, manuscript pp. 91–94, PDF. Those version-specific pages support an inclusive-process statement, not arbitrary cuts. Vetoes, resolved small transverse momentum or spectator-sensitive measurements require their own factorization analysis.

Parton luminosities

Define the ordered luminosity

$$\mathcal L_{ij}(\tau,\mu_F)= \int_\tau^1\frac{dx}{x}\, f_{i/A}(x,\mu_F) f_{j/B}\!\left(\frac{\tau}{x},\mu_F\right), \qquad \tau=\frac{\hat s}{s}.$$

Insert $1=\int d\tau\,\delta(\tau-x_1x_2)$ into a double convolution. Integrating $x_2$ gives the Jacobian $1/x_1$ and hence

$$\sigma=\sum_{ij}\int_{\tau_{\min}}^1d\tau\, \mathcal L_{ij}(\tau,\mu_F)\, \widehat\sigma_{ij}(\tau s)$$

whenever the remaining partonic quantity depends on $x_1,x_2$ only through $\hat s$. Rapidity cuts or asymmetric measurements generally retain additional dependence and should be kept in the original two-dimensional convolution.

Because $\mathcal L_{ij}$ is ordered, summing all $i,j$ already includes both beam assignments. If a symmetrized luminosity is introduced, its factor of $1/(1+\delta_{ij})$ and the matching channel convention must be stated explicitly. Many factor-of-two errors come from combining an ordered hard sum with a symmetrized luminosity.

Worked analytic check

Take identical synthetic beam distributions $f(x)=2(1-x)$ on $0<x<1$. Direct integration gives

$$\begin{aligned} \mathcal L(\tau) &=4\int_\tau^1\frac{dx}{x} (1-x)\left(1-\frac{\tau}{x}\right)\\ &=-8+8\tau-4(1+\tau)\ln\tau. \end{aligned}$$

At $\tau=\tfrac14$,

$$\mathcal L\!\left(\frac14\right)=-6+5\ln4.$$

The expression is positive for $0<\tau<1$ and tends to zero as $\tau\to1$. These are strong checks on the Jacobian and integration limits. This analytic fixture is synthetic; it is not a numerical statement about a released PDF set.

Photon-mediated Drell–Yan at Born level

Consider $A+B\to\ell^-\ell^++X$ through $q\bar q\to\gamma^*\to\ell^-\ell^+$. Neglect beam, quark and lepton masses in the hard kinematics, use $N_c=3$, and let $Q_q$ denote the quark charge in units of the positive coupling $e$, with $\alpha=e^2/(4\pi)$. The pair momentum is $q^\mu=k_-^\mu+k_+^\mu$ and its invariant mass is $Q^2=q^2>0$. The hard mass must be large compared with hadronic and retained particle-mass scales.

Keep only photon exchange and the Born electromagnetic coupling, with fixed $\alpha$ for this example. There is no $Z$ exchange, photon-initiated channel, QED radiation or QCD correction. These omissions define the benchmark; they do not establish that photon exchange dominates at every $Q$. PDFs are unpolarized and color summed, specified at a common fixed $\mu_F$ in a fixed-flavor $\overline{\mathrm{MS}}$ setup. At this order the hard coefficient has no explicit QCD factorization logarithm. Inclusive Drell–Yan and the additional threshold specialization used in a soft-collinear treatment are distinguished in Schwartz 2014, §36.5.1, p. 796.

Spin, color and two-body flux

The photon couples diagonally to the incoming quark color. Averaging over the $N_c^2$ independent initial color choices gives

$$\frac{1}{N_c^2}\sum_{a,b=1}^{N_c} |\delta_{ab}|^2=\frac1{N_c}.$$

This is an initial-state color average, unlike the factor $N_c$ from summing final-state quark colors in $e^+e^-\to q\bar q$. The initial spin average is a separate factor $1/4$. With $p_1,p_2$ the incoming parton momenta, the spin traces give

$$\begin{aligned} \overline{|\mathcal M|^2} &=\frac{e^4Q_q^2}{4N_c\hat s^2} \operatorname{tr}(p_2\!\!\!/\gamma^\mu p_1\!\!\!/\gamma^\nu) \operatorname{tr}(k_-\!\!\!/\gamma_\mu k_+\!\!\!/\gamma_\nu)\\ &=\frac{2e^4Q_q^2}{N_c} \frac{\hat t^2+\hat u^2}{\hat s^2} =\frac{e^4Q_q^2}{N_c}(1+\cos^2\theta). \end{aligned}$$

Here $\hat t=(p_1-k_-)^2=-\hat s(1-\cos\theta)/2$ and $\hat u=(p_1-k_+)^2=-\hat s(1+\cos\theta)/2$; $\theta$ is measured from the incoming quark in the partonic center-of-momentum frame. The trace identity and colorless pair-annihilation normalization are derived in the canonical QED calculation; see also Schwartz 2014, §§13.3.1–13.3.2, pp. 231–233, Eqs. (13.62)–(13.69).

For two distinct massless final particles, the incoming flux is $2\hat s$ and the azimuth-integrated phase space is $\mathrm d\Phi_2=\mathrm d\cos\theta/(16\pi)$. There is no identical-particle factor for $\ell^-\ell^+$. Consequently

$$\begin{aligned} \frac{\mathrm d\widehat\sigma_{q\bar q}}{\mathrm d\cos\theta} &=\frac{\overline{|\mathcal M|^2}}{32\pi\hat s} =\frac{\pi\alpha^2Q_q^2}{2N_c\hat s}(1+\cos^2\theta),\\ \widehat\sigma_{q\bar q}(\hat s) &=\frac{4\pi\alpha^2Q_q^2}{3N_c\hat s}. \end{aligned}$$

The integral $\int_{-1}^1(1+c^2)\,\mathrm dc=8/3$ checks the last factor. This hard cross section includes the complete lepton solid angle.

Pair mass, rapidity and the two beams

Work in the hadronic center-of-momentum frame, with beam $A$ along positive $z$ and $s=(P_A+P_B)^2$. At Born level $q=x_1P_A+x_2P_B$ and the pair has zero transverse momentum. Its rapidity is $Y=\tfrac12\ln[(q^0+q^3)/(q^0-q^3)]=\tfrac12\ln(x_1/x_2)$. Thus

$$x_1=\frac{Q}{\sqrt s}e^Y,\qquad x_2=\frac{Q}{\sqrt s}e^{-Y},\qquad 0<Q^2<s,\qquad |Y|\le\ln\frac{\sqrt s}{Q}.$$

The support follows from both $x_1\le1$ and $x_2\le1$, not from a cut on either beam alone. Differentiating the inverse map gives

$$\left|\frac{\partial(x_1,x_2)}{\partial(Q^2,Y)}\right| =\left|-\frac{x_1x_2}{Q^2}\right|=\frac1s, \qquad \mathrm dx_1\,\mathrm dx_2=\frac{\mathrm dQ^2\,\mathrm dY}{s}.$$

Equivalently, insert $\delta(Q^2-sx_1x_2)$ and $\delta(Y-\tfrac12\ln(x_1/x_2))$ in the original two-beam convolution. The same determinant integrates out $x_1,x_2$. Evaluating the hard cross section at $\hat s=Q^2$ yields

$$\begin{aligned} \frac{\mathrm d\sigma_{\mathrm B}}{\mathrm dQ^2\,\mathrm dY} ={}&\frac{4\pi\alpha^2}{3N_cQ^2s} \sum_q Q_q^2\bigl[ f_{q/A}(x_1,\mu_F)f_{\bar q/B}(x_2,\mu_F)\\ &\hspace{7em} +f_{\bar q/A}(x_1,\mu_F)f_{q/B}(x_2,\mu_F)\bigr]. \end{aligned}$$

The subscript $\mathrm B$ means Born. The flavor sum runs over the specified quarks once; the two terms are the two distinct ordered beam assignments. They contribute incoherently as different incoming partonic channels. No further factor of two or symmetry divisor belongs in this expression.

At fixed $Q^2$, $\tau=Q^2/s$ and $\mathrm dY=\mathrm dx_1/x_1$. Integrating over the full allowed rapidity range therefore gives precisely the ordered luminosities:

$$\frac{\mathrm d\sigma_{\mathrm B}}{\mathrm dQ^2} =\frac{4\pi\alpha^2}{3N_cQ^2s} \sum_qQ_q^2 \left[\mathcal L_{q\bar q}(\tau,\mu_F) +\mathcal L_{\bar q q}(\tau,\mu_F)\right].$$

The luminosity is dimensionless, $\mathrm d\sigma/\mathrm dQ^2$ has mass dimension $-4$, and an invariant-mass-bin integral has dimension $-2$. These checks expose a missing $1/s$ immediately.

A complete synthetic invariant-mass-bin rate

Use one quark flavor, and assign the same $f(x)=2(1-x)$ to $q$ and $\bar q$ in each beam at the chosen fixed scale. Each beam then has zero net valence for this flavor. A spectator gluon with the same $f$ can complete the momentum sum rule, since each species has $\int_0^1xf(x)\,\mathrm dx=1/3$; it does not enter this Born channel. These are synthetic boundary inputs, not proton PDFs.

Both ordered luminosities equal the already computed $\mathcal L(\tau)$. Defining $K=4\pi\alpha^2Q_q^2/(3N_c)$, the mass spectrum and its integral over $as\le Q^2\le bs$, with $0<a<b\le1$, are

$$\frac{\mathrm d\sigma_{\mathrm B}^{\mathrm{toy}}}{\mathrm dQ^2} =\frac{2K}{Q^2s}\mathcal L(Q^2/s), \qquad \sigma_{[a,b]}^{\mathrm{toy}} =\frac{2K}{s}\int_a^b\frac{\mathcal L(\tau)}{\tau}\,\mathrm d\tau.$$

An antiderivative of the dimensionless integrand is

$$\mathcal A(\tau) =12\tau-8\ln\tau-2\ln^2\tau-4\tau\ln\tau, \qquad \mathcal A'(\tau)=\frac{\mathcal L(\tau)}{\tau}.$$

For example, integrate the pair mass from $\sqrt s/2$ to $\sqrt s$ and all allowed rapidities:

$$\sigma_{[1/4,1]}^{\mathrm{toy}} =\frac{2K}{s} \left[9-9\ln4+2\ln^24\right].$$

The bracket is positive. This is a fully specified Born observable with exact normalization, rather than the luminosity alone. The lower mass cutoff keeps the massless photon calculation away from $Q=0$. The upper endpoint is integrable because the toy PDFs vanish at $x=1$; this mathematical fact does not remove enhanced threshold corrections in QCD. A precision prediction would require appropriate higher-order hard coefficients, consistently evolved PDFs and control of all relevant hierarchies.

A supplied correlated input uncertainty

To make covariance propagation concrete without violating the toy sum rules, let $H=A,B$ have small zero-mean normalization parameters $a_H$ and take

$$f_{q/H}=f_{\bar q/H}=(1+a_H)f, \qquad f_{g/H}=(1-2a_H)f.$$

The valence number remains zero and the total momentum remains one in each beam. Restrict the variations so that $-1<a_H<1/2$ if positivity is required. The gluon does not contribute to this Born hard channel; the entire bin rate is multiplied by $(1+a_A)(1+a_B)$.

The beams have the same central distributions in this example, but distinct, correlated nuisance parameters $a_A,a_B$. As a supplied synthetic covariance, choose standard deviations $0.04$ and $0.06$ with correlation $-1/2$:

$$V=\operatorname{Cov} \begin{pmatrix}a_A\\a_B\end{pmatrix} =\begin{pmatrix}0.0016&-0.0012\\-0.0012&0.0036\end{pmatrix}.$$

Linearizing about the central prediction $\sigma_0$ gives $\delta\sigma/\sigma_0=a_A+a_B$. The propagated relative variance and standard deviation are

$$\frac{\operatorname{Var}(\sigma)}{\sigma_0^2} \simeq(1,1)V\begin{pmatrix}1\\1\end{pmatrix} =0.0028,\qquad \frac{\Delta\sigma}{\sigma_0}\simeq5.29\%.$$

Discarding the anticorrelation would give $\sqrt{0.0052}=7.21\%$. This is a first-order propagation of a supplied covariance, not a PDF-fit error or a confidence claim about data. Nonlinear uncertainty propagation requires more information about the joint input distribution. Each common draw of $(a_A,a_B)$ must be used in both ordered channels before forming their summed rate; assigning them independent draws would create false averaging.

Scales, channels, and thresholds

$\mu_R$ belongs to the coupling and hard coefficient; $\mu_F$ separates initial-state collinear radiation from the PDFs. Their variations probe different missing terms and should be performed using a documented correlated prescription. If resummation introduces additional profile or rapidity scales, those are separate nuisance directions rather than aliases for $\mu_R$ and $\mu_F$.

For every channel record:

Field	Required content
incoming channel	ordered species $i,j$ and beam identities
hard definition	process, perturbative order, masses, electroweak inputs, and subtraction scheme
PDF definition	release identifier, factorization scheme, $n_f$, evolution order, and covariance representation
scales	central dynamic or fixed choices and allowed correlated variations
threshold treatment	fixed- or variable-flavor scheme and all matching orders
observable	cuts, bins, jet algorithm, recombination, decays, and measurement function
numerical integration	mappings, random seed policy, tolerance, and bin covariance

Flavor channels are correlated because they arise from one fitted PDF ensemble and share sum rules and data. A “dominant luminosity” does not justify varying that channel independently or neglecting interference and subleading channels without a quantified truncation.

Validation and failure modes

Born reconstruction. For a color-singlet of mass $Q$ and rapidity $Y$ at Born level,

$$x_{1,2}=\frac{Q}{\sqrt s}e^{\pm Y}.$$

Check $0<x_{1,2}\le1$ and the Jacobian of the transformation. Extra radiation changes the reconstruction, so do not impose the Born identity on a higher-multiplicity event.

Inclusive reduction. Set $\mathcal M=1$ and compare with a known inclusive convolution at the same order. Then restore cuts one at a time to isolate acceptance or subtraction problems.

Scale cancellation. Differentiate the PDF–coefficient convolution with respect to $\ln\mu_F^2$. DGLAP evolution of each beam must cancel the corresponding collinear logarithm in the coefficient through the claimed order. The explicit coefficient-cancellation example shows the sign for one incoming distribution. The Born Drell–Yan coefficient has no such QCD logarithm: its residual PDF scale variation is of order $\alpha_s$ relative to the Born result and is only compensated when the appropriate higher-order coefficient is included. This toy rate is not a demonstration of NLO cancellation.

PDF correlation. Recompute every bin and channel for each replica or eigenvector, then form the covariance from the resulting total predictions. Adding channelwise PDF errors in quadrature discards anticorrelations.

Endpoint and small-$x$ checks. A large invariant mass probes large $x$ and may enhance threshold logarithms; a highly asymmetric rapidity can probe small $x$. The collider energy alone does not determine either regime.

Factorization boundary. A veto or low recoil can require hard–beam–soft or TMD factorization. Spectator-sensitive colored observables need an explicit Glauber analysis.

Common pitfalls

Using luminosity as the cross section. It contains no matrix element, spin/color factor, cuts, or units of a rate. It is the incoming flux weight for a particular PDF setup.

Double counting beam order. Decide whether $ij$ is ordered or symmetrized and use the same choice in the hard-channel sum.

Applying cuts after convolution. Fiducial cuts act on phase space and can correlate with parton fractions, channel, and recoil. Keep the measurement function inside the event integral.

Exercises

A rapidity cut before convolution. At a fixed $\tau$, retain only $|Y|\le Y_c$ in the toy model, with $Y_c\ge0$. Derive the luminosity for one ordered channel and show that it returns the inclusive expression when the cut reaches the kinematic boundary.

Solution

Set $y_*=\min(Y_c,-\tfrac12\ln\tau)$. The product of the two toy distributions is $4[1+\tau-2\sqrt\tau\cosh Y]$, so

$$\begin{aligned} \mathcal L_{\mathrm{cut}}(\tau) &=\int_{-y_*}^{y_*}4[1+\tau-2\sqrt\tau\cosh Y]\,\mathrm dY\\ &=8[(1+\tau)y_*-2\sqrt\tau\sinh y_*]. \end{aligned}$$

At $y_*=-\tfrac12\ln\tau$, use $2\sqrt\tau\sinh y_*=1-\tau$ to recover $-8+8\tau-4(1+\tau)\ln\tau$. The cut mass spectrum is $2K\mathcal L_{\mathrm{cut}}/(Q^2s)$, including both ordered channels. Integrating to $Y_c$ when it exceeds the support would sample an unphysical momentum fraction rather than increase the accepted rate.

The same PDF in identical beams. If the two beams use the same uncertain toy PDF, set $a_A=a_B=a$, with standard deviation $u$. Find the first-order relative error on the rate and compare it with sampling the two beams independently.

Solution

The rate is $\sigma=\sigma_0(1+a)^2$, so $\delta\sigma/\sigma_0=2a$ and $\Delta\sigma/\sigma_0\simeq2u$. Independent beam draws would instead give $\sqrt2u$, an underestimate. Sharing one PDF ensemble between the beams requires sharing each replica or nuisance draw; the two partons being drawn from different hadrons does not make the fitted distribution’s uncertainty independent.

Using the collider prediction

A reproducible collider result passes forward

$$\left\{ \begin{gathered} \mathcal M,\ d\widehat\sigma_{ij}^{(k)},\ f_{i/A},f_{j/B},\ \mu_R,\mu_F,\mathcal S,n_f,\\ \text{threshold matches},\ \text{channel/bin covariance}, R_{\mathrm{pow}} \end{gathered} \right\}.$$

Use QCD radiation, jets, and event shapes when the measurement clusters colored radiation. Use the QCD prediction and uncertainty record to combine scale, PDF, parameter, nonperturbative, and numerical components without duplicating shared variations.

References

• Collins, John C., Davison E. Soper, and George Sterman. “Factorization of Hard Processes in QCD.” arXiv:hep-ph/0409313v1, 2004. Version cited: the manuscript with its own pagination. Versioned record. Open PDF.
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014. DOI.