Influence of Photon Inverse Emission on Forward-Backward Asymmetry in Dilepton Production at the LHC

This paper presents a detailed calculation and numerical analysis of the impact of photon inverse emission on dilepton production cross sections and forward-backward asymmetry at the LHC's Run 3 and HL-LHC regimes, utilizing an effective technique of additive relative corrections to assess these radiative contributions across a wide kinematic region.

The Gist

The Big Picture: Looking for Ghosts in the Machine

Imagine the Large Hadron Collider (LHC) as the world's most powerful "smashing machine." Scientists crash protons together at nearly the speed of light to see what breaks apart. Usually, they are looking for brand-new, heavy particles (like finding a new type of Lego brick that nobody has ever seen before).

However, sometimes the "new physics" isn't a giant new brick; it's a tiny, subtle wobble in how the existing bricks fit together. This paper is about measuring those tiny wobbles very precisely to see if the rules of the universe (the Standard Model) are slightly broken.

The Main Event: The "Dilepton" Dance

The scientists are watching a specific event: two protons collide, and out pops a pair of heavy particles called muons (one positive, one negative). Think of this as a dance where two partners (the muons) are created and fly off in opposite directions.

In the "standard" version of this dance (called the Drell-Yan process), the partners are created by a direct handshake between a quark and an antiquark inside the protons. This is the main act, and we know the choreography very well.

The Plot Twist: The "Photon Inverse Emission"

This paper focuses on a sneaky, background dancer that often gets ignored. It's called Photon Inverse Emission.

The Analogy: The Over-enthusiastic Fan
Imagine the main dancers (the quarks) are trying to perform their routine. But, just as they are about to collide, one of them gets so excited they accidentally throw a flashbulb (a photon) at the other person before the main collision happens. This flashbulb changes the momentum of the dancers slightly.

In physics terms, this is "inverse emission." Instead of a particle emitting a photon and losing energy, a photon from the "crowd" (the proton's cloud of particles) hits a quark, changing the way the collision happens.

Most physicists treat this as a tiny, boring background noise. But this paper argues: "Wait a minute! If we are looking for tiny deviations caused by New Physics, we need to know exactly how loud this background noise is, or we might mistake the noise for a new discovery."

The Specific Question: Forward vs. Backward

The authors aren't just counting how many muons are made; they are looking at where they go.

• Forward: The muons fly in the same direction the original proton was moving.
• Backward: The muons fly in the opposite direction.

They measure the Forward-Backward Asymmetry. It's like asking: "Do the dancers prefer to spin clockwise or counter-clockwise?"

The Standard Model predicts a very specific ratio of clockwise to counter-clockwise spins. If a "New Physics" force is hiding in the data, it will tilt this ratio slightly.

The Problem: The "Mathematical Noise"

The paper calculates exactly how much this "Photon Inverse Emission" (the over-enthusiastic fan throwing flashbulbs) changes the spin ratio.

They found that at low energies, this effect is tiny and doesn't matter. But at ultra-high energies (like the future runs of the LHC, where protons smash with 13.6 TeV of energy), this effect becomes significant.

The "Additive" Secret
The authors developed a clever mathematical trick. Instead of trying to recalculate the entire universe every time they add a new effect, they calculate the "relative correction" (the percentage change) for each effect separately.

• Think of it like baking a cake. You calculate how much sugar changes the taste. Then you calculate how much salt changes the taste. Because they are "additive," you can just add the sugar-change and salt-change together to get the total flavor shift. This makes the math much faster and more accurate.

The Results: When Does It Matter?

The paper crunches the numbers for the CMS experiment (one of the big detectors at the LHC).

1. Low Energy (Below 3 TeV): The "flashbulb" effect is negligible. The standard dance is fine.
2. High Energy (Above 3 TeV): The effect grows. At these extreme speeds, the "Photon Inverse Emission" changes the Forward-Backward asymmetry by about 1%.

Why is 1% a big deal?
In high-energy physics, a 1% shift is massive. It is right on the edge of what current detectors can measure. If the scientists don't account for this 1% shift caused by the "flashbulbs," they might think they found a new particle when they actually just forgot to subtract the background noise.

The Conclusion

This paper is a "calibration manual." It tells the LHC scientists: "When you look at the highest energy collisions in the future, you must subtract this specific 1% effect caused by photon inverse emission. If you do, your measurements will be clean enough to spot the real, tiny signals of New Physics. If you don't, you might get fooled by the noise."

It's a reminder that to find the extraordinary, you must first understand the ordinary with extreme precision.

Technical Summary

1. Problem Statement

The paper addresses the need for high-precision theoretical predictions for dilepton (specifically dimuon, $\mu^-\mu^+$) production at the Large Hadron Collider (LHC), particularly in the ultra-high energy regime (Run 3 and High-Luminosity LHC).

• Context: While the Drell-Yan process ($q\bar{q} \to \ell^+\ell^-$) is well-studied, precision measurements of the Forward-Backward Asymmetry ($A_{FB}$) at large invariant masses ($M > 3$ TeV) are sensitive to New Physics (NP).
• The Gap: Minor deviations from Standard Model (SM) predictions are the primary signature for NP. To distinguish these, theoretical uncertainties must be minimized. The paper focuses on Photon Inverse Emission (Photon IE) mechanisms (specifically $\gamma q \to \ell^+\ell^- q$ and $\gamma \ell \to \ell^+\ell^- \ell$), which are often treated as background or higher-order corrections but are of the same perturbative order as electroweak radiative corrections (RCs) and two-photon fusion.
• Objective: To calculate the contribution of photon inverse emission to the dilepton cross-section and forward-backward asymmetry, and to quantify its impact on $A_{FB}$ in the kinematic region relevant for the CMS experiment.

2. Methodology

The author employs a rigorous theoretical framework combining analytical calculations with numerical analysis using the READY (Radiative corrEcutions to lArge invariant mass Drell–Yan process) program.

• Theoretical Framework:

  • Processes Analyzed: The study considers four main mechanisms: Drell-Yan (DY), Two-Photon Fusion ($\gamma\gamma$), Gluon Inverse Emission ($gIE$), and Photon Inverse Emission ($\gamma IE$).
  • Parton-Level Calculation: The amplitude for $\gamma q \to \ell^+\ell^- q$ is calculated using Feynman diagrams involving intermediate $\gamma$ and $Z$ bosons. The squared amplitudes are computed using Casimir's trick, averaging over initial spins and summing over final spins.
  • Kinematics: The calculation utilizes six partonic Lorentz invariants ($s, t, u, s_1, t_1, u_1$) and defines the phase space for the $2 \to 3$ process.
  • Leading Logarithm (LL) Approximation: To handle collinear singularities (where the emitted photon/quark is collinear with the parent), the author derives expressions in the LL approximation. This involves introducing a splitting parameter $\eta$ and utilizing Altarelli-Parisi splitting functions ($P_{\gamma q}$ and $P_{\gamma \ell}$).
  • Mass Singularity Handling: To remove unphysical dependence on quark masses (quark singularities), the MS-subtraction technique is applied. A specific "QS-term" mimicking the LL structure is subtracted from the exact cross-section, ensuring the physical result is independent of the arbitrary quark mass parameter.

• Numerical Implementation:

  • PDFs: Uses the NNPDF4.0 set (via LHAPDF 6.5.5) with a natural scale $Q = M$.
  • Experimental Constraints: Simulates CMS Run 3/HL-LHC conditions: $\sqrt{S} = 13.6$ TeV, lepton pseudorapidity $|y| < 2.5$, and transverse momentum $p_T \ge 20$ GeV.
  • Asymmetry Calculation: The Forward-Backward Asymmetry is defined as $A_{FB} = (\sigma_F - \sigma_B) / (\sigma_F + \sigma_B)$. The author introduces an additive relative correction technique:
    $$\delta^c_{\pm} = \frac{\sigma^c_F \pm \sigma^c_B}{\sigma^0_F \pm \sigma^0_B}$$
    This allows multiple correction sources (QCD, EW, $\gamma IE$, etc.) to be summed linearly to obtain the net correction to the asymmetry, avoiding complex multiplicative interference issues.

3. Key Contributions

1. Detailed Calculation of Photon IE: The paper provides a complete, first-principles calculation of the photon inverse emission contribution to dilepton production, distinguishing between quark-initiated ($\gamma q$) and lepton-initiated ($\gamma \ell$) channels.
2. Additive Correction Formalism: The author demonstrates that relative corrections to $A_{FB}$ are additive. This simplifies the combination of various radiative corrections (QCD, Electroweak, Inverse Emission) into a single predictive model.
3. Mass Independence Verification: Through the MS-subtraction method, the paper proves that the physical cross-section and asymmetry corrections are independent of the quark mass over a wide range ($10^{-6}$ to $10^{-3}$ of the dilepton mass), validating the robustness of the calculation.
4. Kinematic Mapping: The study explicitly maps the partonic variables to the hadronic observables (invariant mass $M$, rapidity $y$, and Collins-Soper angle $\theta^*$) required for experimental comparison.

4. Results

• Magnitude of Corrections:
  • Quark Channel ($\delta_{\gamma IE, q}$): The relative corrections are small (negative, around -0.1% to -0.5%) and grow slowly with invariant mass $M$.
  • Lepton Channel ($\delta_{\gamma IE, \mu}$): This is the dominant contribution. The correction is negative and grows rapidly with $M$. At $M \approx 5$ TeV, the correction becomes significant.
  • Interference: The interference term is negligible.
• Comparison with Other Mechanisms:
  • Photon IE effects are generally smaller than Gluon IE effects (due to the stronger QCD coupling and higher gluon PDFs) but become non-negligible at ultra-high masses.
  • Photon IE contributions become noticeable primarily in the region $M > 3$ TeV.
• Impact on Forward-Backward Asymmetry ($A_{FB}$):
  • The total relative corrections ($\delta_{\pm}$) to the asymmetry show significant mutual compensation between different mechanisms.
  • The inclusion of all radiative corrections (including Photon IE) leads to a deviation from the Born-level asymmetry that is significant at large $M$.
  • The predicted effect size is at the 1% level.

5. Significance

• New Physics Searches: The 1% level of correction is comparable to the expected statistical and systematic uncertainties of the CMS experiment in the HL-LHC era. Therefore, neglecting Photon Inverse Emission could lead to false signals of New Physics or mask genuine deviations.
• Precision Physics: The work establishes that for precision measurements of $A_{FB}$ at invariant masses exceeding 3 TeV, Photon IE must be included in the theoretical baseline.
• Methodological Utility: The additive correction technique proposed provides a practical tool for experimentalists to combine various theoretical corrections efficiently when analyzing LHC data.
• Validation of Tools: The results validate the READY program as a reliable tool for estimating electroweak and QCD corrections in high-mass Drell-Yan processes.

In conclusion, this paper provides a critical theoretical update for the LHC physics program, demonstrating that photon inverse emission is a necessary component of precision Standard Model predictions for dilepton production at ultra-high energies.