1. keycode 0 = 2. keycode 1 = 3. keycode 2 = 4. keycode 3 = 5. keycode 4 = 6. keycode 5 = 7. keycode 6 = 8. keycode 7 = 9. keycode 8 = BackSpace BackSpace 10. keycode 9 = Tab Tab 11. keycode 10 = 12. keycode 11 = 13. keycode 12 = Clear 14. keycode 13
110.81.18.141 = .217.183 = 8909 This is my one day test agent ip, speed is quite slow, these are available to everyone hope you can help 222.125.87.207 = .71.215 = .41.144 = .238.28 = 8909 host-120.155-43-
这篇文章主要介绍了利用jquery.qrcode在页面上生成二维码且支持中文.需要的朋友可以过来参考下,希望对大家有所帮助 实例如下: &!DOCTYPE html PUBLIC &-//W3C//DTD XHTML 1.0 Transitional//EN& &http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd&& &html xmlns=&http://www.w3.org/
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今天发现了个记录Google IP地址的网站,谁知晚上访问时变成了这样子: Sorry! Google doesn't want to publish these ip addresses. We have to close this site. Bye! 很后悔当时没有把那些IP记下来,马上上网查,幸好有人已经这样做了.我也记录下来吧,方便以后查阅. 来源地址:/justjavac/Google-IPs Google 全球 IP 地址库 IP 地址来源:ht
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1. Overview In my last blog topic, I realize a network sniffer in Ubuntu, here I rewrite the code in Windows XP, and add a new function to find all network adapter. 2. Developing enviroment - Windows xp - MinGW 5.1.6 - Gcc 3.4.5 - WinPcap 4.1.1 - Erl
I write a single list of turns, met some very interesting things. The same piece of code, in the win and linux is completely under the two different results. #include &stdio.h& #include &stdlib.h& #define SIZE 64 typedef struct info{
Recently passed a crazy &MongoDB GridFS data access efficiency benchmark& of the post, first seen in the greader, as one said, the collection of, did not take too much notice. We are to force the boss is .. &This we can do.& Well .. th
This article explained the Java serialization mechanism and principle. From the text you can learn how to serialize an object, and when needed, as well as Java serialization serialization algorithm. The Java object serialization and deserialization c
Serialization is the process of saving an object's state to deserialization is the process of rebuilding those bytes into a live object. The Java Serialization API provides a standard mechanism for developers to handle object ser
Java serialization mechanism and principle: the output from the document object serialization analysis shows that the process of object serialization /cwbwebhome/article/article8/862.html Java serialization algorithm dialysis Ser
Transfer from: /cwbwebhome/article/article8/862.html This article explains the Java serialization mechanism and principle. From the article you can learn how to serialize an object, when you need Java serialization and serializat
Callback instance method Principle: an implementation of the callback class is defined as a class interface (interface can save), then multiple threads in each class are injected into a callback object. When the thread is finished, through the callba
有关Java对象的序列化和反序列化也算是Java基础的一部分,下面对Java序列化的机制和原理进行一些介绍 Java序列化算法透析 Serialization(序列化)是一种将对象以一连串的字节描述的过程:反序列化deserialization是一种将这些字节重建成一个对象的过程.Java序列化API提供一种处理对象序列化的标准机制.在这里你能学到如何序列化一个对象,什么时候需要序列化以及Java序列化的算法,我们用一个实例来示范序列化以后的字节是如何描述一个对象的信息的. 序列化的必要性 Ja
RESTful的CoAP协议 CoAP: 嵌入式系统的REST 引自维基百科上的介绍,用的是谷歌翻译... 受约束的应用协议(COAP)是一种软件协议旨在以非常简单的电子设备,使他们能够在互联网上进行交互式通信中使用.它特别针对小型低功率传感器,开关,阀门和需要被控制或监督远程,通过标准的Internet网络类似的组件. COAP是一个应用层协议,该协议是用于在资源受限的网络连接设备,例如无线传感器网络节点使用. COAP被设计为容易地转换为HTTP与Web简化集成,同时也能满足特殊的要求,例如
Serialization is the process of saving an object's state to deserialization is the process of rebuilding those bytes into a live object. The Java Serialization API provides a standard mechanism for developers to handle object ser
本文讲解了Java序列化的机制和原理.从文中你可以了解如何序列化一个对象,什么时候需要序列化以及Java序列化的算法. 有关Java对象的序列化和反序列化也算是Java基础的一部分,下面对Java序列化的机制和原理进行一些介绍. Java序列化算法透析 Serialization(序列化)是一种将对象以一连串的字节描述的过程:反序列化deserialization是一种将这些字节重建成一个对象的过程.Java序列化API提供一种处理对象序列化的标准机制.在这里你能学到如何序列化一个对象,什么时候
这些命令程序是: snmptrap.snmpinform和snmptrapd.其中: snmptrap:可以模拟snmp agent发送一个trap到snmp管理端(一般称为网管,snmp manager或snmp client): snmpinform: 可以模拟snmp agent发送一个inform request到snmp管理端(Trap是发送给SNMP管理者的通知网络状况等的警告消息,而Inform是需要SNMP管理者确认接收的Trap. 与Inform 相比较,Trap通知方式为不可
FILE SIGNATURES TABLE 3 January 2013 This table of file signatures (aka &magic numbers&) is a continuing work-in-progress. I have found little information on this in a single place, with the exception of the table in Forensic Computing: A Practi
/** * 转载请注明作者longdick
**/ Java 序列化算法透析 Serialization (序列化)是一种将对象以一连串的字节描述的过程:反序列化 deserialization 是一种将这些字节重建成一个对象的过程. Java 序列化 API 提供一种处理对象序列化的标准机制.在这里你能学到如何序列化一个对象,什么时候需要序列化以及 Java 序列化的算法,我们用一个实例来示范序列化以后的字节是如何描述一个对象的信息的. 序列
问题来源: /del/archive//1131232.html#1133889 答案: WebBrowser1.OleObject.Document.ParentWindow.变量名; 譬如有这样一个页面(lancernig 举的例子), 如何提取其中的 pvs 呢: &html& &head& &title&测试页&/title& &script language=&java
使用MINA2.0.9搭建的SOCKET服务器,最近在线上一直抛异常: org.apache.mina.filter.codec.ProtocolDecoderException: java.nio.charset.MalformedInputException: Input length = 1 (Hexdump: FF FF F5 41 4C 49 56 45 0D 0A 7B 22 72 65 71 75 65 73 74 43 6D 64 54 79 70 65 22 3A 31 33
1. Material Design Preloader Want to create a Google's latest Material Design look-alike pre-loader? If yes, this jQuery plugin created by Aaron Lumsden will help you do so and you can use it in your mobile apps or websites. Demo | Download 2. jQuery
在论坛上看到有位朋友希望对中文按拼音进行排序,刚好最近有点空,贴一份原来一个同事写的一个排序类,仅稍微改动了下下,拿出来分享下. 废话不多说,看例子: &?xml version=&1.0& encoding=&utf-8&?& &mx:Application xmlns:mx=&/2006/mxml& layout=&absolute& fontSize=&
以下链接是本人整理的关于计算机视觉(ComputerVision, CV)相关领域的网站链接,其中有CV牛人的主页,CV研究小组的主页,CV领域的paper,代码,CV领域的最新动态,国内的应用情况等等.打算从事这个行业或者刚入门的朋友可以多关注这些网站,多了解一些CV的具体应用.搞研究的朋友也可以从中了解到很多牛人的研究动态.招生情况等.总之,我认为,知识只有分享才能产生更大的价值,真诚希望下面的链接能对朋友们有所帮助. (1)googleResearch:http://research.go
俗话说做程序就要很强的逻辑思维能力,可以测试下你的逻辑思维,做做,看你能做对多少道,有空也完善答案,好久没做了,呵呵 75道逻辑思维题 [1] 假设有一个池塘,里面有无穷多的水.现有2个空水壶,容积分别为5升和6升.问题是如何只用这2个水壶从池塘里取得3升的水. [2] 周雯的妈妈是豫林水泥厂的化验员. 一天,周雯来到化验室做作业.做完后想出去玩. &等等,妈妈还要考你一个题目,&她接着说,&你看这6只做化验用的玻璃杯,前面3只盛满了水,后面3只是空的.你 能只移动1只玻璃杯
superword是一个Java实现的英文单词分析软件,主要研究英语单词音近形似转化规律.前缀后缀规律.词之间的相似性规律等等. 1.单词 hadoop 的匹配文本: Subash D'Souza is a professional software developer with strong expertise in crunching big data using Hadoop/HBase with Hive/Pig. Apache Flume Distributed Log Collect
superword是一个Java实现的英文单词分析软件,主要研究英语单词音近形似转化规律.前缀后缀规律.词之间的相似性规律等等. 1101.单词 uuids 的匹配文本: For example, when accessing /_uuids, you get a list of UUIDs from the system. Getting Started with CouchDB The images in the replicated glance server preserve the u
superword是一个Java实现的英文单词分析软件,主要研究英语单词音近形似转化规律.前缀后缀规律.词之间的相似性规律等等. 1701.单词 jpql 的匹配文本: You will learn how to implement the various database operations using JPA's EntityManager interface and JPQL (which is similar to HQL) Pro Spring 3 In another situat
统计书籍: 1.ActiveMQ in Action 2.Next Generation Open Source Messaging with Apollo 3.BookKeeper 4.Durability with BookKeeper 5.Namenode High Availability 6.Serving millions of journals with Apache BookKeeper 7.Cassandra A Decentralized Structured Storage
统计书籍: 1.ActiveMQ in Action 2.Next Generation Open Source Messaging with Apollo 3.BookKeeper 4.Durability with BookKeeper 5.Namenode High Availability 6.Serving millions of journals with Apache BookKeeper 7.Cassandra A Decentralized Structured Storage
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: 1. The project team to use your source code management tool for the what? Should be used. VSS, CVS, PVCS, ClearCase, CCC / Harvest, FireFly can. My choice is VSS. 2. Your use of the project team of the defect management system, you? Should be used.
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According to port number can be divided into three broad categories: (1) recognized the port (Well Known Ports): from 0 to 1023, they are tightly bound (binding) on a number of services. Typically, these ports communication clearly shows a service ag
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Disclaimer: This article reproduced, provenance unknown, only individual learning 1, static Please read the following this procedure: 1. 2. Public class Hello ( 3. Public static void main (String [] args) (/ / (1) 4. System. Out.println ( &Hello, wor
Start of Tomcat 1 - Tomcat Server component 1.1 - Server A Server element represents the entire Catalina servlet container. (Singleton) 1.2 - Service A Service element represents the combination of one or more Connector components that share a single
This example is through the use of Struts in FormFile to write to MySQL,. . . Select a picture with the user, and then press submit into the database can be One must first create a table: create table test (name varchar (20), pic blob); the test libr
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Keywords: Pretreatment, scouring agent, one-step, linen fabric Gaga Shen (Zhejiang University, Hangzhou 310018) contains large amounts of flax fiber hemicellulose, lignin, pectin, wax and bark, etc., before the treatment is difficult, especially diff
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Used computer communications protocol port Port (port) in terms of hardware devices can be USB ports, COM serial port or a switch, router equipment, the external connection port. The terms of the port for software interface between the communications
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Two-pion exchange three-nucleon potential O(q^4) chiral expansion
Two-pion exchange three-nucleon potential: O(q 4) chiral expansionS. Ishikawa1 and M. R. Robilotta21Department of Physics, Science Research Center,Hosei University, 2-17-1 Fujimi, Chiyoda, Tokyo 102-8160, JapanarXiv: [nucl-th] 5 Apr 20072Instituto de F? ?sica, Universidade de S?o Paulo, a C.P. 6-970, S?o Paulo, SP, Brazil a(Dated: February 1, 2008)AbstractWe present the expansion of the two-pion exchange three-nucleon potential (TPE-3NP) to chiral order q 4 , which corresponds to a subset of all possibilities at this order and is based on the πN amplitude at O(q 3 ). Results encompass both numerical corrections to strength coe?cients of previous O(q 3 ) terms and new structures in the pro?le functions. The former are typically smaller than 10% whereas the latter arise from either loop functions or non-local gradients acting on the wave function. The in?uence of the new TPE-3NP over static and scattering three-body observables has been assessed and found to be small, as expected from perturbative corrections.PACS numbers: 13.75.Cs, 21.30.Fe, 13.75.Gx, 12.39.Fe1I.INTRODUCTIONThe research programme for nuclear forces, outlined more than ?fty years ago by Taketani, Nakamura, and Sasaki [1], treats pions and nucleons as basic degrees of freedom. This insight proved to be very fruitful. On the one hand, it implies the interconnection of all nuclear processes, both among themselves and with a class of free reactions. On the other, it determines a close relationship between the number of pions involved in a given interaction and its range. As a consequence, the outer components of nuclear forces are dominated by just a few basic subamplitudes, describing either single (N → πN) or multipion (ππ → ππ, πN → πN, πN → ππN , ...) interactions. Nevertheless, it took a long time for a theoretical tool to be available which allows the precise treatment of these amplitudes. Nowadays, owing to the development of chiral perturbation theory (ChPT) in association with e?ective lagrangians [2, 3], the roles of pions and nucleons in nuclear forces can be described consistently. The rationale for this approach is that the quarks u and d, which have small masses, dominate low-energy interactions. One then works with a two-?avor version of QCD and treats their masses as perturbations in a chiral symmetric lagrangian. The systematic inclusion of quark mass contributions is performed by means of chiral perturbation theory, which incorporates low-energy features of QCD into the nuclear force problem. In performing perturbative expansions, one uses a typical scale q, set by either pion four-momenta or nucleon three-momenta, such that q ? 1 GeV. Nuclear forces are dominated by two-body (NN) interactions and leading contributions pion exchange potential (TPEP) begins at O(q 2 ) and, at present, there are two independent are due to the one-pion exchange potential (OPEP), which begins [4] at O(q 0 ). The two-expansions up to O(q 4 ) in the literature, based on either heavy baryon [5] or covariant [6, 7] ChPT. The TPEP is closely related with the o?-shell πN amplitude and, at this order, two-loop diagrams involving intermediate ππ scattering already begin to contribute. In proper three-nucleon (3N) interactions, the leading term is due to the process known as TPE-3NP, in which the pion emitted by a nucleon is scattered before being absorbed by another one. It has been available since long [8C10], involves only tree-level interactions available NN forces demands the extension of the chiral series for the 3NP up to O(q 4 ). 2 and has the longest possible range. This contribution begins at O(q 3 ) and consistency withHowever, the implementation of this programme is not straightforward, since it requires the evaluation of a rather large number of diagrams. With the purpose of exploring the magnitude of O(q 4 ) e?ects, in this work we concentrate on the particular subset of processes which still belong to the TPE-3NP class. Our presentation is divided as follows. In section II we display the general relationship between the TPE-3NP and the πN amplitude, in order to discuss how it a?ects chiral power counting in the former. The πN amplitude relevant for the O(q 4 ) potential is derived in section III and used to construct the three-body interaction in section IV. We concentrate on numerical changes induced into both potential parameters and observables in sections V and VI, whereas conclusions are presented in section VII. There are also four appendices, dealing with kinematics, πN subthreshold coe?cients, loop integrals and non-local terms.II.GENERAL FORMULATIONPotentials to be used into non-relativistic equations can be derived from ?eld theory by means of the T -matrix. In the case of three-nucleon potentials, one startsfrom the non-relativistic transition matrix describing the process N(p1 ) N(p2 ) N(p3 ) → N(p′1 ) N(p′2 ) N(p′3 ), which includes both kernels and their iterations. The former correspond to proper interactions, represented by diagrams which cannot be split into two pieces by cutting positive-energy nucleon lines only, whereas the latter are automatically gener? ated by the dynamical equation. Therefore, just the kernels, denoted collectively by t3 , are included into the potential. The transformation of a T -matrix into a potential depends on both the dynamical equation adopted and conventions associated with o?-shell e?ects. The latter were discussed in a comprehensive paper by Friar [11]. Here we use the kinematical variables de?ned in ? ? Appendix A and relate t3 to the momentum space potential operator W by writing [12] ? ? p′1 , p′2 , p′3 |W | p1 , p2 , p3 = ?(2π)3 δ 3 (P ′ ?P ) t3 (p′1 , p′2 , p′3 , p1 , p2 , p3 ) . In con?guration space, internal dynamics is described by the function √ W (r′ , ρ′ ; r, ρ) = ? 2/ 36(1)dQr dQρ dq r dq ρ (2π)3 (2π)3 (2π)3 (2π)3 (2)′ ′ ′ ′ ? ×ei[Qr ?(r ?r )+ Qρ ?(ρ ?ρ)+q r ?(r +r )/2+q ρ ?(ρ +ρ)/2] t3 (Qr , Qρ , q r , q ρ ) ,3which is to be used in a non-local version of the Schr¨dinger equation: o ? √ 1 2 1 ?r′ ? ?2′ ? ? ψ(r ′ , ρ′ ) = ? 3/2 m m ρ3dr dρ W (r′ , ρ′ ; r, ρ) ψ(r, ρ) .(3)Non-local e?ects are associated with the variables Qr and Qρ . When these e?ects are not too strong, they can be represented by gradients acting on the wave function and the potential W is rewritten as √ W (r ′ , ρ′ ; r, ρ) = δ 3 (r ′ ?r) δ 3 (ρ′ ?ρ) 2/ 33V (r, ρ) .(4)The two-pion exchange three-nucleon potential is represented in Fig. 1a. It is closely related with the πN scattering amplitude, which is O(q) for free pions and becomes O(q 2 ) within the three-nucleon system. As a consequence, the TPE-3NP begins at O(q 3 ) and, at this order, it also receives contributions from interactions (c) and (d), which have shorter range. into processes that already contribute at O(q 3 ) and the evaluation of many new amplitudes, especially those associated with diagram (b). The extension of the chiral series to O(q 4 ) requires both the inclusion of single loop e?ects(a)(b)(c)(d)FIG. 1: (Color online) Classes of three-nucleon forces, where full and dashed lines represent nucleons an diagram (a) corresponds to the TPE-3NP.In this paper we concentrate on the particular set of processes which belong to the TPE3NP class, represented by the T -matrix Tππ and evaluated using the kinematical conditions given in Fig. 2. The coupling of a pion to nucleon i = (1, 2) is derived from the usual lowest order pseudo-vector lagrangian L(1) and the Dirac equation yields the equivalent forms for the vertex (gA /2fπ ) [τ u (p′ ?p) γ5 u] ?(i)= (mgA /fπ ) [τ u γ5 u](i) , ?(5)4where gA , fπ and m represent, respectively, the axial nucleon decay, the pion decay and the nucleon mass.P1k,aP? 1P3 k?, bP?3P2P? 2FIG. 2: (Color online) Two-pion exchange three-nucleon potential.The amplitude for the intermediate process π a (k)N(p) → π b (k ′ )N(p′ ) has the isospin structure Tba = δab T + + i?bac τc T ? and Fig. 2 yields Tππ = ? mgA fπ2(6)[? γ5 u](1) [? γ5 u](2) u u1 k 2 ??21 k ??2′2τ (1) ?τ (2) T + ? i τ (1) ×τ (2) ?τ (3) T ?(3),(7)propagators are O(q ?2 ). As a consequence, in the O(q 4 ) expansion of the potential one needs Tππ to O(q) and T ± to O(q 3 ). For on-shell nucleons, the sub amplitudes T ± can be written as i (8) σ?ν (p′ ?p)? K ν B ± u(p) , 2m with K = (k ′+k)/2. The dynamical content of the πN interaction is carried by the functions T ± = u(p′ ) D ± ? ? D ± and B ± and their main properties were reviewed by H¨hler [13]. The chiral structure o of these sub amplitudes was discussed by Becher and Leutwyler [14, 15] a few years ago, in the framework of covariant perturbation theory, and here we employ their results. Asi ? [ 2m u(p′ ) σ?ν (p′ ?p)? K ν u(p)](3) → O(q 2 ), indicating that one needs the expansions of D ±? being the pion mass. Results in Appendix A show that [? γ5 u](i) → O(q), whereas pion ufar as power counting is concerned, in Appendix A one ?nds [?(p′ ) u(p)](3) → O(q 0 ) and uand B ± up to O(q 3 ) and O(q) respectively.5At low and intermediate energies, the πN amplitude is given by a nucleon pole superimposed to a smooth background. One then distinguishes the pseudovector (PV) Born term from a remainder (R) and writes± ± T ± = Tpv + TR .(9)The former contribution depends on just two observables, namely the nucleon mass m and the πN coupling constant g, as prescribed by the Ward-Takahashi identity [16]. The calculation of these quantities in chiral perturbation theory may involve loops and other coupling constants but, at the end, results must be organized so as to reproduce the physical values± of both m and g in Tpv [17]. For this reason, one uses the constant g, instead of (gA /fπ ),since the former is indeed the observable determined by the residue of the nucleon pole [13, 15, 18]. The pv Born sub amplitudes are given by+ Dpv =g2 2m+ Bpv = ?g 2 ? Dpv =g2 2m? Bpv = ?g 2k ′ ?k k ′ ?k + , s ? m2 u ? m2 1 1 ? , 2 s?m u ? m2 k?k ′ k?k ′ ν , ? ? 2 2 s?m u?m m 1 1 1 + + , 2 2 s?m u?m 2m2(10) (11) (12) (13)where s and u are the usual πN Mandelstam variables. In the case of free pions, their chiral+ + ? ? orders are respectively [Dpv , Bpv , Dpv , Bpv ] → O[q 2 , q ?1 , q, q 0 ], but important changes dooccur when the pions become o?-shell.± The amplitudes TR receive contributions from both tree interactions and loops. Theformer can be read directly from the basic lagrangians and correspond to polynomials in t = (k ′ ?k)2 and ν = (p′ +p)?(k ′ +k)/4m, with coe?cients given by renormalized LECs [15]. The latter are more complex and depend on Feynman integrals. In the description of πN amplitudes below threshold, one approximates both types of contributions by polynomials and writes [13, 19] XR = xmn ν 2m tn , (14)+ + ? ? where XR stands for DR , BR /ν, DR /ν or BR . The subthreshold coe?cients xmn have thestatus of observables, since they can be obtained by means of dispersion relations applied to scattering data. As such, they constitute an important source of information about the values of the LECs to be used in e?ective lagrangians. 6The isospin odd subthreshold coe?cients include leading order terms, which implement the predictions made by Weinberg [20] and Tomozawa [21] for πN scattering lengths, given by ν 1 ? , BW T = 2 . (15) 2 2fπ 2fπ ? ? For free pions, one has [DW T , BW T ] → O[q, q 0 ], but these orders of magnitude also change? DW T =when pions become virtual. Quite generally, the ranges of nuclear interactions are determined by t-channel exchanges. has the longest possible range. Another t-channel structure becomes apparent at O(q 4 ), associated with the pion cloud of the nucleon, which gives rise to both scalar and vector form factors [18]. These e?ects extend well beyond 1 fm [22, 23] and a limitation of the power series given by Eq. (14) is that they cannot accommodate these ranges, since Fourier transforms of polynomials yield only δ-functions and its derivatives. In the description of the πN amplitude produced by Becher and Leutwyler [15], one learns that the only sources of medium range (mr) e?ects are their diagrams k and l, which contain two pions propagating in the t-channel. In our derivation of the TPE-3NP, the loop content of these diagrams is not approximated by power series and, for free pions, the non-pole subamplitudes are written as+ + ? ? DR = Dmr (t) + d+ + d+ ν 2 + d+ t 00 10 01 + + BR = Bmr (t) + b+ ν 00 (2)At O(q 3 ), the TPE-3NP involves only single-pion exchanges among di?erent nucleons and? + d+ ν 4 + d+ ν 2 t + d+ t2 20 11 02(3),(16) (17)(1),(1)? ? 2 DR = Dmr (t) + ν/(2fπ )? ? + d? ν + d? ν 3 + d? νt 00 10 01(0)(3),(18) (19)? ? 2 BR = Bmr (t) + 1/(2fπ ) + ?? b00+ b? ν 2 + ?? t b01 10(1),where the labels (n) outside the brackets indicate the presence of O(q n ) leading terms and mr denotes terms associated with the nucleon pion cloud. The bar symbol over some coe?cients indicates that they do not include both Weinberg-Tomozawa and medium range± ± contributions, which are accounted for explicitly. The functions DR and BR depend on the ? parameters fπ , gA , ?, m and on the LECs ci and di , which appear into higher order termsof the e?ective lagrangian. The subthreshold coe?cients are the door through which LECs enter our calculation and their explicit forms are given in Appendix B. The dynamical content of the O(q 3 ) πN amplitude is shown in Fig. 3. The ?rst two diagrams correspond to P V Born amplitudes, whereas the third one represents the Weinberg7=+++11 00 11 00++FIG. 3: (Color online) Representation of the πN amplitude used in the construction of the TPE3NP.Tomozawa contact interaction, all of them with physical masses and coupling constants. The fourth graph summarizes the terms within square brackets in Eqs. (16-19) and depends on the LECs. Finally, the last two diagrams describe medium range e?ects owing to the nucleon pion cloud, associated with scalar and vector form factors. This decomposition of the πN amplitude has also been used in our derivation of the two-pion exchange components of the NN interaction [6, 7] and hence the present calculation is consistent with those results.III.INTERMEDIATE πN AMPLITUDEThe combination of Figs. 2 and 3 gives rise to the TPE-3NP, associated with the six diagrams shown in Fig. 4. In the sequence, we discuss their individual contributions to the subamplitudes D ± and B ± . We are interested only in the longest possible component of the potential and numerators of expressions are systematically simpli?ed by using k 2 → ?2′and k 2 → ?2 . In con?guration space, this corresponds to keeping only those terms which contain two Yukawa functions and neglecting interactions associated with Figs. 1 (c) and 1 (d). ? diagrams (a) and (b): The crosses in the nucleon propagators of Figs. 4 (a) and 4 (b) indicate that they do not include forward propagating components, so as to avoid double counting when the potential is used in the dynamical equation. The covariant evaluation of these contributions is based on Eqs. (10-13). Denoting by p the momenta of the propagating ? 8=++(a)(b)(c)+111 000 111 000 111 000 111 000(d)++(e)(f)FIG. 4: (Color online) Structure of the O(q 4 ) two-pion exchange three-nucleon potentialnucleons, the factors 1/(s?m2 ) and 1/(u?m2) are decomposed as 1 ? (?0 )2 ? E 2 p ? with E = = 1 1 ? p0 ? E) ? 2E(?0 + E) , ? ? p ? 2E(? (20)? m2 + p2 . The ?rst term represents forward propagating nucleons, associated withthe iteration of the OPEP, whereas the second one gives rise to connected contributions. Discarding the former and using the results of Appendix A, one has √ √ 1/({s } ? m2 ) → ?1/ 4m2 + 3q 2 +q 2 /3+16Q2 /3 ± 10q r ?Qρ / 3 ? 2q ρ ?Qr / 3 u r ρ ρ After appropriate truncation, one obtains+ Dab = ?. (21)g2 (2?2 ?t) → O(q 2 ) , 8m3 g2 ν → O(q 2 ) , 2m2(22) (23) (24) (25)+ Bab → O(q 2 ) , ? Dab = ?? Bab → O(q 2 ) ,where we have used the fact that, in the case of virtual pions, ν → O(q 2 ). ? diagrams (c) and (d): These contributions are purely polynomial, can be read directly 9from Eqs. (16-19), and are given by+ Dcd = ?g4 ? 4 c1 2 c3 ? + 2 + A 4 (2 ?2 ?t) → O(q 2 ) , 2 fπ fπ 16 πfπ(26) (27) (28) (29)+ Bcd → O(q 2 ) , ? Dcd =1 ν → O(q 2 ) , 2 2fπ4 2 c4 m gA m? 1 + ? → O(q 0 ) . 2 2 4 2fπ fπ 8 πfπ? Bcd =? diagrams (e) and (f ): The medium range components of the intermediate πN amplitude are+ De = 2 gA ? (2t??2 ) (1?t/2?2) Πt ? 2π → O(q 3 ) , 4 64π 2 fπ(30) (31) (32)+ Def → O(q 4 ) , ? Be 2 gA m? = (1?t/4?2) Πt ? π → O(q) , 2f 4 16π πwhere Πt is the dimensionless Feynman integral1Πt =0?2 F (a) da t?M 2←M = 2?/a ,√ 1?a2 8 ?1 ma F (a) = 2 tan a ? (1?a2/2).(33)? The amplitude Def , proportional to ν, is O(q 3 ) for free pions and here becomes O(q 4 ). Thus,? full results: The Golberger-Treiman relation g/m = gA /fπ is valid up to O(q 2 ) and can be used in diagrams (a) and (b). One then has D+ = where2 2 σ(2?2 ) (2?2 ?t) g2 gA ? g 2 (1+gA)? + ? A + c3 + A ? (1?2t/?2) Πt 2 2 2 2 fπ fπ 8m 16πfπ 128π 2fπin fact, diagram (f ) does not contribute to the TPE-3NP at O(q 4 ).,(34)2 3gA ?3 (35) σ(t = 2? ) = ?4 c1 ? ? 2 32πfπ is the value of the scalar form factor at the Cheng-Dashen point [14]. The remaining ampli2 2tudes read B + → O(q 2 ) , D? = B? = (36) (37) (38)2 1?gA ν, 2 2fπ2 2 g 2 m? 1 + 4 c4 m gA (1+2gA)m? ? + A 2 4 (1?t/4?2) Πt . 2 4 2fπ 16 πfπ 16 π fπ10The subamplitudes D ± and B ± begin at O(q 2 ) and one needs just the leading terms in the spinor matrix elements of Eq. (8), which is rewritten as T + = 2m D + , T ? = 2m D ? + i σ (3) ?k′ ×k B ? , with D + → O(q 2 )+O(q 3), D ? → O(q 2 ), and B ? → O(q 0 )+O(q). (39) (40)? O(q 3 ) reduction: In order to compare our amplitudes with previous O(q 3 ) results, one notes that, in case corrections are dropped, one would have D+ = B? = σ(0) (2?2 ?t) + 2 2 fπ fπ 1 2 c4 m + . 2 2 2fπ fπ ?2 gA + c3 8m,(41) (42)These expressions agree with those derived directly from a chiral lagrangian [24], except for the terms within square brackets in both D + and B ? . The former corresponds to a Born contribution whereas the latter is due to diagram (c) in Fig. 4, associated with the Weinberg-Tomozawa term.IV.TWO-PION EXCHANGE POTENTIALThe expansion of the TPE-3NP up to O(q 4 ) requires only leading terms in vertices and propagators. In order to derive the non-relativistic potential in momentum space, one divides √ √ Eq. (7) by the relativistic normalization factor 2E ? 2m for each external nucleon leg and writes1 g2 1 1 ? t3 = A2 2 σ (1) ?k σ (2) ?k′ 2 k′ 2 +?2 4fπ k +? × τ (1) ?τ (2) D + ? i τ (1) × τ (2) ?τ (3) D ? + The con?guration space potential has the form V3 (r, ρ) = τ (1) ?τ (2) V3+ (r, ρ) + τ (1) × τ (2) ?τ (3) V3? (r, ρ) + cyclic permutations,1i (3) ′ σ ?k ×k B ? 2m.(43)(44)One notes that this expression is identical with Eq. (33) of Ref. [10] divided by 8m3 .11with+ ? ? V3+ (r, ρ) = C1 σ (1) ? x31 σ (2) ? x23 U1 (x31 ) U1 (x23 ) + + C2 (1/9) σ(1) ?σ(2) [U(x31 )?U2 (x31 )] [U(x23 )?U2 (x23 )]? ? + (1/3) σ(1) ? x23 σ (2) ? x23 [U(x31 )?U2 (x31 )] U2 (x23 ) ? ? + (1/3) σ(1) ? x31 σ (2) ? x31 U2 (x31 ) [U(x23 )?U2 (x23 )] ? ? ? ? + σ (1) ? x31 σ (2) ? x23 x31 ? x23 U2 (x31 ) U2 (x23 )+ + C3 σ (1) ??I σ (2) ??I ?I ??I I 0 ? 2 I 1 , 31 23 31 23(45)? V3? (r, ρ) = C1 (1/9) σ(1) ×σ (2) ?σ (3) [U(x31 )?U2 (x31 )] [U(x23 )?U2 (x23 )]? ? + (1/3) σ(3) ×σ (1) ? x23 σ (2) ? x23 [U(x31 )?U2 (x31 )] U(x23 ) ? ? + (1/3) σ(1) ? x31 σ (2) ×σ (3) ? x31 U2 (x31 ) [U(x23 )?U2 (x23 )] ? ? ? ? + σ (1) ? x31 σ (2) ? x23 σ (3) ? x31 × x23 U2 (x31 ) U2 (x23 )wf wf ? + C2 σ (1) ? i?31 ?i?23? σ (2) ? x23 [U(x31 )?U2 (x31 )] U1 (x23 ) U1 (x31 ) [U(x23 )?U2 (x23 )]wf wf ? + σ (1) ? x31 σ (2) ? i?31 ?i?23wf wf ? ? ? x + 3 σ (1) ? x31 σ (2) ? x23 i?31 ?i?23 ? [? 31 U2 (x31 ) U1 (x23 ) + x23 U1 (x31 ) U2 (x23 )] I I I I ? + C3 σ (1) ??31 σ (2) ??23 σ (3) ??31 ×?23 I 0 ? I 1 /4 .(46)The pro?le functions are written in terms of the dimensionless variables xij = ? rij and read U(x) = e?x , x e?x , x e?x , xn(47) (48) (49) 1 1 Πt (t) . 2 k′2 +?2 k +?21 U1 (x) = ? 1 + x U2 (x) =n1+3 3 + 2 x x16π I =? 2 ?dk dk′ i(k?r31+k′ ?r 23 ) t e (2π)3 (2π)3 ?2 12(50)The last function involves the loop integral given in Eq. (33) and is discussed further in Appendix C. The gradients ?I act on the functions I n , whereas the ?wf act only on the ij ij wave function and give rise to non-local interactions, as discussed in Appendix D. The strength coe?cients are the following combinations of the basic coupling constants+ C1 = 2 gA ?4 σ(2?2) , 4 64 π 2 fπ 2 gA ?6 4 32 π 2 fπ m 4 gA ?7 , 6 4096 π 3fπ 2 gA ?6 4 256 π 2 fπ m(51) , (52) (53)+ C2 =?2 2 gA g 2 (1+gA )m? + m c3 + A 2 8 16πfπ+ C3 =? C1 =1 + 4m c4 ?2 2 gA (1+2gA)m? 2 8πfπ,(54) (55) (56)? C2 =2 2 gA (gA ?1) ?6 , 4 768 π 2 fπ m 4 gA ?7 . 6 2048 π 3 fπ? C3 = ?V.STRENGTH COEFFICIENTSThe strength constants of the potential involve a blend of four well determined parameters, namely m = 938.28 MeV, ? = 139.57 MeV, gA = 1.267 and fπ = 92.4 MeV, with the scalar form factor at the Cheng-Dashen point and the LECs c3 and c4 , which are less precise. As far as σ(2?2) is concerned, we rely on the results [25] σ(2?2 ) ? σ(0) = 15.2 ± 0.4 MeV, σ(0) = 45 ± 8 MeV, and adopt the central value σ(2?2) = 60 MeV. The values quoted for the LECs in the literature vary considerably, depending on the empirical input employed and the chiral order one is working at. A sample of values is given in Table I. Our work is based on the O(q 3 ) expansion of the intermediate πN amplitude and, for the sake of consistency, one must use LECs extracted at the same order. The kinematical conditions of the three-body interaction are such that the variable ν is O(q 2 ), an order of magnitude smaller than the threshold value, ν = ?. This makes information encompassed in the subthreshold coe?cients better suited to this problem and we use results from Appendix13TABLE I: Some values of the LECs c3 and c4 ; m is the nucleon mass. Reference Chiral order [26] [26] [27] [15] [15] tree this work 3 3 3 4 4 2 3 πN input amplitude at ν = 0, t = 0 m c3 m c4?5.00 ± 1.43 3.62 ± 0.04 ?5.69 ± 0.04 3.03 ± 0.16 -3.4 -4.2 -3.6 -4.9 2.0 2.3 2.0 3.3amplitude at ν = 0, t = 2?2 /3 ?5.01 ± 1.01 3.62 ± 0.04 scattering amplitude subthreshold coe?cients scattering lengths subthreshold coe?cients subthreshold coe?cientsB in order to write2 m c3 = ?m fπ d+ ? 01 4 2 gA m ? 77 gA m ? ? , 2 2 16 π fπ 768 π fπ 2 1 g 2 (1+gA ) m ? . ? + A 2 4 16 π fπ(57) (58)m c4 =2 fπ b? 00 2Adopting the values for the subthreshold coe?cients given by H¨hler [13], namely d+ = o 01 1.14 ± 0.02 ??3 and b? = 10.36 ± 0.10 ??2, one ?nds the ?gures shown in the last row of 00 Table I. These, in turn, produce the strength coe?cients displayed in Table II. For the sake of comparison, we also quote values employed in our earlier calculation [10] and in two TM’ versions [28] of the Tucson-Melbourne potential [8].TABLE II: Strength coe?cients in MeV. reference this work Brazil [10]+ C1 + C2 + C3 ? C1 ? C2 ? C30.794 -2.118 0.034 0.691 0.014 -0.067 0.92 -1.99 0.67 0.58 0.61 -TM’(93) [28] 0.60 -2.05 TM’(99) [28] 0.91 -2.26Changes in these parameters represent theoretical progress achieved over more than two decades and it is worth investigating their origins in some detail. With this purpose in mind, we compare present results with those of our previous O(q 3 ) calculation [10]. At the 14chiral order one is working here, new qualitative e?ects begin to show up, associated with both loops and non-local interactions. They are represented by terms proportional to the+ ? ? coe?cients C3 , C2 and C3 in Eqs. (45) and (46). 2 2 The πN coupling is now described by gA ?2 /fπ = 3.66 whereas, previously, the factorg 2 ?2 /m2 = 3.97 was used. From a conceptual point of view, the latter should be preferred, since g is indeed the proper coupling observable. In chiral perturbation theory, the di?erence Goldberger-Treiman discrepancy [15]. As this is a O(q 2 ) e?ect, both forms of the coupling subject to larger uncertainties and the form based on gA is more precise. Our present choice accounts for a decrease of 8% in all parameters.+ + ? ′ The relations C1 ? Cs , C2 ? Cp and C1 ? ?Cp allow one to compare Eqs. (45)between both forms is ascribed to the parameter ?GT = ?2d18 ?2 /g, which describes the become equivalent in the present calculation. On the other hand, the empirical value of g isand (46) with Eq. (67) of Ref. [10]. One notes that the latter contains an unfortunate′ misprint in the sign of the term proportional to Cp , as pointed out in Ref. [29]. In theearlier calculation, the coe?cient Cs was based on a parameter [30] ασ = 1.05??1 , which+ corresponds to σ(2?2 ) = 64 MeV. The results of Table II show that the values of C2 and ? ′ C1 are rather close to those of Cp and ?Cp . This can be understood by rewriting Eqs. (52)and (54) in terms of the subthreshold coe?cient d+ and b? as follows 01 00+ C2 = ? ? C1 = 2 gA ?6 4 32 π 2 fπ m 2 m fπ d+ + 01 2 2 gA 29gA m? + 2 8 768πfπ,(59) (60)2 gA ?6 4 128 π 2fπ m2 fπ b? + 002 gA m? 2 16πfπ.+ ? Numerically, this amounts to C2 = ?(1.845 + 0.110 + [0.163]) MeV and C1 = (0.624 +[0.067]) MeV. The second term in the former equation was overlooked in Ref. [10] and should have been considered there. The square brackets2 correspond to next-to-leading+ order contributions and yield corrections of about 8% and 11% to the leading terms in C2 ? and C1 , respectively.3 As the model used in Ref. [10] was explicitly designed to reproducethe subthreshold coe?cients quoted by H¨hler [13], it produces the very same contributions o as the ?rst terms in Eqs. (59) and (60).23These factors can be traced back to loop diagrams in Fig. 3 and are dynamically related with the term ± proportional to C3 , as we discuss in Appendix C. When comparing the new coe?cients with those in the second row of Table II, one should also take into account the 8% e?ect due to the Goldberger-Treiman discrepancy.15VI.NUMERICAL RESULTS FOR THREE-NUCLEON SYSTEMSIn order to test the e?ects of the TPE-3NP at O(q 4 ), in this section, we present some numerical results of Faddeev calculations for three-nucleon bound and scattering states. The calculations are based on a con?guration space approach, in which we solve the Faddeev integral equations [31C33], Φ3 = Ξ12,3 + 1 E + i? ? H0 ? V12 × [V12 (Φ1 + Φ2 ) + W3 (Φ1 + Φ2 + Φ3 )] , (and cyclic permutations), (61)where Ξ12,3 , which does not appear in the bound state problem, is an initial state wave function for the scattering problem, H0 is a three-body kinetic operator in the center of mass, V12 is a nucleon-nucleon (2NP) potential between nucleons 1 and 2, and W3 is the 3NP displayed in Fig. 2. Partial wave states of a 3N system, in which both NN and 3N forces act, are restricted to those with total NN angular momenta j ≤ 6 for bound state calculations, and j ≤ 3 for scattering state calculations. The total 3N angular momentum (J) is truncated at J = 19/2, while 3NP is switched o? for 3N states with J & 9/2 for scattering calculations. These truncation procedures are con?rmed to give converged results for the purposes of the present work. ? When just local terms are retained, t3 in Eq. (43) can be cast in the conventional form [8C10]′2 2 2 ?3 = ? gA F (k ) F (k ) (σ (1) ? k)(σ(2) ? k′ ) t 2 4fπ k2 + ?2 k′2 + ?2× (τ (1) ? τ (2) ){a + b(k ? k′ )} ?(iτ (1) × τ (2) ? τ (3) )(iσ (3) ? k′ × k)d , (62)where the coe?cients, a, b, and d are related with our potential strength coe?cients by+ + ? [C1 , C2 , C1 ] =1 (4π)2gA 2fπ2[?a?4 , b?6 , ?d?6 ] .(63)as BR-O(q 4 ). In this table, the values for the older version of the Brazil TPE-3NP, BR(83) [10], and the potential up to O(q 3 ) given by Eqs. (41-42), BR-O(q 3 ), are shown as well. 16The values of the coe?cients, a, b, and d for the TPE-3NP at O(q 4 ) are shown in Table III,TABLE III: Coe?cients a, b, and d of the TPE-3NP. 3NP a? b ?3 d ?3BR-O(q 4 ) -0.981 -2.617 -0.854 BR-O(q 3 ) -0.736 -3.483 -1.204 BR(83) -1.05 -2.29 -0.768In Eq. (62), the function F (k2 ) represents a πNN form factor. We apply a dipole form factor with the cut o? mass Λ, and U2 (x) in Eqs. (47-49) as U(x) = e?x e?Λx ? x x? Λ2 + k Λ2 ??2 22, which modi?es the pro?le functions U(x), U1 (x), ? Λ2 ? 1 ? x , 2Λ 1 e?x ? 2 +Λ 1+ ? x Λx 1+(64) e?Λx ? Λx (65)?1 U1 (x) = ? 1 + x ?2 ? 1 Λ ? + e?Λx , 2 3 3 e?x U2 (r) = 1 + + 2 x x xe?Λx 3 3 ? ?Λ3 1 + ? + ? 2 ? Λx (Λx) Λx ? ? Λ(Λ2 ? 1) 1 ? ? 1+ ? e?Λx , 2 Λx ? with Λ = Λ/?.?(66)We choose the Argonne V18 model (AV18) [34] for a realistic NN potential, by which the triton binding energy (B3 ) becomes 7.626 MeV, underbinding it by about 0.9 MeV compared to the empirical value, 8.482 MeV. As it is well known, the introduction of the TPE-3NP remedies this de?ciency. The amount of attractive contribution depends on the cuto? mass Λ, as shown in Fig. 5. The solid curve shows the dependence of B3 on Λ for the calculation with the BR-O(q 4) 3NP in addition to the AV18 2NP (AV18+BR-O(q 4)). In the ?gure, the empirical value and the AV18 result are displayed by the dashed and dotted horizontal lines, respectively. Due to the strong attractive character of the 3NP, B3 is reproduced by choosing a rather small value of Λ, namely 660 MeV. In the same ?gure, the Λ-dependence of B3 for AV18+BR-O(q 3) is displayed by a dashed curve and that for the AV18+BR(83) by a dotted curve. From these curves we see that AV18+BR-O(q 3 ) reproduces B3 for Λ = 620 MeV and AV18+BR(83) for Λ = 680 MeV. In other words, the BR-O(q 4) 3NP is slightly 17more attractive than the BR(83) 3NP and a large attractive e?ect occurs when one moves from the TPE O(q 4 ) 3NP to the O(q 3 ) 3NP. This tendency is strongly correlated with the magnitude of the coe?cient b, as shown in Table III. This can be understood as a dominant contribution to B3 from the component of the TPE-3NP associated with the coe?cients b. This dominance is shown in Table IV, where we tabulate calculated B3 for the AV18 plus the BR-O(q 4 ) 3NP and plus each term of the BR-O(q 4) coming from the coe?cients a, b, and d.9.59.0B3 (MeV)8.58.07.5 550 600 650L (MeV)700750800FIG. 5: (Color online) The triton binding energy B3 as functions of the cuto? mass Λ of the πN N dipole form factor. The solid curve denotes the result for AV18+BR-O(q 4 ), the dashed curve for AV18+BR-O(q 3 ), and the dotted curve for AV18+BR(83). The horizontal lines denote the AV18 result (dotted line) and the empirical value (dashed line).In Fig. 6, we compare six calculated observables for proton-deuteron elastic scattering, namely di?erential cross sections σ(θ), vector analyzing powers of the proton Ay (θ) and of the deuteron iT11 (θ), and tensor analyzing powers of the deuteron T20 (θ), T21 (θ), andlab lab T22 (θ), at incident proton energy EN = 3.0 MeV, (or incident deuteron energy Ed = 6.0MeV,) with experimental data of Ref. [35, 36]. In the ?gure, the solid curves designate the AV18 calculations and the dashed curves the AV18+BR-O(q 4 ) calculations, which are almost indistinguishable from the AV18+BR-O(q 3 ) and AV18+BR(83) calculations, once 18TABLE IV: Triton binding energy for the AV18 2NP plus the BR-O(q 4 ) 3NP for each term of the BR-O(q 4 ) 3NP with Λ = 660 MeV. ?B3 means the di?erence of the calculated binding energy from that of the AV18 calculation. B3 (MeV) ?B3 (MeV) AV18+BR-O(q 4 ) AV18+BR-O(q 4 )-a AV18+BR-O(q 4 )-b AV18+BR-O(q 4 )-d 8.492 7.673 8.241 7.787 0.866 0.047 0.615 0.161the cut o? masses are chosen so that B3 is reproduced. It is reminded that the TPE-3NF gives minor e?ects on the vector analyzing powers. This happens because the exchange of pions gives essentially scalar and tensor components of nuclear interaction in spin space, which are not so e?ective to the vector analyzing powers.lab On the other hand, as is noticed in Refs. [37, 38], at EN = 3.0 MeV, the TPE-3NP gives awrong contribution to the tensor analyzing power T21 (θ) around θ = 90? . In Fig. 7, we compare calculations of observables in neutron-deuteron elastic scatteringlab at EN = 28.0 MeV with experimental data of proton-deuteron scattering Ref. [39]. Atthis energy, discrepancies between the calculations and the experimental data in the vector analyzing power iT11 (θ) appear at θ ? 100? , where iT11 (θ) has a minimum, and at θ ? 140?, where iT11 (θ) has a maximum, which are not compensated by the introduction of the TPE3NP. On the other hand, while the AV18 calculation almost reproduces the experimental data of T21 (θ) at θ ? 90? , the introduction of the TPE-3NP gives a wrong e?ect, as in thelab EN = 3 MeV case.These results set the stage for the introduction of terms associated with the coe?cients± 3NP. Terms proportional to C3 , which include the rather complicated function I(r31 , r23 ) + ? ? C3 , C2 , and C3 , Eqs. (44-45), which are new features of the O(q 4 ) expansion of the TPE-given in Appendix C, arise from a loop integral, Eq. (33). On the other hand, the term with? C2 corresponds to a non-local potential and includes the gradient operator ?wf , which acts ijon the wave function and arises from the kinematical variable ν. Both kinds of contributions are not expressed in the conventional local form shown in Eq. (62), which involves only+ + ? the coe?cients C1 , C2 , and C1 , and the full evaluation of their e?ects would require an19EN400lab= 3.0 MeV (Ed0.05 0.04lab= 6.0 MeV)0.03Ay0.02iT11300I (mb/sr)0.03 200 0.02 0.01 100 0.00 0.02 0.00 0.000.01Gcm (deg)0.02 0.01 0.00Gcm (deg)Gcm (deg)-0.01 0.01-0.01 -0.02 -0.03 -0.04 0.00 -0.02T2060 120 Gcm (deg) 180 0T2160 120 Gcm (deg)-0.03T2260 120 Gcm (deg) 18001800lab FIG. 6: (Color online) Proton-deuteron elastic scattering observables at EN = 3.0 MeV. Solidcurves are calculations for the AV18 potential, and dashed curves for the AV18+BR-O(q 4 ). Experimental data are taken from Refs. [35, 36].extensive rebuilding of large numerical codes. However, the coe?cients of the new terms are small, and in this exploratory paper we estimate their in?uence over observables as follows. The function I(r31 , r23 ) is approximated by Eq. (C11), which amounts to replacing Πt (t) namely 1 ? 2t/?2 and 1 ? t/4?2 , are approximately evaluated by putting t ≈ 2?2 , which by a factor ?π. Further, the kinematical factors in front of Πt (t) in Eqs. (34) and (38),+ ? into C2 and C1 , or in b and d respectively, and one has + + ?C2 = ?3C3 ,+ ? yields ?3 and 1/2, respectively. By this procedure, the coe?cients C3 and C3 are absorbed? ? ?C1 = C3 /2. 1 C+ 20 2(67)? 1 ? 20 C1 , or ?b = ?0.125(??3) and ?d = 0.042(??3). The net change produced in the triton + ? binding energy is +0.026 MeV (+0.037 MeV from ?C2 and -0.011 MeV from ?C1 ), just+ Numerically, this corresponds to ?C2 = ?0.102 MeV ?? and ?C1 = ?0.034 MeV ?about 1/30 of the total increase in B3 due to the local terms of the BR-O(q 4 ) TPE-3NP.? The non-local term proportional to C2 is more involved and we restrict ourselves to arough assessment of its role. We replace the variable ν by a constant ν and assume, for 20EN100 80lab= 28.0 MeV (Ed0.2lab= 56.0 MeV)0.3 0.2Ay0.0iT11I (mb/sr)600.1400.020 -0.2-0.10 0.3 0.2 0.1 0.0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6-0.2Gcm (deg)0.2 0.1Gcm (deg)0.0Gcm (deg)-0.1 0.0 -0.1-0.2T2060 120 Gcm (deg)-0.2 -0.3T2160 120 Gcm (deg) 180-0.3T220 60 120 Gcm (deg) 18001800lab FIG. 7: (Color online) Nucleon-deuteron elastic scattering observables at EN = 28.0 MeV. Curvesare calculations for neutron-deuteron scattering. Solid curves denote calculations for the AV18 potential and dashed curves for the AV18+BR-O(q 4 ). Experimental data are those for protondeuteron scattering taken from Ref. [39].example, that ν =?2 . 4m? This changes the C2 term in Eq. (46) into the very simple form? ? ?? ? ? V3? (r, ρ) = C1 (? ? ? ) + iC2 σ (1) ? x31 σ (2) ? x23 U1 (x31 )U1 (x23 ) + C3 (? ? ? ) ,(68)with2 2 g 2 1 ? gA ?4 g 2 (1 ? gA )?6 ?? C2 = ? A2 ν =? A = 0.021 MeV . 2 4 4fπ 2fπ (4π)2 512π 2 fπ m(69)+ Except for the isospin factor, this term is similar to that with C1 (or a), which adds ? about 0.05 MeV to the triton binding energy. Since the potential strength C ? is about 3 % 2of+ C1 ,its contribution to the binding energy may be estimated to be a tiny 0.001 MeV.VII.CONCLUSIONSIn the framework of chiral perturbation theory, three-nucleon forces begin at O(q 3 ), with a long range component which is due to the exchanges of two pions and relatively simple. At O(q 4 ), on the other hand, a large number of di?erent processes intervene and a full 21description becomes rather complex. For this reason, here we concentrate on a subset of O(q 4 ) interactions, namely that which still involves the exchanges of just two pions. This part of the 3NP is closely related with the πN amplitude, and the expansion of the former up to O(q 4 ) depends on the latter at O(q 3 ). Our expressions for the potential are given in Eqs. (44-56) and the new chiral layer of the TPE-3NP considered in this work gives rise to both numerical corrections to strength coef+ + ? ?cients of already existing terms (C1 , C2 , C1 ) and new structures in the pro?le functions.Changes in numerical coe?cients lay in the neighborhood of 10% and can be read in Tables II and III. New structures, on the other hand, arise either from loop functions representing form factors or the non-local terms associated with gradients acting on the wave function.+ ? ? They correspond to the terms proportional to the parameters C3 , C2 and C3 , which aresmall and compatible with perturbative e?ects. In order to insert our results into a broader picture, in Table V we show the orders at which the various e?ects begin to appear, including the drift potential derived recently [40].TABLE V: Chiral picture for two- and three-body forces. beginning O(q 0 ) O(q 2 ) O(q 3 ) O(q 4 ) TWO-BODY? ? OPEP: VT , VSS ? OPEP: VD + + ? TPEP: VC ; VT , VSS + + ? ? ? TPEP: VLS , VT , VSS ; VC , VLS + + ? TPEP: VD ; VQ , VD ? + + TPEP: C1 ; C1 , C2 ? ? + TPEP: C2 ; C3 , C3TWO-BODYTHREE-BODYThe in?uence of the new TPE-3NP over three-body observables has been assessed in both static and scattering environments, adopting the Argonne V18 potential for the two-body interaction. In order to reproduce the empirical triton binding energy, the O(q 4 ) potential requires a cuto? mass of 660 MeV. Comparing this with the value of 680 MeV for the 1983 Brazil TPE-3NP, one learns that the later version is more attractive. In the study of proton-deuteron elastic scattering, we have calculated cross sections σ(θ), vector analyzing powers Ay (θ) of the proton and iT11 (θ) of the deuteron, and tensor analyzing powers T20 (θ), T21 (θ), and T22 (θ) of the deuteron, at energies of 3 and 28 MeV. Results are displayed in Figs. 6 and 7, where it is possible to see that there is little sensitivity to the changes induced in the strength parameters when one goes from O(q 3 ) to O(q 4 ). Old 22problems, as the Ay (θ) puzzle, remain unsolved. The present version of the TPE-3NP contains new structures, associated with loop integrals an non-local operators. Their in?uence over observables has been estimated and found to be at least one order of magnitude smaller than other three-body e?ects. A more detailed study of this part of the force is being carried on.APPENDIX A: KINEMATICSThe coordinate describing the position of nucleon i is ri and one uses the combinations R = (r 1 +r2 +r 3 )/3 , which yield r1 = R ? ρ r ? √ , 2 2 3 r2 = R + ρ r ? √ , 2 2 3 ρ r3 = R + √ . 3 (A2) r = r 2 ?r 1 , √ ρ = (2 r3 ?r 1 ?r 2 )/ 3 , (A1)The momentum of nucleon i is pi and one de?nes P = p1 +p2 +p3 , pr = (p2 ?p1 )/2 , √ pρ = (2 p3 ?p1 ?p2 )/2 3 . (A3)Initial momenta p and ?nal momenta p′ are used in the combinations Q = (P ′ +P )/2 , Qr = (p′r +pr )/2 , Qρ = (p′ρ +pρ )/2 , q = (P ′ ?P ) , q r = (p′r ?pr ) , q ρ = (p′ρ ?pρ ) . (A4) (A5) (A6)In the CM, one has P = 0 and the three-momenta are given by √ p1 = ?(Qr ?q r /2) ? (Qρ ?q ρ /2)/ 3 , √ p2 = (Qr ?q r /2) ? (Qρ ?q ρ /2)/ 3 , √ p3 = 2(Qρ ?q ρ /2)/ 3 , √ p′1 = ?(Qr +q r /2) ? (Qρ +q ρ /2)/ 3 , (A7) √ p′2 = (Qr +q r /2) ? (Qρ +q ρ /2)/ 3 , (A8) √ p′3 = 2(Qρ +q ρ /2)/ 3 .(A9)Energy conservation for on-shell particles yield the non-relativistic constraint Qr ?q r + Qρ ?q ρ = 0 . 23 (A10)The momenta of the exchanged pions are written as k = p1 ? p′1 , k ′ = p′2 ? p2 , (A11) (A12) (A13)√ √ k 0 = ?(q r +q ρ / 3)?(Qr +Qρ / 3)/m , √ √ ′ k 0 = (q r ?q ρ / 3)?(Qr ?Qρ / 3)/m ,√ k = q r +q ρ / 3 , √ k′ = q r ?q ρ / 3 ,and the Mandelstam variables for nucleon 3 read √ √ √ s = (p3 +k)2 = m2 ? (q r +q ρ / 3) ? (q r +2 Qr ?q ρ / 3+2 3 Qρ ) + O(q 4 ) , √ ν = (s?u)/4m = ?2 q r ?Qρ / 3 + O(q 4 ) . (A14) (A15) (A16)√ √ √ u = (p3 ?k ′ )2 = m2 ? (q r ?q ρ / 3) ? (q r +2 Qr +q ρ / 3?2 3 Qρ ) + O(q 4 ) ,In the evaluation of the intermediate πN amplitude, one needs [?(p′ ) u(p)](3) ? 2m + O(q 2 ) , u [ (A17) (A18)√ i u(p′ ) σ?ν (p′ ?p)? K ν u(p)](3) ? 2 i σ(3) ?q ρ ×q r / 3 + O(q 4 ) . ? 2mThe πN vertex for nucleon 1 is associated with √ [?(p′ ) γ5 u(p)](1) ? σ (1) ?(q r +q ρ / 3) + O(q 3 ) , u and results for nucleon 2 are obtained by making q r → ?q r .APPENDIX B: SUBTHRESHOLD COEFFICIENTS± The polynomial parts of the amplitudes TR , Eqs. (30-35), are determined by the sub-(A19)threshold coe?cients of Ref. [15]. The terms relevant to the O(q 3 ) expansion are written as24[6]4 2 2 (2c1 ? c3 ) ?2 8 gA ?3 3 gA ?3 + + , 2 4 4 fπ 64 π fπ 64 π fπ mr 4 2 48 gA ? 77 gA ? c3 ? , d+ = ? 2 ? 01 4 4 fπ 768 π fπ 768 π fπ mr 2 193 gA d+ = , 02 4 15360 π fπ ? mr 1 + O(q 2 ) , d? = 00 2 2 fπ W T 4 1 2 c4 m g A m ? g2 m ? b? = + ? ? A 4 , 00 2 2 4 2 fπ W T fπ 8 π fπ 8 π fπ mr 2 gA m , b? = 01 4 96 π fπ ? mrd+ = ? 00(B1) (B2) (B3) (B4) (B5) (B6)? where the parameters ci and di are the usual coupling constants of the chiral lagrangians of order 2 and 3 respectively [41] and the tilde over the latter indicates that they were renormalized [15]. Terms within square brackets labeled (mr) in these results are due to the medium range diagrams shown in Fig. 3 and have been included explicitly into the± ± functions Dmr and Bmr . Terms bearing the (WT ) label were also explicitly considered inEqs. (15-19). The subthreshold coe?cients are determined from πN scatterig data and a set of experimental values is given in Ref. [13].APPENDIX C: FUNCTIONS I nThe functions I n , describing loop contributions, are given by 16π I (r 31 , r23 ) = ? 2 ?ndk dk′ i(k?r 31 +k′ ?r23 ) t e (2π)3 (2π)3 ?2nn1 1 Πt (t) . 2 k′2 +?2 k +?2(C1)Using the de?nition Eq. (33) and the Jacobi variables Eq. (A1), one writes 4 ?2 ρ I (r 31 , r23 ) = 2 3?n 1I(r 31 , r 23 ) ,?1(C2)I(r31 , r23 ) = 128π0da tan√ ma 1?a2 ? (1?a2/2)√L(a; r, ρ)(C3) (C4)L(a; r, ρ) =1 1 dq dQ ei(Q?r? 3 q ?ρ/2) . 3 (2π)3 2 q 2 +4?2 2 +?2 ] [(Q+q)2 +?2 ] (2π) a [(Q?q)The numerical evaluation of the function L is can be simpli?ed by using alternative representations. 25? form 1: One uses the Feynman procedure for manipulating denominators, which yields1L(a; r, ρ) =0db1dq dQ ei(Q?r ? 3 q ?ρ/2) 1 2 3 (2π)3 2 q 2 +4?2 2 /4+?2)?(1?2b)q ? Q]2 (2π) a [(Q +q db dq ei[(1?2b) r? 3 ρ]?q /2 e?Θ r , (2π)3 a2 q 2 +4?2 Θ (C5)√√1 = 8π Θ=0?2 +b(1?b) q 2 .Performing the angular integration over q, one has 1 L(a; r, ρ) = 16 π 31db0√ e?Θ r sin q [(1 ? 2b) r ? 3 ρ]/2 √ dq q . Θ (a2 q 2 +4?2 ) [(1 ? 2b) r ? 3 ρ]/2 1 = 2 k +?2 e??x 4π x(C6)? form 2: The Fourier transform dx e?ik?x (C7)allows one to write L(a; r, ρ) = 1 1 64π 3 a2 dz e??|r 31+z | e??|r 23?z | e?2? z/a . |r 31 +z| |r23 ?z| z (C8)These results may be further simpli?ed by means of approximations. baryon case, one uses F (a) → 4π/a2 in Eq. (33) and Eqs. (C5) and (C7) yield, respectively, √ ∞ e?Θ r sin q [(1 ? 2b) r ? 3 ρ]/2 2 1 ?1 q √ db dq tan I(r 31 , r23 ) ? , (C9) π 0 2? ? Θ [(1 ? 2b) r ? 3 ρ]/2 0 I(r 31 , r23 ) ? 1 π dz e??|r 31+z | e??|r 23?z | e?2? z . |r31 +z| |r 23 ?z| 2? z 2 (C10) ? heavy baryon approximation: In the limit m → ∞, corresponding to the heavy? multipole approximation: The integrand in Eq. (C10) is peaked around z = 0 and a multipole expansion of the Yukawa functions produces I(r31 , r 23 ) ? U(x31 ) U(x23 ) + ? ? ? . (C11)t2 /80?4 + ? ? ? ], valid for low t, directly into Eq. (C1).The same result can also be obtained by using the expansion Πt (t) ? ?π[1 + t/12?2 +26APPENDIX D: NON-LOCAL TERMIn con?guration space, the variable Qρ corresponds to a non-local operator, represented ? by a gradient acting on the wave function. In order to make the dependence of t3 on Qρ explicit, one writes ? t3 = [Qρ ]i Xi (q r , q ρ ) , where X is a generic three-vector, and evaluates the matrix element ψ |W |ψ = ? 1 (2π)′(D1)12dr′ dρ′ dr dρ ψ ? (r′ , ρ′ ) ψ(r, ρ)′ ′ ′dQr dQρ dq r dq ρ? × ei[Qr ?(r ?r )+ Qρ ?(ρ ?ρ)+q r ?(r +r )/2+q ρ ?(ρ +ρ)/2] t3 (Qr , Qρ , q r , q ρ ) =? × 1 (2π)6dr dρi i ? ψ ? (r, ρ) ψ(r, ρ) + ψ ? (r, ρ) ? ? ψ(r, ρ) ρ ρ 2 2 iidq r dq ρ ei[q r ?r+q ρ ?ρ] Xi (q r , q ρ ) .(D2)This yields the potential√ [2/ 3]3 V3 (r, ρ) = ? (2π)6? → ←i ? ? ? 2 ρdq r dq ρ ei[q r ?r +q ρ ?ρ] Xi (q r , q ρ ) ,i(D3)where the operator ? = ? ? ? acts only on the wave function. An alternative form can be obtained by integrating Eq. 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