Is the Standard Model consistent?
$$ m_W^2 \left(1 - \frac{m_W^2}{m_Z^2} \right) = \frac{\pi \alpha}{\sqrt{2} G_\mu} \left(1 + \Delta r(m_t, m_H,\ldots) \right) $$Is the Standard Model consistent?
$$ m_W^2 \left(1 - \frac{m_W^2}{m_Z^2} \right) = \frac{\pi \alpha}{\sqrt{2} G_\mu} \left(1 + \Delta r(m_t, m_H,\ldots) \right) $$
Is the Standard Model consistent?
Turn it around: $m_t$ determination
The SM measurement
Template fits in $p_T^l$, $p_T^\nu$, $m_T$ in electron and muon channels
Data-driven measurement "instead" of SM measurement
- Calibration of various detector components with respect to momenta and energy based on $c\bar{c}$ and $b\bar{b}$ meson resonances ($J/\Psi$, $\Upsilon$)
Data-driven measurement "instead" of SM measurement
- Calibration of various detector components with respect to momenta and energy based on $c\bar{c}$ and $b\bar{b}$ meson resonances ($J/\Psi$, $\Upsilon$)
- Differential Z-boson measurements used as further calibration and tuning!
Precision predictions!
NNLL+NLO ($\alpha_s$)
ResBos: Balázs, C.-P. Yuan, Ladinsky '94; '97The kinematic properties of W and Z boson production and decay are simulated using the RESBOS program, which calculates the differential cross section with respect to boson mass, transverse momentum, and rapidity for boson production and decay. The calculation is performed at next-to-leading order in perturbative quantumchromodynamics (QCD), along with next-to-next-to-leading logarithm resummation of higher-order radiative quantum amplitudes. RESBOS offers one of the most accurate theoretical calculations available for these processes.
The model-dependent nature of the analysis implies that the robustness of the measurement is dependent on the reliability of the models in use, which, although being the state-of-the-art, are continuously being updated and improved. The quoted uncertainties from those models are estimated using plausible assumptions in the context of the models themselves and, therefore, do not cover the possibility of significant updates.
But also public and well supported!
“If a theoretical calculation is done, but it can not be used by any experimentalist, does it make a sound?”
Theory uncertainties
- Fixed-order expansions in QCD and EW
- Higher-order resummation
- Parton showers
- Non-perturbative effects, PDFs, TMDs
- Higher power/twist terms in factorization
- Understanding universality of tuning
- Numerical precision
- ...
(Fully theory-driven SM measurement very far away from 10 MeV,
even with state-of-the-art results)
State-of-the-art theory
see Milano workshop "Precision calculations for Drell-Yan processes", Nov. '22
Improved power counting $$ \log(m_Z/q_T) \sim 1/\alpha_s $$
$$ \alpha_s \left(\alpha_s \log\left(\frac{m_Z}{q_T}\right)\right)^n$$ is the new $\alpha_s$ (NNLL)
Fixed order predictions
$$\alpha_s \log(Q^2/q_T^2)$$
$$ \sigma = \color{red}{\hat\sigma} \otimes f \otimes f + \, \mathcal{O}(\Lambda_\text{QCD} / m_Z) $$
$$ \sigma = \color{red}{\hat\sigma} \otimes f \otimes f + \, \mathcal{O}(\Lambda_\text{QCD} / m_Z) $$
- Loops $1/\epsilon$
- Legs $ \int \mathrm{d}\text{PS} |\mathcal{M}|^2 = \infty$
Legs:
$ \int_{q_T^\text{cut}} \mathrm{d}q_T |\mathcal{M}|^2 = \text{finite!}$
Legs:
$ \int_{q_T^\text{cut}} \mathrm{d}q_T |\mathcal{M}|^2 = \text{finite!}$
Factorization at small $q_T$:
$$ \mathrm{d}\sigma_{ij} \sim \int \mathrm{d}\xi_1 \mathrm{d}\xi_2\, \mathrm{d}\sigma^0_{ij} \cdot H(\xi_1 p_1, \xi_2 p_2, \mu) \cdot $$ $$ \cdot \int \mathrm{d}^2 x_\perp e^{-iq_\perp x_\perp} (x_T^2 Q^2)^{-F(x_\perp,\mu)} \cdot B_i(\xi_1,x_\perp,\mu) \cdot B_j(\xi_2,x_\perp,\mu) $$
Our results up to N$^3$LO
Neumann, Campbell '22
Resummation
Resummation for N$^3$LL'
- three-loop beam functions M.-x. Luo, T.-Z. Yang, H. X. Zhu, Y. J. Zhu '19, '20; Ebert, Mistlberger, Vita '20
Fixed-order Z+jet NNLO calculation
- 1. via 1-jettiness slicing Boughezal, Focke, Liu, Petriello; Boughezal, Campbell, Ellis, Focke, Giele, Liu, Petriello '15
- 2. via antenna subtractions Gehrmann-De Ridder, Gehrmann, Glover, Huss, Morgan '15
State of the field: antenna NNLO Z+jet interfaced to resummation:
- with DYturbo Camarda, Cieri, Ferrera '21
- with RadISH Bizon, Gehrmann-De Ridder, Gehrmann, Glover, Huss, Monni, Re, Rottoli, Walker '19; Chen, Gehrmann, Glover, Huss, Monni, Re, Rottoli, Torrielli '22
Moving beyond
Fixed-order $\alpha_s^3$ and logarithmic $\alpha_s^3$ accuracy while counting $\log(q_T^2/Q^2) \sim 1/\alpha_s$:
N$^4$LL+N$^3$LO up to N$^3$LO PDF's!
- Use independent Z+jet NNLO calculation (via 1-jettiness slicing)Boughezal, Focke, Liu, Petriello; Boughezal, Campbell, Ellis, Focke, Giele, Liu, Petriello '15
- N$^4$LL: Four loop rapidity anomalous dimensionDuhr, Mistlberger, Vita '22; Moult, H.X. Zhu, Y. J. Zhu '22
- e.g. Four-loop collinear anomalous dimensionAgarwal, von Manteuffel, Panzer, Schabinger '21
- Massive three-loop axial singlet contributions Chen, Czakon, Niggetiedt '22
But also public and well supported!
mcfm.fnal.gov
“If a theoretical calculation is done, but it can not be used by any experimentalist, does it make a sound?”
See also "Computational Challenges for Multi-loop Collider Phenomenology:
A Snowmass 2021 White Paper"
Febres Cordero, von Manteuffel, Neumann '22
New power counting $$\log(Q^2/q_T^2) \stackrel{!}{=} \mathcal{O}(1/\alpha_s)$$
Results
Fiducial results in comparison with CMS 13 TeV data
CMS measurement: 1909.04133
Fiducial results in comparison with CMS 13 TeV data
Cure for Jacobian peak in lepton $q_T$
Total fiducial cross-sections
This brings up the question of realistic uncertainites.
We find $2.5\%$, only other estimate with RadISH resummation: $1\%$ (2203.01565).
Question of PDFs!
$$ x^{a_1} (1-x)^{a_2} P(\sqrt{x}, a_j) $$
MSHT approximate N$^3$LO PDFs. Approximations for four-loop splitting functions, known information on small and large $x$, available Mellin moments.
These MHO effects are included in Hessian procedure as nuisance parameters.
How about for the Tevatron (W-mass)...
This doesn't look too bad...
How about for the Tevatron (W-mass)...
