We calculate non-singlet quark operator matrix elements of deep-inelastic scattering in the chiral limit including operators with total derivatives. This extends previous calculations with zero-momentum transfer through the operator vertex which provides the well-known anomalous dimensions for the evolution of parton distributions, as well as calculations in off-forward kinematics utilizing conformal symmetry. Non-vanishing momentum-flow through the operator vertex leads to mixing with total derivative operators under renormalization. In the limit of a large number of quark flavors $n_f$ and for low moments in full QCD, we determine the anomalous dimension matrix to fifth order in the perturbative expansion in the strong coupling $α_s$ in the $\overline{\mbox{MS}}$-scheme. We exploit consistency relations for the anomalous dimension matrix which follow from the renormalization structure of the operators, combined with a direct calculation of the relevant diagrams up to fourth order.
The renormalization constant $Z_J$ of the flavor-singlet axial-vector current with a non-anticommuting $γ_5$ in dimensional regularization is determined to order $α_s^3$ in QCD with massless quarks. In addition, the equality between the $\overline{\mathrm{MS}}$ renormalization constant $Z_{F\tilde{F}}$ associated with the operator $\big[F \tilde{F} \big]_{R}$ and that of $α_s$ is verified explicitly to hold true at 4-loop order.
We match the electroweak chiral Lagrangian with two singlet scalars to the next-to-minimal composite Higgs model with $ SO(6)/SO(5) $ coset structure and extract the scalar divergences to one loop. Assuming the additional scalar to be heavy, we integrate it out and perform a matching to the well-established electroweak chiral Lagrangian with one light Higgs.
Splitting functions govern the scale evolution of parton distribution functions. Through a Mellin transformation, they are related to anomalous dimensions of twist-two operators in the operator product expansion. We study off-shell operator matrix element, where the physical operators mix under renormalization with other gauge-variant operators of the same quantum numbers. We devise a new method to systematically extract the Feynman rules resulting from those operators without knowing the operators themselves. As a first application of the new approach, we independently reproduce the well-know…
The framework of the electroweak chiral Lagrangian with a light Higgs is extended by an additional scalar and then generalized to N scalars in the Higgs sector. Divergences from scalar fluctuations are renormalized up to one loop using the background field method. The results are crosschecked against the case of one scalar. A subset of the divergences is demonstrated and crosschecked diagrammatically. Together with the complete one-loop renormalization of the electroweak chiral theory with one light Higgs conducted previously, this constitutes a renormalization framework of any pure scalar ext…
We investigate the nucleon's transverse spin-dependent structure function g_2(x, Q^2) in the framework of the operator product expansion and the renormalization group. We construct the complete set of the twist-3 operators for the flavor singlet channel, and give the relations among them. We develop an efficient, covariant approach to derive the anomalous dimension matrix of the twist-3 singlet operators by computing the off-shell Green's functions. As an application, we investigate the Q^2-evolution of g_2(x, Q^{2}) for the lowest moment case, and discuss its experimental implication.
We introduce a new method for calculating the mixing matrix for non-singlet quark operators including total derivatives, based solely on their renormalization structure in the chiral limit. As input, the method requires the well-known forward anomalous dimensions, which determine the evolution of parton distribution functions, and a calculation of the matrix elements of operators without total derivatives. Assuming a large number of quark flavors $n_f$, we are able to calculate the mixing matrix to fifth order in the strong coupling $α_s$ in the $\overline{\mbox{MS}}$-scheme.
We use the BLM procedure to eliminate the renormalization scale ambiguity in the evolution equation for the non-singlet deep-inelastic structure function $F_2^{\text NS}(x,Q).$ The scale of the QCD coupling in the $\overline{\text{MS}}$ scheme has the form $Q^*(x) = Q (1-x)^{1/2} / x~f(x) $, where $x$ is the Bjorken variable and $f(x)$ is a smoothly varying function bounded between 0.30 to 0.45. Equivalently, the evolution of the $n$th moment of the structure function should contain an effective $Λ_{\text{QCD}}$ pattern, with $Λ_n \sim n^{1/2}$. This variation of $Λ_n$ agrees with experimental…
We present a general method for the solution of the renormalization group equations for the non-forward parton distributions on the two-loop level in the flavour singlet channel based on an orthogonal polynomial reconstruction. Using this formalism we study the effects of the evolution on recently proposed model distribution functions.
We provide the $\bar{\mathrm{MS}}$ factor of the renormalization constant of the singlet axial-current operator in dimensional regularization at $\mathcal{O}(α_s^5)$ in perturbative Quantum Chromodynamics (QCD). The result is obtained from a formula derived using the Adler-Bell-Jackiw equation in terms of renormalized operators. The required input consists of the complete four-loop results for operator renormalization constants obtained by explicit diagrammatic computation in a previous publication.
We determine the anomalous dimension matrix for the transversity operator mixing into total derivative operators in the limit of a large number of quark flavors $n_f$ to fourth order in the strong coupling $α_s$ in the $\overline{\text{MS}}$-scheme. This is achieved by using a consistency relation between the operator anomalous dimensions, which follows from the renormalization structure of the operators in the chiral limit.
We study the decay of a heavy Higgs boson into a light Higgs pair at one loop in the singlet extension of the Standard Model. To this purpose, we construct several renormalization schemes for the extended Higgs sector of the model. We apply these schemes to calculate the heavy-to-light Higgs decay width at next-to-leading order electroweak accuracy, and demonstrate that certain prescriptions lead to gauge-dependent results. We comprehensively examine how the NLO predictions depend on the relevant singlet model parameters, with emphasis on the trademark behavior of the quantum effects, and how …
We calculate the two loop correction to the quark 2-point function with the non-zero momentum insertion of the flavour singlet axial vector current at the fully symmetric subtraction point for massless quarks in the modified minimal subtraction (MSbar) scheme. The Larin method is used to handle $γ^5$ within dimensional regularization at this loop order ensuring that the effect of the chiral anomaly is properly included within the construction.
The parameter space of the simplest extension of the standard model, is studied in the light of the 125 GeV Higgs boson discovery. The Hill model extends the scalar sector of the standard model with a real singlet, that mixes with the SM Higgs boson. The two-loop SM renormalization group equations are completed with the one-loop Hill equations. Stability up to the Planck scale is possible without tension with the other parameters. An extension with more singlet fields, in particular a higher dimensional (HEIDI) field is presented.
We use the large N_f self consistency formalism to compute the $O(1/N_f)$ critical exponent corresponding to the renormalization of the flavour non-singlet twist two Wilson operators which arise in the operator product expansion of currents in deep inelastic processes. Expanding the $d$-dimensional expression in powers of $ε$ $=$ $(4-d)/2$ the coefficients of $ε$ agree with the known two loop structure of the corresponding renormalization group function and we deduce analytic expressions for all moments, $n$, at three and higher orders in perturbation theory in the $\overline{\mbox{MS}}$ schem…
We provide the renormalization constants of the axial current operators, both singlet and non-singlet, in dimensional regularization up to four-loop order in QCD, determined using the off-shell Ward-Takahashi identity for an axial current with a non-anticommuting $γ_5$. A possible application of the result for the singlet axial current operator is the extraction of the non-decoupling mass logarithms in the axial quark form factors.
We present the next-to-leading-order calculation of the partial decay widths of light CP-even Higgs bosons decaying into four fermions in the Two-Higgs-Doublet Model and a Singlet Extension of the Standard Model. Different renormalization schemes are applied in the calculation, which is implemented into the analysis tool PROPHECY4F. Some sample results on the Higgs->4fermions decay widths illustrate how the corrections reduce the dependence on the renormalization scale and the choice of the scheme.
The Next-to-Minimal 2-Higgs-Doublet Model (N2HDM) is an interesting benchmark model for a Higgs sector consisting of two complex doublet and one real singlet fields. Like the Next-to-Minimal Supersymmetric extension (NMSSM) it features light Higgs bosons that could have escaped discovery due to their singlet admixture. Thereby, the model allows for various different Higgs-to-Higgs decay modes. Contrary to the NMSSM, however, the model is not subject to supersymmetric relations restraining its allowed parameter space and its phenomenology. For the correct determination of the allowed parameter …
The importance of Yukawa contributions to the renormalization of the spectrum in non-minimal supersymmetric models is illustrated in the cases of explicit lepton number violation (leading to the possibility of singly produced sneutrinos at LEP energies), an intermediate scale singlet neutrino and negative mass squared parameters (possibly modifying fine-tuning considerations), and a grand-unified sector. The relevance of model-dependent renormalization to the supersymmetric flavor problem is emphasized. Sources of the flavor problem, some of which are newly identified, as well as possible solu…
We investigate the nucleon's transverse spin-dependent structure function g_2(x, Q^{2}) in the framework of the operator product expansion and the renormalization group. We construct the complete set of the twist-3 operators for the flavor singlet channel, and give the relations among them. We develop an efficient, covariant approach to derive the anomalous dimension matrix of the twist-3 singlet operators by computing the off-shell Green's functions. As an application, we investigate the renormalization mixing for the lowest moment case, including the operators proportional to the equations o…
We consider new physics contributions to the top-quark forward-backward asymmetry from a neutral $V^0_8$ or charged $V^+_8$ color-octet vector exchanged in the $t$-channel. We study the phenomenological constraints on these particles arising from the Tevatron and LHC7 measurements and compare them with those on their color singlet counterparts $Z^\prime$ and $W^\prime$. We find that the color octets fare better than the singlets in that they generate a lower $A_C$, a lower high-invariant mass cross-section at LHC7 and a lower same sign top-pair cross-section. However, they also generate a lowe…
In the paper, we calculate the fragmentation functions for $c \to η_c$ and $b \to η_b$ up to next-to-leading-order (NLO) QCD accuracy. The ultraviolet divergences in the real corrections are removed through operator renormalization under the modified minimal subtraction scheme. We then obtain the fragmentation functions $D_{c \to η_c}(z,μ_F)$ and $D_{b \to η_b}(z,μ_F)$ up to NLO QCD accuracy, which are presented as figures and fitting functions. The numerical results show that the NLO corrections are significant. The sensitives of the fragmentation functions to the renormalization scale and th…
We resum the leading ultrasoft logs of the singlet and octet static QCD potentials within potential NRQCD. We then obtain the complete three-loop renormalization group improvement of the singlet QCD static potential. The discrepancies between the perturbative evaluation and the lattice results at short distances are slightly reduced.
The prescription for the $γ_5$-matrix within dimensional regularization in multiloop calculations is elaborated. The three-loop anomalous dimension of the singlet axial current is calculated.
We compute the three-loop corrections to the quark axial vector form factor in massless QCD, focusing on the pure-singlet contributions where the axial vector current couples to a closed quark loop. Employing the Larin prescription for $γ_5$, we discuss the UV renormalization of the form factor. The infrared singularity structure of the resulting singlet axial-vector form factor is explained from infrared factorization, defining a finite remainder function.
The CDF and D0 collaborations have recently reported a large forward-backward asymmetry in the ttbar system which deviates from the next to-leading order QCD standard model prediction. We study the asymmetry in the ttbar system within the framework of singlet extensions of the Minimal Supersymmetric Standard Model. For this purpose, we introduce non-renormalizable couplings between first and third generation of quarks to scalars. We analyze two limiting cases of the model, characterized by the size of the supersymmetric mass for the singlet superfield. We study both the small and large limits …
The renormalization constant $Z_J$ of the flavor-singlet axial-vector current with a non-anticommuting $γ_5$ in dimensional regularization is determined to order $α_s^3$ in QCD with massless quarks. The result is obtained by computing the matrix elements of the operators appearing in the axial-anomaly equation $\big[\partial_μ J^μ_{5} \big]_{R} = \frac{α_s}{4 π}\, n_f\, {\displaystyle \mathrm{T}_{F}} \, \big[F \tilde{F} \big]_{R}$ between the vacuum and a state of two (off-shell) gluons to 4-loop order. Furthermore, through this computation, the conjectured equality between the $\overline{\mat…
Closed expressions are presented for the contributions to QCD non-singlet forward evolution kernels $P(z)$ for the DGLAP equation and to $V(x,y)$ for non-forward (ER-BL) evolution equation for a certain class of diagrams which include renormalon chains. Calculations are performed in covariant $ξ$-gauge, in the \MSbar scheme. The assumption of "naive nonabelianization" approximation for kernel calculations is discussed, and a special choice of the gauge parameter $ξ=-3$ is analyzed in this context. Partial solutions to the ER-BL evolution equation are obtained.
Potential non-relativistic QCD provides a convenient framework to study the three heavy-quark system. We have studied its symmetries under exchange of the heavy quarks and determined the leading ultrasoft contributions to the singlet static potential and to the singlet static energy. Specializing to an equilateral geometry, we have solved the renormalization group equations in order to resum the leading ultrasoft logarithms for the singlet static potential, which is thus known at NNLL accuracy.
We calculate the leading twist contribution to near-forward proton-proton (and proton-antiproton) elastic scattering with large momentum transfer, in the multiple scattering (Landshoff) mechanism. The amplitude in the near-forward region is dominated by singlet exchange for all three valence quark-quark scatterings. We assume the existence of a hard singlet quark-quark amplitude, which we estimate to be ${\cal O}(α_s^2/t)$. For a three-quark state whose transverse size is less than $1/Λ_{\rm QCD}$, Sudakov resummation accounts for both approximate $\d σ^{pp}/ \d t \sim t^{-8}$ at moderate $t$,…
It is well known that the effect of top quark loop corrections in the axial part of quark form factors (FF) does not decouple in the large top mass or low energy limit due to the presence of the axial-anomaly type diagrams. The top-loop induced singlet-type contribution should be included in addition to the purely massless result for quark FFs when applied to physics in the low energy region, both for the non-decoupling mass logarithms and for an appropriate renormalization scale dependence. In this work, we have numerically computed the so-called singlet contribution to quark FFs with the exa…
We present the results of the first complete one-loop matching calculation between the real singlet scalar extension of the Standard Model and the Standard Model effective field theory (SMEFT) at dimension six. Beyond their immediate relevance to the precision calculation of observables in singlet extensions of the Standard Model, our results illustrate a variety of general features of one-loop matching. We explore the interplay between non-supersymmetric non-renormalization theorems, the logarithmic dependence of Wilson coefficients, and the relevance of mixed diagrams in theories with large …
We consider an extension of the Standard Model by a real singlet scalar field with a $Z_2$-symmetric Lagrangian and spontaneous symmetry breaking with vacuum expectation value for the singlet. Considering the lighter of the two scalars of the theory to be the 125GeV Higgs particle, we parametrize the scalar sector by the mass of the heavy Higgs boson, a mixing angle $α$, and a scalar Higgs self-coupling $λ_{12}$. Taking into account theoretical constraints from perturbativity and vacuum stability, we compute next-to-leading-order electroweak and QCD corrections to the decays $h\to WW/ZZ \to 4$…
We study the large-$t$ small angle behavior of quark-quark elastic scattering. We employ a factorization procedure previously developed for fixed angle scattering, which depends on the color structure of the factorized hard subprocess. We find an evolution in $t$ that (in leading logarithmic approximation) becomes diagonal in a singlet-octet basis in the $t$-channel as $s\rightarrow \infty$. Octet exhange in the hard scattering is associated with the familiar `reggeized', $s^{α_g(t)}$ behavior, which arises from $s$-dependence in Sudakov suppression. In contrast, Sudakov suppression for $t$-ch…
A method for calculating coefficient functions of the operator product expansion, which was previously derived for the non-singlet case, is generalized for the singlet coefficient functions. The resulting formula defines coefficient functions entirely in terms of corresponding singlet composite operators without applying to elementary (quark and gluon) fields. Both "diagonal" and "non-diagonal" gluon coefficient functions in the product expansion of two electromagnetic currents are calculated in QCD. Their renormalization properties are studied.
We study the next-to-next-to-leading order (NNLO) evolution of flavour non-singlet quark densities and structure functions in massless perturbative QCD. Present information on the corresponding three-loop splitting functions is used to derive parametrizations of these quantities, including Bjorken-x dependent estimates of their residual uncertainties. Compact expressions are also provided for the exactly known, but rather involved two-loop coefficient functions. The size of the NNLO corrections and their effect on the stability under variations of the renormalization scale are investigated. Th…
We analyse non-local rotating observables in holography corresponding to spinning bound states. To renormalize their energies and momenta we suggest and discuss different holographic renormalization schemes motivated by the static non-local observables. Namely the holographic renormalization and the rotating color singlet mass subtraction scheme. In the holographic renormalization we identify the infinite boundary terms and subtract them. In the mass subtraction scheme we evaluate the energy of a spinning trailing string corresponding to the color charged singlet which experiences dragging phe…
We compute one-loop renormalization group equations for non-singlet twist-four operators in QCD. The calculation heavily relies on the light-cone gauge formalism in the momentum fraction space that essentially rephrases the analysis of all two-to two and two-to-three transition kernels to purely algebraic manipulations both for non- and quasipartonic operators. This is the first brute force calculation of this sector available in the literature. Fourier transforming our findings to the coordinate space, we checked them against available results obtained within a conformal symmetry-based formal…
An upperbound on the mass of the lightest neutral scalar Higgs boson is calculated in an extended version of the minimal supersymmetric standard model that contains an additional Higgs singlet. We integrate the renormalization group equations of the model, and impose low energy boundary conditions consistent with present experimental results, and ultra-violet conditions following from triviality. Radiative corrections induced by a large top quark Yukawa coupling are included in our analysis, and we find the allowed values for the mass of the Higgs boson as a function of the mass of the top qua…
We study the effects of next-to-leading order corrections on the evolution of the twist-two non-forward parton distribution functions in the flavour non-singlet sector. It is found that the deviation from leading order evolution is small for all values of the parton momentum fraction variable for moderately large values of the scale parameter.
We discuss renormalization effects on neutrino masses and mixing angles in a supersymmetric string-inspired SU(4) X SU(2)_L X SU(2)_R X U(1)_X model, with matter in fundamental and antisymmetric tensor representations and singlet Higgs fields charged under the anomalous U(1)_X family symmetry. The quark, lepton and neutrino Yukawa matrices are distinguished by different Clebsch-Gordan coefficients. The presence of a second U(1)_X breaking singlet with fractional charge allows a more realistic, hierarchical light neutrino mass spectrum with bi-large mixing. By numerical investigation we find a …
We discuss the general operator product expansion of a non-forward unequal mass virtual compton scattering scattering amplitude. We find that the expansion now should be done in double moments with new moment variables. There are in the expansion new sets of leading twist operators which have overall derivatives, and they mix under renormalization. We compute the evolution kernels from which the anomalous dimensions for these operators can be extracted. We also obtain the lowest order Wilson Coefficients. In the high energy limit we find the explicit form of the dominant contributing anomalous…
We compute a variety of operator-operator correlation functions to third order in the MSbar scheme in the chiral limit. These include combinations of quark bilinear currents with gauge invariant operators such as moments n = 2 and 3 of the flavour non-singlet Wilson operators of deep inelastic scattering and moment n = 2 of the transversity operator, as well as the correlation functions of the latter operators with themselves. The explicit values of these gauge independent correlation functions are required to assist with non-perturbative matching to lattice regularized calculations of the sam…
We develop a framework for the reconstruction of the non-forward kernels which govern the evolution of twist-two distribution amplitudes and off-forward parton distributions beyond leading order. It is based on the knowledge of the special conformal symmetry breaking part induced by the one-loop anomaly and conformal terms generated by forward next-to-leading order splitting functions, and thus avoids an explicit two-loop calculation. We demonstrate the formalism by applying it to the chiral odd and flavour singlet parity odd sectors.
A careful and complete discussion is given of the renormalization of the singlet axial anomaly equation in a vector-like nonabelian gauge theory such as QCD regularized by ordinary dimensional regularization. Pseudotensorial structures are treated with the 't Hooft-Veltman prescription. A general framework for calculations is developed, and subsequently verified by explicit computations through two loops. This is followed by a discussion of the matrix elements obtained.
We study the evolution of the flavour non-singlet deep-inelastic structure functions F_{2,NS} and F_3 at the next-to-next-to-next-to-leading order (N^3LO) of massless perturbative QCD. The present information on the corresponding three-loop coefficient functions is used to derive approximate expressions of these quantities which prove completely sufficient for values x > 10^{-2} of the Bjorken variable. The inclusion of the N^3LO corrections reduces the theoretical uncertainty of alpha_s determinations from non-singlet scaling violations arising from the truncation of the perturbation series t…
In the paper, we calculate the fragmentation functions for a quark to fragment into a spin-singlet quarkonium, where the flavor of the initial quark is different from that of the constituent quark in the quarkonium. The ultraviolet divergences in the phase space integral are removed through the operator renormalization under the modified minimal subtraction scheme. The fragmentation function $D_{q \to η_Q}(z,μ_F)$ is expressed as a two-dimensional integral. Numerical results for the fragmentation functions of a light quark or a bottom quark to fragment into the $η_c$ are presented. As an appli…
We compute the next-to-next-to-leading order (NNLO) contributions to the three splitting functions governing the evolution of unpolarized non-singlet combinations of quark densities in perturbative QCD. Our results agree with all partial results available in the literature. We find that the correct leading logarithmic (LL) predictions for small momentum fractions x do not provide a good estimate of the respective complete results. A new, unpredicted LL contribution is found for the colour factor d^{abc}d_{abc} entering at three loops for the first time. We investigate the size of the correctio…
We study the symmetries of the three heavy-quark system under exchange of the quark fields within the effective field theory framework of potential non-relativistic QCD. The symmetries constrain the form of the matching coefficients in the effective theory. We then focus on the color-singlet sector and determine the so far unknown leading ultrasoft contribution to the static potential, which is of order alpha_s^4 ln(μ), and consequently to the static energy, which is of order alpha_s^4 ln(alpha_s). Finally, in the case of an equilateral geometry, we solve the renormalization group equations an…
We compute the anomalous dimensions of the flavour non-singlet twist-2 Wilson operators in the RI'/SMOM scheme at two loops in an arbitrary linear covariant gauge. In addition we provide the full Green's function for these operators inserted in a quark 2-point function at the symmetric subtraction point. The three loop anomalous dimensions in the Landau gauge are also derived.
We compute Green's functions with a bilinear quark operator inserted at non-zero momentum for a generalized momentum configuration to two loops. These are required to assist lattice gauge theory measurements of the same quantity in matching to the high energy behaviour. The flavour non-singlet operators considered are the scalar, vector and tensor currents as well as the second moment of the twist-2 Wilson operator used in deep inelastic scattering for the measurement of nucleon structure functions.
We conjecture that QCD can be tuned to an infrared limit cycle in the three-nucleon system by adjusting the up and down quark masses to critical values at which the binding energies of the deuteron and its spin-singlet partner are tuned to zero. At the critical point, the triton would have infinitely many arbitrarily-shallow excited states with the ratio of the binding energies of successive states approaching a universal constant close to 515.
QCD in non-integer $d=4-2ε$ space-time dimensions enjoys conformal invariance at the special fine-tuned value of the coupling. Counterterms for composite operators in minimal subtraction schemes do not depend on $ε$ by construction, and therefore the renormalization group equations for composite operators in physical (integer) dimensions inherit conformal symmetry. This observation can be used to restore the complete evolution kernels that take into account mixing with the operators containing total derivatives from their eigenvalues (anomalous dimensions). Using this approach we calculate the…
The explicit breaking of chiral symmetry of the Wilson fermion action results in additive quark mass renormalization. Moreover, flavour singlet and non-singlet scalar currents acquire different renormalization constants with respect to continuum regularization schemes. This complicates keeping the renormalized strange quark mass fixed when varying the light quark mass in simulations with $N_f=2+1$ sea quark flavours. Here we present and validate our strategy within the CLS (Coordinated Lattice Simulations) effort to achieve this in simulations with non-perturbatively order-$a$ improved Wilson …
We compute the order $α_s^2$ QCD corrections to the $b$-quark forward-backward asymmetry in $e^+e^-\to b{\bar b}$ collisions at the $Z$ boson resonance, taking the non-zero mass of the $b$ quark into account. We determine these corrections with respect to both the $b$-quark axis and the thrust axis definition of the asymmetry. We compute also the distributions of these axes with respect to the electron beam. If one neglects the flavor singlet contributions to the $b$-quark asymmetry, as was done in previous computations for massless $b$ quarks, then the second-order QCD corrections for $m_b\ne…
