This thesis computes order αs corrections on these boundaries, increasing the precision of the resulting constraints on the physical form factors. The jet functions in bHQET are then performed from m to Γ, adding the powers of some logarithms of the ratio m/Γ.
Constraining exclusive B decay measurements
It is often used to parametrize unknown parameters and relate them to other unknown parameters through general symmetry arguments. It is often possible to remove some physics we do not understand by taking its effects from somewhere else.
Regulating SCET in the infrared
In that chapter, the corrections to these limits are calculated, increasing the accuracy of the resulting constraints on the physical form factors.
Precision top quark physics
This can seriously reduce the accuracy of the calculation, as these logs effectively increase the coupling constant. The jet functions in bHQET are then run from m to Γ, summing the powers of the individual logarithms of the m/Γ ratio.
Jet definitions in SCET
Since these scales are so different, any log of them appearing in a perturbative calculation will be quite large. At the m scale the top quark mass is integrated by matching the SCET jet functions to a boosted version of the heavy quark effective theory (bHQET).
Plan of the thesis
This chapter is a brief overview of some aspects of the heavy quark effective theory (HQET). This exchange of momentum can change the velocity of the heavy quark by only an order of magnitude.
Leading-order in the heavy quark expansion
The presence of this projector in the heavy quark propagator results in a change in the interaction between heavy quarks and gluons. Since this vertex is always sandwiched between two heavy quark propagators, the effective vertex is justvµ.
Lagrangian beyond leading order in 1/m Q
In this chapter, we discuss the sum rules of the effective theory of heavy quarks and perturbative enhancement. The heavy quark symmetry [4] relates the form factors ¯B→D(∗)l¯ν to the corresponding Isgur-Wise function, with the result FD∗(w) =FD(w) =ξ(w) in the heavy quark limit of QCD.
Derivation of the generic sum rule
The order αs corrections to the Wilson coefficient of the leading operator are given by a matching calculation that shows the diagrams in Figs. Because HQET loop graphs do not change the matrix structures of inserted operators, perturbative corrections to matrix elements of the other stream cancel those of the leading operator in the OPE.
Vector and axial vector sum rules
As with the vector sum rule, the masses of the intermediate states are denoted by m(ℓ)(n)ℓ±1/2. Derivatives of the vector sum rule with p+q = 2 give expressions for σ2, while the additional factors of (wx−1) in the axial rule require p+q= 3 for curvature relations.
Physical bounds
When considering the limits in Eq. 3.45) and (3.46) and their dependence on ∆, it must be remembered that the logarithms of the perturbing corrections are only small if ∆, mb and mc are roughly of the same order. As in the derivation of the sum rules here, focusing on specific resonances provides form factor constraints.
Summary
This is a brief overview of some aspects of fuzzy collinear effective theory (SCET). With these definitions and scaling, we can proceed with the extension of the quark propagator. When we derive the leading-order Lagrangian SCET in the next section, it will be easy to confirm this form of the propagator.
Lagrangian
Just as the slight decrease of freedom in HQET could not change the velocity of a heavy quark, the mild degrees of freedom here cannot change the scale of the collinear quark's momentum. While the large fluctuations of the collinear quark field are fully removed from our Lagrangian, the same is not true of the collinear gluons. Note that only one term in the Lagrangian above contains interactions of the collinear quark field with ultrasoft gluons.
Label operators
In contrast, the collinear gluon still has large fluctuations that must be removed just as we did for the collinear quark:. For now, let's go ahead and write the Lagrangian with the ultrasoft gluons dropped where possible:. This expression allows us to easily verify the collinear quark propagator derived above by expanding the QCD propagator with the SCET power count.
Gauge invariance
Since ultrasoft fluctuations are on a much lower momentum scale (or a much larger distance scale) than collinear fluctuations, we can think of the ultrasoft gauge field as a classical background field for collinear degrees of freedom. At the same time, collinear fields must be transformed under collinear gauge transformations, just as they do in QCD to the background field gauge:. 4.41). Now we note that the problem with the Lagrangian form above is that it is not invariant due to the derivative term in the denominator.
Collinear-ultrasoft decoupling
One is usually interested in describing the interactions between these collinear degrees of freedom with nonperturbative degrees of freedom at rest that satisfy pµ ∼ (ΛQCD,ΛQCD,ΛQCD). Interactions between usoft and collinear degrees of freedom are contained in the leading Lagrangian of SCETI. However, the same field redefinition must be performed on the external operators in a given problem, and this reproduces the interactions with the infrequent degrees of freedom.
Matching from SCET I onto SCET II
Although the off-shellness terms do not agree logarithmically in the two theories, we argue that this is due to unregulated IR differences in SCETII. To determine the matching coefficient at one loop, we calculate matrix elements of the current in the two theories. This mixed divergence disappears in the sum of the two diagrams and we find, after adding also the contributions of the wave function.
Infrared regulators in SCET
Problems with known IR regulators
Other divergences arise if the three-momentum of the gluon goes to infinity or θ goes to π. The situation is different in SCETII, since the shelling of the light quark does not enter diagram (a). To check if the IR divergences of the two theories match, one needs a regulator that regulates all IR divergences in both SCETI and SCETII.
A new regulator for SCET
Note that these terms retain the immutability of the theory under the field redefinitions given in Eq. It follows from this discussion that the presence of the soft collinear messenger mode depends on the precise implementation of the IR regulator in the theory. Because the definition of an effective theory must be independent of the controller used for an explicit computation, one can think of the soft-collinear messenger mode as part of the IR controller.
Summary
But these widely separated scales also lead to QCD enhancements of the cross-section corrections in the form of logs, which must be summed. The Wilson coefficients of the second matching step can then be scaled down to the final scale, Γ. This process sums all relevant logs and the cross section can then be calculated in bHQET to give the final rescore.
QCD at one loop
Sudakov's logarithms are summed up by running the current in this theory up to the degree m. This result includes wave function contributions which cancel the poles of the vertex graph, and so the counterterm is zero here. For example, this result confirms that the one-mass SCET reproduces the full infrared of bulk QCD.
SCET I at one loop
Now we repeat the one-loop calculation above, but this time we take the outer particles to be nearly in the layer: ∆2 ≡ p2 −m2 ≈ 0. The result of the one-loop vertex corrections is the sum of the collinear vertex corrections given in two the first graphs of Fig. This is a check that one-measure SCET reproduces one-measure QCD and confirms the correctness of SCETm.
SCET I cross section
- Ultrasoft-collinear factorization
- Final state invariant mass constraints
- Specifying jet momenta and SCET I jet functions
- Computing SCET I jet functions
Furthermore, the usoft-collinear decoupling property of the SCETI Lagrangian will allow us to factorize the cross section so that the top quark and antiquark are decoupled from each other. In this section, we use the well-known property usoft-collinear decoupling of the leading order SCETILagrangian to decouple the top quark and antiquark from each other. It is given in terms of two decoupled jet functions, Jn and Jn¯, that describe the dynamics of the top quark and the antiquark moving in the ~n and ~n¯ directions, respectively.
Boosted HQET
The equations to match the SCETI jet functions Kn,¯n to the bHQET jet functions K(vn,vn¯) are given by. 6.54). The Wilson coefficients to match the SCETI jet functions to the bHQET jet functions in Eq. The evolution of the Wilson coefficients Cn,¯n under m is given by the anomalous dimension of the jet functions K(vn,vn¯)(y). 6.58).
Summary
We have an expression for moments of the cross section in the boundary that corresponds to toyn,¯n~1. The momentum of the jet is required to be equal to the total momentum of all collinear parts in the same direction as the hadronic jet. In this chapter we derive an expression for the jet P+ distribution in the semileptonic or radiative decays of a heavy meson, under the assumption explained above.
Review of radiative and semileptonic B decay in SCET
The heavy quark bv has momentum mbv+k, the track width boson has q and the total momentum of the unsoft Wilson line Yn is l. Thus, for the process B →Xsγ the photon energy spectrum is given by the expectation value hk+−l+i which is given by the well-known form function. The first is residual momentum, which classically describes the small recoil of the heavy b-quark inside a B-meson at rest.
Standard observables
The second is momentum l, which gives the total momentum lost by the energetically light quark due to soft radiation. Three steps are required to obtain expressions for the scalar functions Wi(f): (1) Matching the QCD fluxes to the SCET fluxes at µ1 ~ mb. 3) Integrate the final hadronic states at µ2 ~ p.
Jet P + distribution
To see this, we go back to the original definition of the matrix element of W. Somewhat schematically, we have. where ǫ is the polarization vector of the photon and the sum over final states includes all the phase space integrals. We can now use this expression to translate the initial position of the second Wilson line and then do the sum over ultrasoft conditions.
Summary
In ref. ref. We have also shown that a soft-collinear messenger mode is required in the definition of the IR regulator if we insist on counting the regulator powers in the same way as the kinetic expressions in the operation. The second and third parts then repeated the derivation of the standard lepton and photon distributions for comparison.
Diagrams contributing to the order α s corrections to the sum rules. The squares
One-loop renormalization of the leading operator in the operator product expansion of
Dispersive constraints on F D derivatives combined with the corrected sum rule bounds
Diagrams in SCET I contributing to the matching. The solid square denotes
Diagrams in SCET II contributing to the matching
Contribution of the additional SCET II mode proposed in Refs. [40]
Tree-level current (a) and the one-loop correction (b) in QCD
One-loop vertex correction in SCET
Forward scattering amplitudes
![Figure 5.3: Contribution of the additional SCET II mode proposed in Refs. [40].](https://thumb-ap.123doks.com/thumbv2/123dok/10400733.0/54.918.212.806.157.248/figure-contribution-additional-scet-ii-mode-proposed-refs.webp)
