A. Jakovac
In the O(N) model for the large N expansion one needs resummation which makes the renormalization of the model difficult. In the paper it is discussed, how can one perform a consistent perturbation theory at zero as well as at finite temperature with the help of momentum dependent renormalization schemes.
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A practical method is suggested for performing renormalized 2PI resummation at finite temperature using specific momentum dependent renormalization schemes. In this method there is no need to solve Bethe-Salpeter equations for 2PI resummation. We examine the consistency of such schemes in the paper. The proposed method is used to perform a two-loop renormalized 2PI resummation in the finite temperature Phi^4 model.
In the naive form of most resummations we get into conflict with order-by-order renormalization. We present a method that is capable to ensure UV consistency of any resummations satisfying certain conditions. The method is based on the observation that resummation is equivalent with a calculation in an adequate perturbation scheme, followed by a renormalization scheme changing. This framework works both in static and momentum-dependent cases. In particular it is possible to establish finite infrared 2PI resummation.
Using resummation in perturbation theories at finite temperature or in non-equilibrium is unavoidable to obtain consistent results. Resummation, however, is often in conflict with renormalization. In this talk we give two possible solutions to problems of this type in static resummation. One is the use of auxiliary finite temperature UV divergent counterterms. The other is to use a renormalization scheme fitted to the environment. We also suggest a way how one could proceed in case of dynamic resummations.
At finite temperature and in non-equilibrium environments we have to resum perturbation theory to avoid infrared divergences. Since resummation shuffles the perturbative orders, renormalizability is a nontrivial issue. In this paper we demonstrate that one can modify any type of resummations -- even if it is momentum dependent, or it resums higher point functions -- in a way that it is renormalizable with the usual zero temperature counterterms. To achieve this goal we reformulate resummation in form of a renormalization scheme. We apply this technique to perform a 2PI resummation in the six d…
Resummation, ie. reorganization of perturbative series, can result in an inconsistent perturbation theory, unless the counterterms are reorganized in an appropriate way. In this paper two methods are presented for resummation of counterterms: one is a direct method where the necessary counterterms are constructed order by order; the other is a general one, based on renormalization group arguments. We demonstrate at one hand that, in mass independent schemes, mass resummation can be performed by gap equations renormalized prior to the substitution of the resummed mass for its argument. On the o…
Temperature Renormalization Group and Resummation
温度重整化群方程（TRGE）进行了比较与$(ϕ^4)_4$ -理论的图解展开。结果表明，单回路TRGE对有效质量的温度主导幂进行求和。需要对TRGE的双环路贡献来对耦合常数进行超前修正。它还表明，高阶TRGE求和子领先的权力的温度。
We present a simple proof of the all-order exponentiation of soft logarithmic corrections to hard processes in perturbative QCD. Our argument is based on proving that all large logs in the soft limit can be expressed in terms of a single dimensionful variable, and then using the renormalization group to resum them. Beyond the next-to-leading log level, our result is somewhat less predictive than previous all-order resummation formulae, but it does not rely on non-standard factorization, and it is thus possibly more general. We use our result to settle issues of convergence of the resummed seri…
We prove the all-order exponentiation of soft logarithmic corrections to prompt photon production in hadronic collisions, by generalizing an approach previously developed in the context of Drell-Yan production and deep-inelastic scattering. We show that all large logs in the soft limit can be expressed in terms of two dimensionful variables, and we use the renormalization group to resum them. The resummed results that we obtain are more general though less predictive than those proposed by other groups, in that they can accommodate for violations of Sudakov factorization.
Renormalization-Group Improved Resummation of Super-Leading Logarithms
本文提出重整化群改进的重求和策略，通过重排演化算符指数化双对数修正，首次实现任意2→M散射中超领头对数的领头阶重求和，并证明高阶Glauber次领头修正受参数抑制。
Previously developed Pade-related method of resummation for QCD observables, which achieves exact renormalization-scale-invariance, is extended so that the scheme-invariance is obtained as well. The dependence on the leading scheme parameter $c_2$ is eliminated by a variant of the method of the principle of minimal sensitivity. The subleading parameter $c_3$ in the approximant is then fixed in such a way that the correct known location of the leading infrared renormalon pole is reproduced. Thus, $beta$-functions which go beyond the last perturbatively calculated order in the observable are use…
We study the kinetics of the distribution function for charged particles of hard momentum in scalar QED. The goal is to understand the effects of infrared divergences associated with the exchange of quasistatic magnetic photons in the relaxation of the distribution function. We begin by obtaining a kinetic transport equation for the distribution function for hard charged scalars in a perturbative expansion that includes hard thermal loop resummation. Solving this transport equation, the infrared divergences arising from absorption and emission of soft quasi-static magnetic photons are manifest…
The resummation of logarithms in Quantum Field Theories is a long tale plenty of successes, yet the resummation of logarithms in non-relativistic theories has remained elusive. This was the most frustrating, since the first quantum field theory log ever computed was the Lamb shift one. We briefly review recent progress on the resummation of logarithms of $α$, which appear in the physics of non-relativistic states, using effective field theories. We put special emphasis on the basic formalism.
The standard Hartree--Fock approximation of the $O(N)-$invariant $ϕ^4$ model suffers from serious renormalization problems. In addition, when the symmetry is spontaneously broken, another shortcoming appears in relation to the Goldstone bosons: they fail to be massless in the intermediate states. In this work, within the framework of out--of--equilibrium Quantum Field Theory, we propose a class of systematic improvements of the Hartree--Fock resummation which overcomes all the above mentioned difficulties while ensuring also exact Renormalization--Group invariance to one loop.
A simple introduction to soft resummation
本文介绍软重新归一化的基本概念与技术，重点讨论了共线因子化、红外因子化及重新归一化群不变性，展示了阈值重新归一化的推导方法，并讨论了不同形式的重新归一化结果及横动量重新归一化。
By exploiting the analyticity properties of the two-point current-current correlator we obtain numerical predictions for the $e^+e^-$ moments in terms of the $τ$ decay rate. We perform a partial resummation of the pertinent perturbative series expansion by solving the renormalization group equation for Adler's function. Our predictions are renormalization scheme independent but depend on the order of the perturbative $β$-function expansion. The analysis involves the unknown five-loop coefficient $k_3$ for which we give some new estimates.
We discuss several methods of calculating the DIS structure functions F_2(x,Q^2) based on BFKL-type small x resummations. Taking into account new HERA data ranging down to small x and low Q^2, the pure leading order BFKL-based approach is excluded. Other methods based on high energy factorization are closer to conventional renormalization group equations. Despite several difficulties and ambiguities in combining the renormalization group equations with small x resummed terms, we find that a fit to the current data is hardly feasible, since the data in the low Q^2 region are not as steep as the…
Accurate determinations of the MS-bar b-quark mass $m_b(m_b)$ from $σ(e^+e^-\to{\rm hadrons})$ experimental data currently contain three comparable sources of uncertainty; the experimental uncertainty from moments of this cross-section, the uncertainty associated with $α_s(M_z)$, and the theoretical uncertainty associated with the renormalization scale. Through resummation of all logarithmic terms explicitly determined in the perturbative series by the renormalization-group (RG) equation, it is shown that the renormalization-scale dependence is virtually eliminated as a source of theoretical u…
We discuss perturbative solutions of renormalization group equations, and propose the use of resummation scale techniques in assessing theoretical uncertainties on the extraction of parton distribution functions from data.
Resummation and renormalization of kinematical effects in inclusive $P$-wave quarkonium production
本文研究 inclusive $P$ 波重夸克偶素产生中运动学效应的形状函数重整化，利用矩阵元紫外发散推导其大动量渐近行为以约束非微扰模型，并证明该形式体系可改善阈值对数处理及与实验的一致性。
We study the factorization and resummation of the $t\bar{t}W^\pm$ production at hadron colliders. The cross section in the threshold limit can be factorized into a convolution of hard and soft functions and parton distribution functions with the soft-collinear effective theory. We calculate the next-to-leading order soft function for the associated production of the heavy quark pair and colorless particle, and we perform the resummation calculation with the next-to-next-to-leading logarithms accuracy. Our results show that the resummation effects reduce the dependence of the cross section on t…
We introduce the method of dynamical renormalization group to study relaxation and damping out of equilibrium directly in real time and applied it to the study of infrared divergences in scalar QED. This method allows a consistent resummation of infrared effects associated with the exchange of quasistatic transverse photons and leads to anomalous logarithmic relaxation of the form $e^{-αT t \ln[t/t_0]}$ which prevents a quasiparticle interpretation of charged collective excitations at finite temperature. The hard thermal loop resummation program is incorporated consistently into the dynamical …
Renormalization-group methods in soft-collinear effective theory are used to perform the resummation of large perturbative logarithms for deep-inelastic scattering in the threshold region x->1. The factorization theorem for the structure function F_2(x,Q^2) for x->1 is rederived in the effective theory, whereby contributions from the hard scale Q^2 and the jet scale Q^2(1-x) are encoded in Wilson coefficients of effective-theory operators. Resummation is achieved by solving the evolution equations for these operators. Simple analytic results for the resummed expressions are obtained directly i…
We study the next-to-next-to-leading logarithmic order resummation for the large $p_T$ Higgs boson production at the LHC in the framework of soft-collinear effective theory. We find that the resummation effects reduce the scale uncertainty significantly and decrease the QCD NLO results by about $11\%$ in the large $p_T$ region. The finite top quark mass effects and the effects of the NNLO singular terms are also discussed.
We use renormalization-group methods in effective field theory to improve the theoretical prediction for the cross section for Higgs-boson production at hadron colliders. In addition to soft-gluon resummation at NNNLL, we also resum enhanced contributions of the form (C_Aπα_s)^n, which arise in the analytic continuation of the gluon form factor to time-like momentum transfer. This resummation is achieved by evaluating the matching corrections arising at the Higgs-boson mass scale at a time-like renormalization point μ^2<0, followed by renormalization-group evolution to μ^2>0. We match our resu…
In this PhD thesis, we analyze and generalize the renormalization group approach to the resummation of large logarithms in the perturbative expansion due to soft and collinear multiparton emissions. In particular, we present a generalization of this approach to prompt photon production. It is interesting to see that also with the more intricate two-scale kinematics that characterizes prompt photon production in the soft limit, it remains true that resummation simply follows from general kinematic properties of the phase space. Also, this approach does not require a separate treatment of indivi…
The quantum evolution equations for the field expectation value are analytically solved to cubic order in the field amplitude and to one-loop level in the lambda phi-fourth model. We adapt and use the renormalization group (RG) method for such non-linear and non-local equations. The time dependence of the field expectation value is explicitly derived integrating the RG equations. It is shown that the field amplitude for late times approaches a finite limit as the time to the power -3/2. This limiting value is expressed as a function of the initial field amplitude.
The consistent description of unstable particles, renormalons, or other Schwinger--Dyson-type of solutions within the framework of perturbative gauge field theories necessitates the definition and resummation of off-shell Green's functions, which must respect several crucial physical requirements. A formalism is presented for resummation of off-shell two-point correlation functions, which is mainly based on arguments of analyticity, unitarity, gauge invariance and renormalizability. The analytic results obtained with various methods, including the background field gauges and the pinch techniqu…
It is easy to sum chain-free self-energy rainbows, to obtain contributions to anomalous dimensions. It is also easy to resum rainbow-free self-energy chains. Taming the combinatoric explosion of all possible nestings and chainings of a primitive self-energy divergence is a much more demanding problem. We solve it in terms of the coproduct $Δ$, antipode S, and grading operator Y of the Hopf algebra of undecorated rooted trees. The vital operator is $S\star Y$, with a star product effected by $Δ$. We perform 30-loop Padé-Borel resummation of 463 020 146 037 416 130 934 BPHZ subtractions in Yukaw…
A new resummation scheme in scalar field theories is proposed by combining parquet resummation techniques and flow equations, which is characterized by a hierarchy structure of the Bethe--Salpeter (BS) equations. The new resummation scheme greatly improves on the approximations for the BS kernel. Resummation of the BS kernel in the $t$ and $u$ channels to infinite order is equivalent to truncate the effective action to infinite order. Our approximation approaches ensure that the theory can be renormalized, which is very important for numerical calculations. Two-point function can also be obtai…
Higher order renormalization in 4D quantum gravity is carried out using dimensional regularization with great care concerning the conformal-mode dependence. In this regularization, resummation can be automatically carried out without making an assumption like that of David, Distler and Kawai. In this paper we consider a model of 4D quantum gravity coupled to QED. Resummation inevitably implies a four-derivative quantum gravity. The renormalizability is directly checked up to $O(e_r^6)$ and $O(t_r^2)$, where $e_r$ and $t_r$ are the running coupling constants of QED and the traceless gravitation…
In this lecture I present some of the new developments concerning the use of Pade Approximants (PA's) for resumming perturbative series in QCD. It is shown that PA's tend to reduce the renormalization scale and scheme dependence as compared to truncated series. In particular it is proven that in the limit where the $β$ function is dominated by the 1-loop contribution, there is an exact symmetry that guarantees invariance of diagonal PA's under changing the renormalization scale. In addition it is shown that in the large $β_0$ approximation diagonal PA's can be interpreted as a systematic metho…
We resum Sudakov threshold enhancements in heavy quark hadroproduction for single-heavy quark inclusive and pair-inclusive kinematics. We expand these resummed results and derive analytical finite-order cross sections through next-to-next-to-leading order. This involves the construction of next-to-leading order matching conditions in color space. For the scale dependent terms we derive exact results using renormalization group methods. We study the effects of scale variations, scheme and kinematics choice on the partonic and hadronic cross sections, and provide estimates for top and bottom qua…
Refined renormalization group improvement for thermally resummed effective potential
本研究提出热重整化有效势的重整化群改进方法，保持热重整后逐阶RG不变性，大幅降低临界温度等相变物理量的标度依赖性，还设计通用重整化方法纳入大对数项与温度相关修正，使精细化的重整化一阶有效势结果与二阶结果在误差内一致。
In this paper a resummation method inspired by the renormalization-group improvement is applied to the one-loop effective potential (EP) in massive scalar $ϕ^4$ model at $T\neq0$. By investigating the phase structure of the model at $T \neq 0$ we get the following observations; i) Starting from the perturbative calculations with the theory renormalized at an arbitrary mass-scale $μ$ and at an arbitrary temperature $T_0$, we can in principle fully resum terms of $O(λT/μ)$ together with terms of $O(λ(T/μ)^2)$. The key idea is to fix the arbitrary RS-parameters so as to make both of one-loop radi…
For many observables, the most difficult part of a single logarithmic resummation is the analytical treatment of the observable's dependence on multiple emissions. We present a general numerical method, which allows the resummation specifically of these single logarithms. Some first applications and new results for the thrust major, the oblateness and the two-jet rate in the Durham algorithm are also presented.
We make progress towards resummation of power-suppressed logarithms in dijet event shapes such as thrust, which have the potential to improve high-precision fits for the value of the strong coupling constant. Using a newly developed formalism for Soft-Collinear Effective Theory (SCET), we identify and compute the anomalous dimensions of all the operators that contribute to event shapes at order $1/Q^2$. These anomalous dimensions are necessary to resum power-suppressed logarithms in dijet event shape distributions, although an additional matching step and running of observable-dependent soft f…
A method is developed for calculating the jet mass distribution at hadron colliders using an expansion about the kinematic threshold. In particular, we consider the mass distribution of jets of size R produced in association with a hard photon at the Large Hadron Collider. Expanding around the kinematic threshold, where all the energy goes into the jet and the photon, provides a clean factorization formula and allows for the resummation of logarithms associated with soft and collinear divergences. All of the large logarithms of jet mass are resummed at next-to-leading logarithmic level (NLL), …
For many event-shape observables, the most difficult part of a resummation in the Born limit is the analytical treatment of the observable's dependence on multiple emissions, which is required at single logarithmic accuracy. We present a general numerical method, suitable for a large class of event shapes, which allows the resummation specifically of these single logarithms. It is applied to the case of the thrust major and the oblateness, which have so far defied analytical resummation and to the two-jet rate in the Durham algorithm, for which only a subset of the single logs had up to now be…
We studied different levels of resummations of the exactly solvable Bloch-Nordsieck model in order to be able to compare the approximations with an exact result. We studied one-loop perturbation theory, 2PI resummation and Schwinger-Dyson equations truncated in a way to maintain Ward-identities. At all levels we carefully performed renormalization. We found that although the 2PI resummation does not exhibit infrared sensitivity at the mass shell (the one-loop perturbation theory does), but it is still far from the exact solution. The method of truncated Schwinger-Dyson equations, however, is e…
We investigate the renormalization structure of scalar Galileons in flat spacetime. We explicitly calculate the ultraviolet divergent one-loop contributions to the 2-point, 3-point, 4-point, and 5-point functions. We discuss the structure of the counterterms and their hierarchy within an effective field theory expansion. We comment on different resummation schemes, including a geometric resummation for which our results could be generalized to arbitrary n-point functions.
We study the O(N) scalar field theory with quartic self-coupling in de Sitter space. When the field is light in units of the expansion rate, perturbative methods break down at very low momenta due to large infrared logarithmic terms. Using the nonperturbative large-N limit, we compute the four-point vertex function in the deep infrared regime. The resummation of an infinite series of perturbative (bubble) diagrams leads to a modified power law which is analogous to the generation of an anomalous dimension in critical phenomena. We discuss in detail the role of high momentum (subhorizon) modes,…
We present first results on the resummation of Next-to-Soft Virtual (NSV) logarithms for the threshold production of pseudoscalar Higgs boson through gluon fusion at the LHC. These results are presented after resumming the NSV logarithms of the kind ${\log}^{i}(1-z)$ to $\overline{\text{NNLL}}$ accuracy and matching them systematically to the fixed order NNLO cross-sections. These results are obtained using collinear factorization, renormalization group invariance and recent developments in the NSV resummation techniques. The phenomenological implications of these NSV resummed results for 13 T…
Solvability, Consistency and the Renormalization Group in Large-$N_c$ Models of Hadrons
我们建立了大$N_{c}$极限下介子-重子相互作用的拉格朗日模型的下列基本原理：1。在$1/N_c$展开式中计算$1$ -介子/ $2$ -重子格林函数的首阶贡献涉及到对无穷类发散费曼图求和。只要裸拉格朗日量正确地服从所有大- $N_c$选择规则，这种全回路重构可以通过求解具有短距离截止的耦合经典场方程来精确地实现。2.重正化的唯一作用是重正化了模型的裸汤川耦合、重子质量和超精细重子质量分裂。3.在此过程中，完全确定了这些裸参数的大- $N_{c}$重整化群流.我们推测，变体的Skyrme模型出现的UV不动点等流量。
Many observables in QCD rely upon the resummation of perturbation theory to retain predictive power. Resummation follows after one factorizes the cross section into the rele- vant modes. The class of observables which are sensitive to soft recoil effects are particularly challenging to factorize and resum since they involve rapidity logarithms. In this paper we will present a formalism which allows one to factorize and resum the perturbative series for such observables in a systematic fashion through the notion of a "rapidity renormalization group". That is, a Collin-Soper like equation is rea…
Starting from the first renormalized factorization theorem for a process described at subleading power in soft-collinear effective theory, we discuss the resummation of Sudakov logarithms for such processes in renormalization-group improved perturbation theory. Endpoint divergences in convolution integrals, which arise generically beyond leading power, are regularized and removed by systematically rearranging the factorization formula. We study in detail the example of the $b$-quark induced $h\toγγ$ decay of the Higgs boson, for which we resum large logarithms of the ratio $M_h/m_b$ at next-to…
Renormalization of the Next-to-Leading-Power Soft Function for the Drell-Yan Off-diagonal Channel
本文对Drell-Yan过程qg非对角道在阈值附近次领头幂展开中的软函数进行重整化，并求解其重整化群方程，为高阶因子化与重求和提供关键输入。
We generalize the concept of Borel resummability and renormalons to a quantum field theory with an arbitrary number of fields and couplings, starting from the known notion based on the running coupling constants. An approach to identify the renormalons is provided by exploiting an analytic solution of the generic one-loop renormalization group equations in multi-field theories. Methods to evaluate the regions in coupling space where the theory is resummable are described. The generalization is then illustrated in a toy model with two coupled scalar fields, representing the simplest extension o…
The measurement of azimuthal correlations of Mueller-Navelet jets is generally considered as a decisive test to reveal the effect of BFKL dynamics at hadron colliders. The first experimental study of these correlations at the LHC has been recently performed by the CMS collaboration. We show that the ratios of cosine moments of the azimuthal distribution are successfully described within our next-to-leading logarithmic BFKL treatment. The whole set of CMS data for the azimuthal correlations can also be consistently described provided that one uses a larger renormalization/factorization scale th…
We show that Dyson resummation schemes based on Baym's $Φ$-derivable approximations can be renormalized with counter term structures solely defined on the vacuum level. First applications to the self-consistent solution of the sunset self-energy in $ϕ^4$-theory are presented.
Gravitational Sommerfeld Effects: Formalism, Renormalization, and Perturbation to $O(G^{10})$
本文提出计算引力Sommerfeld因子的系统框架，结合EFT与黑洞扰动理论方法，精确求解到O(G^{10})，并建立新的重整化群方程以改进波形重缩放。
Two-Loop Anomalous Dimension for Non-Global and Clustering Logarithms
本文在有效场论中推导了两圈反常维度，用于次领头非全局及聚类对数的重求和，提出修正偶极子减除方案以避免额外维度问题，并在Marzili框架中验证了喷流间隙截面的方案无关性。
We study the production of two forward jets with a large interval of rapidity at hadron colliders, which was proposed by Mueller and Navelet as a possible test of the high energy dynamics of QCD, within a complete next-to-leading logarithm framework. We show that using the Brodsky-Lepage-Mackenzie procedure to fix the renormalization scale leads to a very good description of the recent CMS data at the LHC for the azimuthal correlations of the jets. We show that the inclusion of next-to-leading order corrections to the jet vertex significantly reduces the importance of energy-momentum non-conse…
We derive and analytically solve renormalization group (RG) equations of gauge invariant non-local Wilson line operators which resum logarithms for event shape observables $τ$ at subleading power in the $τ\ll 1$ expansion. These equations involve a class of universal jet and soft functions arising through operator mixing, which we call $θ$-jet and $θ$-soft functions. An illustrative example involving these operators is introduced which captures the generic features of subleading power resummation, allowing us to derive the structure of the RG to all orders in $α_s$, and provide field theory de…
We show how the renormalons emerge from the renormalization group equation with a priori no reference to any Feynman diagrams. The proof is rather given by recasting the renormalization group equation as a resurgent equation studied in the mathematical literature, which describes a function with an infinite number of singularities in the positive axis of the Borel plane. Consistency requires a one-to-one correspondence between the existence of such kind of equation and the actual (generalized) Borel resummation of the renormalons through a one-parameter transseries. Our finding suggests how no…
The jet shape is a classic jet substructure observable that probes the average transverse energy profile inside a reconstructed jet. The studies of jet shapes in proton-proton collisions have served as precision tests of perturbative Quantum Chromodynamics (QCD). They have also recently become the baseline for studying the in-medium modification of parton showers in ultra-relativistic nucleus-nucleus collisions. The jet shape is a function of two angular parameters $R$ and $r$, which can be at hierarchical scales. Its calculation suffers from large logarithms of the ratio between the two scale…
The renormalization-group (RG) functions for the three-dimensional n-vector cubic model are calculated in the five-loop approximation. High-precision numerical estimates for the asymptotic critical exponents of the three-dimensional impure Ising systems are extracted from the five-loop RG series by means of the Pade-Borel-Leroy resummation under n = 0. These exponents are found to be: γ= 1.325 +/- 0.003, η= 0.025 +/- 0.01, ν= 0.671 +/- 0.005, α= - 0.0125 +/- 0.008, β= 0.344 +/- 0.006. For the correction-to-scaling exponent, the less accurate estimate ω= 0.32 +/- 0.06 is obtained.
The RG expansions for renormalized coupling constants g_6 and g_8 of the 3D n-vector model are calculated in the 4-loop and 3-loop approximations respectively. Resummation of the RG series for g_6 by the Pade-Borel-Leroy technique results in the estimates for its universal critical values g_6^*(n) which are argued to deviate from the exact numbers by less than 0.3%. The RG expansion for g_8 demonstrates stronger divergence being much less suitable for getting reliable numerical estimates.
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